ALBERT

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Yujuan Chen〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work, we investigate a cooperative parabolic system 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉λ〈/mi〉〈mo〉−〈/mo〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉b〈/mi〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉Δ〈/mi〉〈mi〉v〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉v〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉μ〈/mi〉〈mo〉−〈/mo〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈mi〉c〈/mi〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 with blow-up initial and boundary values over a smooth bounded domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉Ω〈/mi〉〈/math〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉. We show existence, nonexistence and uniqueness of solutions. We also provide an exact estimate of the behavior of the solutions near the parabolic boundary.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Yang Cao, Zhiyong Wang, Jingxue Yin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the Cauchy problem for the semilinear pseudo-parabolic equation with initial data with compact support. We find the life span of the solution, when the power exponent 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉p〈/mi〉〈/math〉 is smaller than the Fujita exponent.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Hongbo Guan, Yong Wang, Huiqing Zhu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Christos Sourdis〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We establish the energy minimality property of a solution to the generalized Painlevé-II equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mi〉Δ〈/mi〉〈mi〉y〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mi〉y〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉2〈/mn〉〈msup〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, which is increasing in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and converges to the positive and negative Hastings–McLeod solutions as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉→〈/mo〉〈mo〉±〈/mo〉〈mi〉∞〈/mi〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): Lili Li, Dongfang Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we investigate the exact and numerical solutions of the time fractional Burgers’ equation. By using Cole–Hopf transformation and the method of variables separation, the exact solution is presented in terms of the Mittag-Leffler functions. Moreover, by using the L1-scheme on graded meshes and the Legendre–Galerkin spectral method, the numerical approximation is obtained. A numerical example is presented to confirm all the results.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Shengbing Deng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is devoted to studying the existence of solutions for the following weighted mean field equation 〈span〉〈span〉(0.1)〈/span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉div〈/mi〉〈mrow〉〈mo fence="true"〉(〈/mo〉〈mrow〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mo fence="true"〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mfrac〉〈mrow〉〈mi〉θ〈/mi〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mi〉e〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mo linebreak="badbreak"〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mi〉e〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉x〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi mathvariant="normal"〉in〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mi〉B〈/mi〉〈mo〉;〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi mathvariant="normal"〉on〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mi〉∂〈/mi〉〈mi〉B〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉B〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is the unit ball in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉θ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is of logarithmic type function. By showing the logarithmic Trudinger-Moser inequalities in the weighted Sobolev spaces, we establish the existence of solutions for problem (0.1) when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mi〉λ〈/mi〉〈/math〉 is in some certain range.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Dandan Shi, Yufeng Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper utilizes ansatz method to find bright and dark solutions of the conformable space–time fractional modified equal width wave (MEW) equation. In addition, a fractional novel 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo〉∕〈/mo〉〈mi〉G〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉-expansion method is used to look for periodic solutions, dark solutions, soliton solutions and soliton-like solutions of the space–time fractional MEW equation for the first time. Meanwhile, some solutions are presented through graphs The obtained results show that the two methods are suitable and more powerful for solving other kinds of nonlinear fractional differential equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Wei Zhang, Wenbin Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study the existence of solutions to the Dirichlet boundary value problem for fractional differential equations with instantaneous and non-instantaneous impulses, establish the existence result based on the variational method, and provide an example that illustrates the main result.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): Bin Pei, Yong Xu, Jiang-Lun Wu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, an averaging principle for multidimensional, time dependent, stochastic differential equations (SDEs) driven by fractional Brownian motion and standard Brownian motion was established. We combined the pathwise approach with the Itô stochastic calculus to handle both types of integrals involved and proved that the original SDEs can be approximated by averaged SDEs in the manner of mean square convergence.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): Feng-Hua Qi, Ye-Hui Huang, Pan Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is focusing in studying an extended (3+1)-dimensional Jimbo–Miwa-like equation. By the Hirota’s method, we construct the solitary-wave solutions and new exact solutions, and discuss the dynamical behavior of those solutions with graphical illustration. The results can be expected to model some phenomena in fluid dynamics.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Liguang Xu, Xiaoyan Chu, Hongxiao Hu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the globally exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses. By establishing some non-autonomous fractional differential inequalities and using the properties of the Mittag-Leffler function, some sufficient criteria on the exponential ultimate boundedness are presented for the systems.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Ji Lin, Yi Xu, Yuhui Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper presents a new meshless scheme based on the compactly supported radial basis functions for solving linear and nonlinear advection–diffusion–reaction equations. The simulation of proposed problems is divided into the rough approximation obtained from the boundary conditions and corrected approximation using compactly supported radial basis functions where a sparse system matrix will be obtained. The behaviors of the method are verified by several examples which indicate its accuracy and efficiency.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Kehan Shi, Zhichang Guo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, an elliptic system consisting of a weighted total variation (TV) equation and a linear equation is proposed as an approximation of Euler’s elastica image inpainting model. The existence of weak solutions for the system is proved by treating the TV term as the limit 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo〉→〈/mo〉〈msup〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mo〉+〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 of the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplacian term. Numerical experiments show the connectivity of the proposed system for image inpainting.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): Jianwei Zhou, Ziwu Jiang, Huantian Xie, Haifeng Niu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study the a posteriori error estimates of Galerkin spectral methods for the singularly perturbed problem on a unit interval. By the generalized orthogonal Jacobi polynomials, the lower bound of orthogonal coefficients, and the upper bound of the a posteriori error estimates are rigorously proposed. Our estimates consist of the truncation error of weighted 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-projection of the right hand function and limit low frequency terms of the coefficients within Jacobi polynomial expansions. While the low frequency coefficients are together with exponentially decaying multiples, the decaying rate of high frequency coefficients of the right hand function can be used to measure the approximation accuracy of the numerical solution.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Xia Ye, Mingxuan Zhu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we consider an initial–boundary value problem of one-dimensional magnetohydrodynamic (MHD) equations. We prove the solution of MHD equations converges uniformly to the solution of non-resistive MHD equations as the resistivity coefficient tends to zero. Compared with the results proved in Jiang and Zhang (2017), the initial vacuum is allowed.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Guifeng Deng, Qiuying Lu, Lili Qian〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the generalized Lyness difference equation under linear perturbation 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak"〉+〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉α〈/mi〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak"〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="badbreak"〉+〈/mo〉〈mi〉β〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉γ〈/mi〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak"〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="badbreak"〉+〈/mo〉〈mi〉δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/mfrac〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉η〈/mi〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈mspace width="2em"〉〈/mspace〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mo〉…〈/mo〉〈mo〉,〈/mo〉〈/mrow〉〈/math〉〈/span〉with initial values 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mo〉,〈/mo〉〈mi〉β〈/mi〉〈mo〉,〈/mo〉〈mi〉γ〈/mi〉〈mo〉,〈/mo〉〈mi〉δ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉γ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉δ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉η〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉. It is proved that the solutions are globally asymptotically stable for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉η〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉. Therefore, it is concluded that the generalized Lyness difference equation under linear perturbation holds the dichotomy property as follows: for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉η〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, all of its solutions are globally asymptotically stable; for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉η〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, almost all of its solutions are diverge and strictly oscillatory.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Stefano De Marchi, Holger Wendland〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The rescaled localized RBF method was introduced in Deparis, Forti, and Quarteroni (2014) for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. It requires the solution of two linear systems with the same sparse matrix, which has a small condition number, due to the scaling of the basis function. Hence, it can be computed using an unpreconditioned conjugate gradient method in linear time. Numerical evidence provided in Deparis, Forti, and Quarteroni (2014) shows that the method produces good approximations for many examples but no theoretical results were provided. In this paper, we discuss the convergence of the rescaled localized RBF method in the case of quasi-uniform data and stationary scaling. As the method is not only interpolatory but also reproduces constants exactly, linear convergence is expected. We can show this linear convergence up to a certain conjecture.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Yinhe Lin, Ning-An Lai, Sen Ming〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we prove the blow-up result for small data Cauchy problem of semilinear wave equation in Schwarzschild spacetime with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉[〈/mo〉〈mfrac〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo〉]〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, without the assumption that the supports of the initial data should be far away from the black hole. What is more, we establish the upper bound of the lifespan estimate.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Shu-Nan Li, Bing-Yang Cao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Anomalous heat diffusion, which is commonly characterized by the nonlinear growth of mean square of displacement (MSD), 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mfenced open="〈" close="〉"〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉Δ〈/mi〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfenced〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∼〈/mo〉〈msup〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/math〉), is usually paired with a length-dependence of effective thermal conductivity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉e〈/mi〉〈mi〉f〈/mi〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈/math〉, namely, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉e〈/mi〉〈mi〉f〈/mi〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∼〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mi〉L〈/mi〉〈/math〉 the system length. In this work, a generic time- and length-dependence of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉e〈/mi〉〈mi〉f〈/mi〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is obtained based on the fractional Fokker–Planck equation (FFPE) with orders 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mfenced open="(" close=")"〉〈mrow〉〈mi〉γ〈/mi〉〈mo〉,〈/mo〉〈mi〉μ〈/mi〉〈/mrow〉〈/mfenced〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, namely, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉e〈/mi〉〈mi〉f〈/mi〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∝〈/mo〉〈msup〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉γ〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉μ〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉. Two existing paradigmatic results, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉e〈/mi〉〈mi〉f〈/mi〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∝〈/mo〉〈msup〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉e〈/mi〉〈mi〉f〈/mi〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∝〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈mo〉∕〈/mo〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, are first unified in our work, which reflect memory effects and nonlocality in energy fluctuations, respectively. We formulate the effective thermal conductivity in terms of entropy generation, which does not rely on the local-equilibrium hypothesis.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Yongshuai Zhang, Shuwei Xu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉By taking the Wadati–Konno–Ichikawa (WKI) equation as an example, this paper refines the Olmedilla’s technique of solving the Gelfand–Levitan–Marchenko (GLM) equation with one higher-order pole. In order to solve the GLM equation, the Laurent’s series and residue theorem are used, and the formulae of the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉N〈/mi〉〈/math〉th order soliton solution and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉N〈/mi〉〈/math〉th order bound-state soliton solution are expressed in a unified determinant form. Moreover, these formulae are simpler and more explicit to calculate solutions of the nonlinear integrable equations, compared with the results of Olmedilla.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Qizhen Xiao, Hongliang Liu, Xiao Wang, Yao Huang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This work focuses on the fixed-time bipartite flocking control problem for multi-agent systems. By employing the structurally balanced signed graph theory and a fixed-time stability theorem, we can prove that all individuals can be partitioned into two disjoint clusters within a fixed time and they move to opposite directions with the same velocity. We find that the settling time merely depends on the parameters and network connectivity. Moreover, the parameters, network connectivity and the initial states have an important influence on the diameter of each disjoint cluster. The results are novel, which are firstly illustrated by both theoretical analysis and numerical simulations.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 99〈/p〉 〈p〉Author(s): Mayank Chadha, Michael D. Todd〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper considers the curvature of framed space curves, their higher-order derivatives, variations, and co-rotational derivatives. We realize that parametrizing rotation tensor using the 〈em〉Gibbs vector〈/em〉 is effective in deriving a closed-form formula that gives the general 〈em〉n〈/em〉-th order derivative of the curvature tensor as the summation of functions of the parametrizing quantity and its own derivatives. We use these results for formulating a linearized updating algorithm for curvature and its derivatives when the configuration of the curve acquires a small increment.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): D. Bertaccini, F. Durastante〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We propose a tensor structured preconditioner for the tensor train GMRES algorithm (or TT–GMRES for short) to approximate the solution of the all–at–once formulation of time-dependent fractional partial differential equations discretized in time by linear multistep formulas used in boundary value form and in space by finite volumes.〈/p〉 〈p〉Numerical experiments show that the proposed preconditioner is efficient for very large problems and is competitive, in particular with respect to the AMEn algorithm.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 25 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Jeta Molla, Markus Schmuck〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider basic and easily extendible transport formulations for lithium batteries consisting of an anode (Li-foil), a separator (polymer electrolyte), and a composite cathode (composed of electrolyte and intercalation particles). Our mathematical investigations show the following novel features: (i) 〈em〉complete and very basic description of mixed transport processes〈/em〉 relying on a neutral, binary symmetric electrolyte resulting in a non-standard Poisson equation for the electric potential together with interstitial diffusion approximated by classical diffusion; (ii)  〈em〉upscaled and basic composite cathode equations allowing to take geometric and material features of electrodes into account〈/em〉; (iii) 〈em〉the derived effective macroscopic model can be numerically solved with well-known numerical strategies for homogeneous domains〈/em〉 and hence does not require to solve a high-dimensional numerical problem or to depend on a computationally involved multiscale discretisation strategies where highly heterogeneous and realistic, nonlinear, and reactive boundary conditions are still unexplored. We believe that the here proposed basic and easily extendible formulations will serve as a basic and simple setup towards a systematic theoretical and experimental understanding of complex electrochemical systems and their optimization, e.g. Li-batteries.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Jiantang Zhao, Yunfeng Jia〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper deals with an autocatalytic reaction–diffusion model with arbitrary order functional response subject to no-flux boundary conditions. We mainly discuss the stability of the steady-state bifurcation which emanates from the unique positive constant equilibrium. On the stability of the bifurcation solution, the conventional way is to consider the sign of the first derivative of a certain function. However, sometimes, the first derivative may be equal to zero. This leads to the uncertainty of the stability. In such case, it needs to break through the common idea. We present an approach which determines the stability of the bifurcation solution.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Zhonglong Zhao, Lingchao He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The bilinear method is employed to construct the multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation in fluid dynamics. The 1-lump solutions, 3-lump solutions and 6-lump solutions are explicitly presented. The centers of the 3-lump wave have a triangular structure, and the 6-lump wave possesses a central peak and five peaks in a ring. The dynamic characteristics of the obtained solutions are analyzed with the aid of numerical simulation.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 91〈/p〉 〈p〉Author(s): Xin-Yi Gao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Plasmas are believed to be possibly the most abundant form of ordinary matter in the Universe, supporting a variety of the wave phenomena, while a dusty plasma is of interest as a non-Hamiltonian system of interacting particles. In this Letter, symbolic computation on an observationally/experimentally-supported (2+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation is done, for certain dust-acoustic, electron-acoustic, positron-acoustic, magneto-acoustic, dust-magneto-acoustic, ion-acoustic, dust-ion-acoustic and/or quantum-dust-ion-acoustic waves in one of the cosmic/laboratory dusty plasmas. Auto-Bäcklund transformation and families of the solitonic solutions are obtained, for the electrostatic wave potential, perturbation of the magnitude of the magnetic field, fluctuation of electron or ion density, or radial-direction component of the velocity of ions or dust particles, relying on such plasma coefficient functions as the nonlinearity, dispersion, dusty-fluid-viscosity/Burgers-dissipation, geometric-effect and diffraction/transverse-perturbation coefficients. Shock structures presented in this Letter are very close to the experimental results previously reported. Future plasma observations/experiments might verify some other effects offered by our analytic results with respect to those plasma coefficient functions.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 91〈/p〉 〈p〉Author(s): Bin Han, Tao Li, Xiaosheng Zhuang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A 〈em〉directional compactly supported〈/em〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉d〈/mi〉〈/math〉〈em〉-dimensional Haar tight framelet〈/em〉 is constructed such that all its high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs and they exhibit totally 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msup〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/math〉 directions in dimension 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉d〈/mi〉〈/math〉. Furthermore, applying the projection method to such a tight framelet, a 〈em〉directional compactly supported box spline tight framelet〈/em〉 with simple geometric structure is built such that all the high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs as well. Moreover, such compactly supported box spline tight framelets can achieve arbitrarily high numbers of directions by using refinable box splines with increasing supports. Their application to pMRI with good performance is presented.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 91〈/p〉 〈p〉Author(s): Jian Liu, Zengqin Zhao, Tongqian Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we aim to prove the existence of solutions for impulsive Hamiltonian systems with first derivative. Two theorems of the existence of triple solutions are obtained via variational methods and three-critical-points theorems.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 91〈/p〉 〈p〉Author(s): Xiaoming Zhu, Dafeng Zuo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we propose some 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mrow〉〈mo〉(〈/mo〉〈mn〉2〈/mn〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉-dimensional nonlocal ‘breaking soliton’-type systems and show that most of them could be decomposed into two 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mrow〉〈mo〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉-dimensional integrable systems. With the use of this, we construct their Darboux transformations and some exact solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 91〈/p〉 〈p〉Author(s): Rumeng Zheng, Xiaoyun Jiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, the L1 Fourier spectral method is considered to solve the time-fractional Navier–Stokes equation with periodic boundary condition. The Fourier spectral method is employed for spatial approximation, and the L1 finite difference scheme is used to discrete the Caputo time fractional derivative. Analysis of stability and convergence are accomplished as well, leading to the conclusion that our numerical method is unconditionally stable, and the solution converges to the exact one with order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉O〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si97.gif"〉〈mi〉τ〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si101.gif"〉〈mi〉N〈/mi〉〈/math〉 are the time step size and polynomial degree, respectively. The numerical example is provided to testify the effectiveness of our scheme, from the results of which, it turns out that the L1 Fourier spectral method is effective for solving the time-fractional Navier–Stokes equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 92〈/p〉 〈p〉Author(s): Jinghui Zhang, Chao Zhou, Salamat Ullah, Yang Zhong, Rui Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, a two-dimensional generalized finite integral transform method is developed for new analytic bending solutions of orthotropic rectangular thin foundation plates. The vibrating beam functions are adopted as the integral kernels to construct the integral transform pairs. By imposing the transform to the governing equation, utilizing some inherent properties of the beam functions, the title problem is converted to that of solving a system of linear algebraic equations, by which the new analytic solutions are elegantly obtained in a straightforward way. Numerical examples validate the present method as well as the solutions yielded by satisfactory agreement with the literature and finite element analysis.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 91〈/p〉 〈p〉Author(s): Miaomiao Gao, Daqing Jiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 91〈/p〉 〈p〉Author(s): Jianxiong Cao, Guojie Song, Jie Wang, Qihong Shi, Sujing Sun〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper investigates the blow-up of solutions for a time fractional nonlinear reaction–diffusion equation with weakly spatial source. We first derive two sufficient conditions under which the solutions may blow up in finite time. Then, we prove the existence of global solution when the initial data are small enough. Moreover, the long time behavior of bounded solutions will be analyzed.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): Jiabao Su, Rushun Tian, Leiga Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Consider the elliptic system with linearly coupled terms 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉i〈/mi〉〈mi〉n〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mi〉v〈/mi〉〈mo〉=〈/mo〉〈mi〉μ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉i〈/mi〉〈mi〉n〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉v〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉o〈/mi〉〈mi〉n〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo〉,〈/mo〉〈mi〉μ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈/math〉 are constants and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉Ω〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 is a smooth bounded domain. We study the local and global bifurcations with respect to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi mathvariant="script"〉T〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≔〈/mo〉〈mrow〉〈mo〉{〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉λ〈/mi〉〈mo〉,〈/mo〉〈mi〉μ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉0〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉}〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉×〈/mo〉〈mi〉X〈/mi〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mi〉X〈/mi〉〈/math〉 is a proper Banach space. Our results are of particular interest for obtaining nontrivial solutions in the case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≠〈/mo〉〈mi〉μ〈/mi〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): D.I. Borisov, D.A. Zezyulin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider a Schrödinger operator on the axis with a bipartite potential consisting of two compactly supported complex-valued functions, whose supports are separated by a large distance. We show that this operator possesses a sequence of approximately equidistant complex-valued wavenumbers situated near the real axis. Depending on its imaginary part, each wavenumber corresponds to either a resonance or an eigenvalue. The obtained sequence of wavenumbers resembles transmission resonances in electromagnetic Fabry–Pérot interferometers formed by parallel mirrors. Our result has potential applications in standard and non-hermitian quantum mechanics, physics of waveguides, photonics, and in other areas where the Schrödinger operator emerges as an effective Hamiltonian.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 12 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Dongmei Shen, Musheng Wei〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉A〈/mi〉〈/math〉 be a diagonalizable complex matrix with two different eigenvalues. We derive all explicit solutions of the Yang–Baxter-like matrix equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mi〉A〈/mi〉〈mi〉X〈/mi〉〈mi〉A〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉X〈/mi〉〈mi〉A〈/mi〉〈mi〉X〈/mi〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): Guofeng Che, Hongxia Shi, Zewei Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the following quasilinear elliptic problem: 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi〉ε〈/mi〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mfrac〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈/mfrac〉〈mo〉=〈/mo〉〈mi〉K〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mtext〉  in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉∈〈/mo〉〈mi〉B〈/mi〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉ε〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 is a small parameter. Under some mild conditions on the nonnegative functions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉K〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, we establish the existence of a ground state solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈/math〉 for the above problem. Moreover, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈/math〉 concentrates on the intersection set of global minimum points of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and maximum points of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉K〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉. The methods are based on the Nehari manifold technique and the Concentration-Compactness Principle of Lions.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 100〈/p〉 〈p〉Author(s): Xin Yu, Zhi-Yuan Sun〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The unconventional characteristic line for a nonautonomous KP equation is constructed, which reveals that the propagation of solitons can be different from the traditional ones. Based on the Painlevé analysis and bilinear method, we obtain the soliton solution, and further study the polarity of characteristic line, the effect of initial phase, as well as the inelastic interaction and soliton resonance. We expect these results to be helpful in understanding relevant phenomena in water waves and dust acoustic waves.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Qin Wang, Jingqin Zhang, Hanchun Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We are concerned with the two-dimensional Riemann problems of the isentropic Euler system for the Chaplygin gas with initial data being two constant states outside a convex cornered wedge. We establish a global theory of existence of self-similar solution. Our results on shock intersections can be applied to solve the shock diffraction problem.〈/p〉〈/div〉

Description: 〈p〉Publication date: April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 102〈/p〉 〈p〉Author(s): Shoji Itoh, Masaaki Sugihara〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉An improved preconditioned conjugate gradient squared (PCGS) algorithm has recently been proposed. This algorithm is more accurate and efficient than the conventional PCGS algorithm, and retains the advantages of the left-PCGS in terms of a solution structure. In this paper, we propose a changing over stopping criterion for the improved PCGS, that results in a higher accuracy than the conventional and the left-PCGS. A series of numerical results illustrate the stable solving statuses and the enhanced effectiveness of the improved PCGS with changing over stopping criterion.〈/p〉〈/div〉

Description: 〈p〉Publication date: March 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 101〈/p〉 〈p〉Author(s): Tianwei Zhang, Lianglin Xiong〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉On the basis of some crucial properties for one-parameter and two-parameter Mittag-Leffler functions, the existence, uniqueness and global exponential stability of periodic solution are discussed for a class of semilinear impulsive fractional functional differential equations with piecewise Caputo derivative. Some better result is achieved and it improves and extends some existing research finding.〈/p〉〈/div〉

Description: 〈p〉Publication date: March 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 101〈/p〉 〈p〉Author(s): Xiujuan Wang, Mingshu Peng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper aims to give a further study of stable regions in some fractional difference equations. Some criteria are established about root distribution of fractional functions. As applications of theoretical results, explicit conditions which guarantee the master–slave synchronization in fractional chaotic maps can be established, which are not discussed in the literature. Numerical simulation gives a solid confirmation of our analysis.〈/p〉〈/div〉

Description: 〈p〉Publication date: March 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 101〈/p〉 〈p〉Author(s): Rui Du, Zixuan Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The fractional advection–diffusion problem coupled with incompressible Navier–Stokes equations is important in science and engineering. In this paper, a fresh lattice Boltzmann model with double-distribution functions for this problem is presented. One distribution function is used for pressure and velocity. The other is applied for concentration. Through Chapman–Enskog expansion, the macroscopic continuum equations can be recovered from the LB model. Numerical simulations are carried out to verify the numerical accuracy and efficiency.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 27 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Shanming Ji, Ming Mei, Zejia Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the time delayed Nicholson’s blowflies equation with degenerate diffusion. We prove the existence and uniqueness of the positive steady state solution under the Dirichlet boundary condition and we show the stability of the nontrivial steady state.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 26 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Ying Sui, Huimin Yu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we investigate the oscillation of a kind of quasilinear equation with mixed arguments and Dirichlet/Robin boundary conditions. By using inequality technique, generalized Riccati transformation and comparison method, we obtain three sufficient conditions. At last, the main results are illustrated by examples.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Jian Huang, Zhongdi Cen, Li-Bin Liu, Jialiang Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper a numerical method is considered for a two-point boundary value problem with a Riemann–Liouville fractional derivative, where the exact solution may have weak singularity. The linear interpolation is used to approximate the functions in the fractional integral transformed from the Riemann–Liouville boundary value problem. In order to capture the singular phenomena of the exact solution, an adaptive mesh is developed by equidistributing a monitor function. The stability is derived by a modified Grönwall inequality. It is shown that the scheme is second-order convergent. Numerical experiments are provided to demonstrate the theoretical results.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 103〈/p〉 〈p〉Author(s): Shangjiang Guo, Shangzhi Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The local stability of the steady state for a general class of a reaction–diffusion model with nonlocal delay effect and nonlinear boundary condition is investigated under natural and minimal hypotheses.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 24 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Yuming Qin, Gaofeng Lv〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we prove the global existence and uniqueness of solutions to the initial boundary value problem of 3D Navier–Stokes equations with Cattaneo law for viscous heat-conductive gas. We know that the change in the heat conduction law affects the energy equation directly, so the key is to prove the existence of the solution of the linear energy equation. To prove the key point, we make great use of the idea about positive definite matrix by the properties and related theorem of parabolic linear equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 102〈/p〉 〈p〉Author(s): Li-Min Song, Zhen-Jun Yang, Xing-Liang Li, Shu-Min Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work, we mainly investigate the propagation of the superimposed field of Laguerre–Gaussian and Hermite–Gaussian solitons in nonlocal nonlinear Schrödinger equation. The propagation expression is obtained and the propagation properties are analyzed. The results show that the propagation properties of the superimposed field are closely related to the input power and the characteristic parameters. For different input powers, the superimposed field will show periodic changes of broadening or compression while propagating. Different characteristic parameters determine which soliton occupies the dominant position in propagation.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Dianli Zhao, Haidong Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper formulates a new switched two species chemostat model and discusses the coexistence behavior in the chemostat. A complete classification on the single-species chemostat is carried out firstly, where the stationary distribution with ergodicity is derived to exist and be unique. Then, based on the obtained stationary distribution and the comparison theorem, we put forward some sufficient conditions for the coexistence of microorganisms in the two species chemostat with Markov switchings. Moreover, when the species coexist in the deterministic chemostat for each state and have the same break-even concentrations for all states, they are proved to coexist still in the switched chemostat, which randomized the results of the classical deterministic chemostat. Results in this paper show that Markov switchings can contribute to coexistence of the two species.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Meiqiang Feng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Applying a new fixed point theorem due to Radu Precup, this paper investigates the existence and multiplicity of nontrivial radial solutions of coupled system of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉k〈/mi〉〈/math〉-Hessian equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Jishan Fan, Liangwei Wang, Yong Zhou〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper proves a regularity criterion for a new density-dependent incompressible Hall-MHD system with positive density.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Yu Tian, Min Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The aim of this paper is to study the existence of solutions for second-order differential equations with instantaneous and non-instantaneous impulses. Applying variational method, the existence result is obtained.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Mengchao Wang, Qi Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is devoted to the existence of solutions for a singular critical semilinear elliptic equation. Some existence and multiplicity results are obtained by using mountain pass arguments and analysis techniques. The results of Ding and Tang (2007) and Kang (2007) and related are improved.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Chao Li, Qilong Guo, Meimei Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, the (2+1)-dimensional time-fractional Schrödinger equation is studied within the Jumarie’s modified Riemann–Liouville derivative framework. By introducing the fractional complex transform, the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mrow〉〈mo〉(〈/mo〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉-expansion method is used to find the explicit analytical solutions of this equation. Abundant new exact solutions including the hyperbolic functions, the trigonometric functions and the rational functions are derived. These solutions are new and significant to reveal the pertinent features of the physical phenomenon.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Jiu Liu, Tao, Liu, Hong-Ying Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉With the help of the Nehari manifold, a Kirchhoff type equation involving two potentials was considered. Under new assumptions, a ground state solution was obtained.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Shujun You, Boling Guo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the initial–boundary value problem for the KdV–Burgers equation posed on a bounded interval 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mrow〉〈mo〉[〈/mo〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉b〈/mi〉〈mo〉]〈/mo〉〈/mrow〉〈/math〉. This problem features non-homogeneous boundary conditions applied at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉x〈/mi〉〈mo〉=〈/mo〉〈mi〉a〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉x〈/mi〉〈mo〉=〈/mo〉〈mi〉b〈/mi〉〈/math〉 and is known to have unique global smooth solution.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Yong Zhou〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The purpose of this paper is to prove the existence theorems of the nonoscillatory solution for higher order neutral dynamic equations on time scales. Our results improve and generalize the recent known results by removing some restrictive conditions.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 93〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Yanfeng Guo, Donglong Li, Junxia Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, traveling waves with different frequencies and velocities can be constructed by three wave method. Some new exact solitary and periodic solitary solutions are obtained for the Fifth-Order Sawada–Kotera equation using three wave type method via Hiröta bilinear form. The solutions investigated by three wave method are more than solutions by others method such as homoclinic test method.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Yanling Zhu, Kai Wang, Yong Ren, Yingdong Zhuang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we investigate the global existence of almost surely positive solution to a stochastic Nicholson’s blowflies delay differential equation with regime switching, and give the estimation of the path. The results presented in this paper extend some corresponding results in 〈em〉Wang et al. Stochastic Nicholson’s blowflies delayed differential equations, Appl. Math. Lett. 87 (2019) 20–26〈/em〉 .〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Jiguang Rao, Jingsong He, Dumitru Mihalache, Yi Cheng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this letter, we investigate the dynamics and various interaction scenarios of localized wave structures in the Kadomtsev–Petviashvili (KP)-based system. By using a combination of the Hirota’s bilinear method and the KP hierarchy reduction method, new families of determinant semi-rational solutions of the KP-based system are derived, including lump solitons and rogue-wave solitons. The generic interaction scenarios between distinct types of localized wave solutions are investigated. Our detailed study reveals different types of interaction phenomena: fusion of lumps and line solitons into line solitons, fission of line solitons into lumps and line solitons, a mixture of fission and fusion processes of lumps and line solitons, and the inelastic collision of line rogue waves and line solitons.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Yibin Ding, Xiang Xu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, a strategy to design a functional for inverse problems of hyperbolic equations is proposed. For an inverse source problem, it is shown that the designed functional is globally strictly convex. For an inverse coefficient problem, we can only prove that it is strictly convex near true solution. This strategy can be generalized to other inverse problems, as long as Lipschitz stability is given.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Thomas Trogdon〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Spectral and numerical properties of classes of random orthogonal butterfly matrices, as introduced by Parker (1995), are discussed, including the uniformity of eigenvalue distributions. These matrices are important because the matrix–vector product with an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉N〈/mi〉〈/math〉-dimensional vector can be performed in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉O〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉log〈/mo〉〈mi〉N〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 operations. In the simplest situation, these random matrices coincide with Haar measure on a subgroup of the orthogonal group. We discuss other implications in the context of randomized linear algebra.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Dumitru Motreanu, Patrick Winkert〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The existence of solutions of opposite constant sign is proved for a Dirichlet problem driven by the weighted 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mi〉q〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉-Laplacian with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉q〈/mi〉〈mo〉〈〈/mo〉〈mi〉p〈/mi〉〈/math〉 and exhibiting a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉q〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉-order term as well as a convection term. The approach is based on the method of sub–supersolution. Extremal solutions in relevant ordered intervals are obtained as well.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Xinwei Wang, Zhangxian Yuan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A comparison is made of the finite differences as a sum-accelerated pseudospectral method (FDSPM) and the discrete singular convolution (DSC) algorithms in free vibration analysis for the first time. Taylor series expansion method is used to eliminate the friction points. Some important issues on these numerical methods are addressed from the practical application point of view.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Shihu Li, Wei Liu, Yingchao Xie〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work we establish the exponential mixing property for stochastic 3D fractional Leray-〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉α〈/mi〉〈/math〉 model with degenerate noise. The main result is also applicable to stochastic 3D hyperviscous Navier–Stokes equations and stochastic 3D critical Leray-〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉α〈/mi〉〈/math〉 model.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Yan Tian, Shuhuang Xiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Prolate spheroidal wave functions of order zero (PSWFs) are widely used in scientific computation. There are few results about the error bounds of the prolate interpolation and differentiation. In this paper, based on the Cauchy’s residue theorem and asymptotics of PSWFs, the convergence rates are derived. To get stable approximation, the first barycentric formula is applied. These theoretical results and high accuracy are illustrated by numerical examples.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Hai-qiong Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The modified Volterra lattice equation with nonholonomic constrain has been considered in this paper. The integrability of the deformed model has been demonstrated by providing a Lax pair. Applying the gauge transformation to the Lax pair, we establish Darboux transformation theorem for the nonholonomic deformation equation. Some analytic solutions of the system are obtained via the one-fold and two-fold Darboux transformations. The deformation on explicit solutions exhibits different curvy profiles and propagation trajectories that were not found in modified Volterra lattice equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Ruihong Ji, Dan Li, Youhua Wei, Jiahong Wu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The stability problem on perturbations near the hydrostatic equilibrium of the Boussinesq equations has recently attracted considerable attention and there are substantial developments. This paper establishes the global 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉-stability for a 2D Boussinesq system with partial dissipation. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. This paper assesses the stability of the temperature gradient when there is a horizontal thermal diffusion.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Maosheng Jiang, Jiansong Zhang, Jiang Zhu, Xijun Yu, Luiz Bevilacqua〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉To illustrate the characteristic states of actin wave formation numerically, such as, spots, spirals and traveling waves, we propose a finite element method for simulating pattern formation in polymerizing actin flocks. Here we use the traditional Galerkin finite element method to solve the average filament polarization equation, and a mass-conservative characteristic finite element method to solve the density of actin equation, which deals well with the convection-dominated problem and keeps the mass balance globally. We study the convergence of this method and give the corresponding error estimate under some regularity assumptions of the solution. Finally some numerical results show the pattern formation in polymerizing actin flocks.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Xuezhong Wang, Maolin Che, Yimin Wei〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The analysis of the existence of solutions for the systems of polynomial equations is much more difficult to handle in general. Nonetheless, we can still confirm the existence of some particular solutions, especially for some special structured tensor equations. In this paper, we theoretically prove that a system of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si283.svg"〉〈msup〉〈mrow〉〈mi mathvariant="script"〉H〈/mi〉〈/mrow〉〈mrow〉〈mo〉+〈/mo〉〈/mrow〉〈/msup〉〈/math〉-tensor equations with the positive right-hand side has a unique positive solution by proposing a positive increasing map. We also prove that this system also has a negative solution. Numerical examples are presented to verify our results are true.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Jiajie Li, Shengfeng Zhu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The distributed shape gradient is shown recently to be more accurate and converges faster than the boundary type shape gradient, when finite element methods are used for discretizations. We propose a new algorithm for shape identification in stokes flow based on the distributed shape gradient. Numerical comparisons show that the proposed algorithm is more effective and efficient than the popular boundary type algorithm.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Ting Rong, Fuyi Li, Zhanping Liang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this study, the Fučik spectrum is used to investigate Kirchhoff-type problems with jumping nonlinearities at infinity, and a new result on the existence of nontrivial solutions is obtained.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Junchao Chen, Xueli Hu, Shundong Zhu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Rational solutions of the (2+1)-dimensional Kaup–Kupershmidt equation are presented in determinant form by taking the long wave limit of soliton solutions. With the appropriate parameters’ form, the soliton solution is firstly changed into the mixed solution consisting of solitons and periodic waves. Then taking the limit of the fixed parameter gives rise to the rational solution, which represents the interaction between the rational line soliton and the lump. It is shown that the rational line soliton exhibits the W-shaped soliton, while the lump is classified into the double-peak and the single-peak patterns. The interactions between two types of rational waves are discussed in different cases.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Xuehui Chen, Weidong Yang, Xinru Zhang, Fawang Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A double fractional Maxwell model is developed to study the boundary layer flow of viscoelastic MHD fluid over a stretching sheet. Based on the Caputo operators, fractional boundary layer equations are solved numerically by the finite difference method. The effects of stress fractional parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si65.gif"〉〈mi〉α〈/mi〉〈/math〉 and strain fractional parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉β〈/mi〉〈mspace width="0.16667em"〉〈/mspace〉〈/math〉on the flow characteristics are analyzed. It is worth mentioning that the viscoelastic MHD fluid described by the double fractional Maxwell model presents solid-like behavior or fluid-like behavior for different 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si65.gif"〉〈mi〉α〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉β〈/mi〉〈mspace width="0.16667em"〉〈/mspace〉〈/math〉, which agree well with the classical work of Friedrich. The results indicate that the greater the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si65.gif"〉〈mi〉α〈/mi〉〈/math〉 is, the stronger the elastic characteristic and the oscillating phenomenon is. However, as the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉β〈/mi〉〈mspace width="0.16667em"〉〈/mspace〉〈/math〉decreasing, the fluid represents viscous fluid-like behavior.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Elisa Francomano, Guido Ala, Marta Paliaga〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper improved fast summations are introduced to enhance a meshfree solver for the evolution of the electromagnetic fields over time. The original method discretizes the time-domain Maxwell’s curl equations via Smoothed Particle Hydrodynamics requiring many summations on the first derivatives of the kernel function and field vectors at each time step. The improved fast Gauss transform is properly adopted picking up the computational cost and the memory requirement at an acceptable level preserving the accuracy of the computation. Numerical simulations in two-dimensional domains are discussed giving evidence of improvements in the computation compared to the standard formulation.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Jun Zhang, Chuanjun Chen, Xiaofeng Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider numerical approximations for a phase-field dendritic crystal growth model, which is a highly nonlinear system that couples the anisotropic Allen–Cahn type equation and the heat equation. By combining the stabilized-Invariant Energy Quadratization method with a novel decoupling technique, the scheme requires solving only a sequence of linear elliptic equations at each time step, making it the first, to the best of the author’s knowledge, totally decoupled, linear, unconditionally energy stable scheme for the model. We further prove the unconditional energy stability rigorously and present various numerical simulations to demonstrate the stability and accuracy.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Feifei Du, Baoguo Jia〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we firstly develop an inequality on the rising function, which help us to deal with the fractional integral (sum) in terms of Hölder inequality. Secondly, a finite-time stability criterion of a class of nonlinear fractional delay difference systems with order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/math〉 is addressed by virtue of a new discrete Gronwall inequality and Jensen inequality. Finally, two numerical examples are presented to demonstrate our theoretical result.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 19 June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Enrique Otárola, Abner J. Salgado〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In two dimensions, we show existence of solutions to the stationary Navier Stokes equations on weighted spaces 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈msubsup〉〈mrow〉〈mi mathvariant="bold"〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉ω〈/mi〉〈mo〉,〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉×〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉ω〈/mi〉〈mo〉,〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, where the weight belongs to the Muckenhoupt class 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉. We show how this theory can be applied to obtain a priori error estimates for approximations of the solution to the Navier Stokes problem with singular sources.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Xiancheng Gao, Hongjun Gao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The main goal of this article is to study the 3D stochastic incompressible Navier–Stokes equations with delays in the convective term. Firstly, we consider the stochastic 3D Navier–Stokes equations with delays and obtain the local existence and uniqueness of the weak solutions, then we obtain the existence of global weak solutions by using the priori estimates.〈/p〉〈/div〉

Description: 〈p〉Publication date: September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 95〈/p〉 〈p〉Author(s): Ying Wang, Qi Li, Liquan Mei〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we propose an efficient numerical scheme for the space-fractional Klein–Gordon–Schrödinger (SFKGS) equations. Motivated by the “Invariant Energy Quadratization” (IEQ) approach, we introduce two auxiliary variables to transform the SFKGS system into a new equivalent system in which the time derivative is discretized by the Crank–Nicolson method, and the space discretization is based on the Fourier spectral method. Consequently, the numerical scheme shares two good features. The first feature is that the nonlinear terms are treated semi-explicitly and a linear symmetric system is solved at each time step. The second feature is the energy conservation at the discrete level. These two advantages are proved by the theoretical analysis and illustrated by a given numerical example.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Long Pei, Yuexun Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local well-posedness in classical Sobolev spaces, using a square-root type transformation to symmetrise the system.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Wenping Fan, Hui Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we do research on the numerical analysis and the inverse problem for the two-dimensional nonlinear space fractional Schrödinger equation. The second order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉θ〈/mi〉〈/math〉 scheme combined with the unstructured mesh finite element method, which can cover the second order Crank–Nicolson method and the second order backward difference method in time, and can deal with the two-dimensional irregular region boundaries better in space, is used to obtain the numerical solution of the considered equation. On this basis, we introduce the Bayesian method to inverse the unknown parameters for the two-dimensional fractional Schrödinger equation. To testify the validity and efficiency of the proposed methods, a numerical example is conducted.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Zhengguang Liu, Xiaoli Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Two novel and efficient modified techniques based on recently developed stabilized invariant energy quadratization (IEQ) approach to deal with nonlinear terms in gradient flows are proposed in this paper. We proved the unconditional energy stability for a class of gradient flows and their semi-discrete schemes carefully and rigorously. One of the contributions for this approach is that we succeeded in finding suitable positive preserving functions in square root and do not need to add a positive constant which cannot be fixed before computing. Secondly, all nonlinear terms can be treated semi-explicitly, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): Xifu Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we discuss the following minimum rank matrix approximation problem in the spectral norm: 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈munder〉〈mrow〉〈mo〉min〈/mo〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/munder〉〈mi〉r〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 subject to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈msub〉〈mrow〉〈mo〉‖〈/mo〉〈mi〉A〈/mi〉〈mo〉−〈/mo〉〈mi〉B〈/mi〉〈mi〉X〈/mi〉〈msup〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mo〉‖〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉A〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mi〉X〈/mi〉〈/math〉 are Hermitian matrices. For its application, we also consider the minimum rank matrix approximation problem: 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈munder〉〈mrow〉〈mo〉min〈/mo〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/munder〉〈mi〉r〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 subject to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈msub〉〈mrow〉〈mo〉‖〈/mo〉〈mfenced open="(" close=")"〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none none none none none none none none none none none none none none none none none none none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="center"〉〈mi〉A〈/mi〉〈/mtd〉〈mtd columnalign="center"〉〈mi〉B〈/mi〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="center"〉〈msup〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈/mtd〉〈mtd columnalign="center"〉〈mi〉X〈/mi〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="center"〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈mo〉‖〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 98〈/p〉 〈p〉Author(s): J.A. Ezquerro, M.A. Hernández-Verón〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The interest of the majorizing (scalar) sequences lies in that, from their convergence, we can deduce the convergence of an iterative method in Banach spaces. We propose a new technique to construct majorizing sequences that generalizes that given by Kantorovich for Newton’s method.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Jinji Yang, Yanling Tian〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The traveling wave solutions connecting two equilibria for a delayed Logistic equation in a cylinder are obtained for any delay 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉τ〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉. We attain our goal by using the approach based on the combination of Schauder fixed point theory and the weak coupled upper-lower solutions method. Moreover, we prove that there is a constant 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈/math〉 that serves as the minimal wave speed of such traveling wave solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 22 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Yang Cao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, shift-splitting preconditioners are studied for a special class of block three-by-three saddle point problems, which arise from many practical problems and are different from the traditional saddle point problems. It is proved that the block three-by-three saddle point matrix is positive stable and the corresponding shift-splitting stationary iteration method is unconditionally convergent, which leads to a nice clustering property of the eigenvalues of the shift-splitting preconditioned matrix. Numerical results show that the proposed shift-splitting preconditioners outperform much better than some existing block diagonal preconditioners studied recently.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 19 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): S.A. Buterin, P.A. Terekhin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider a nonlinear integral equation of a special type that appears in the inverse spectral theory of integro-differential operators and whose unique solvability in the class of square-integrable functions is known. However, for some applied issues in order to construct effective algorithms for solving equations of this type, it is required to establish their solvability in the class of analytic functions. Assuming the free term of the nonlinear equation under consideration to be an entire function of exponential type, we prove that so is its solution. Leaning on this result we provide a constructive procedure for solving this equation in the class of square-integrable functions, which can be easily implemented numerically.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Xiangpeng Xin, Linlin Zhang, Yarong Xia, Hanze Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, using the standard truncated Painlevé analysis, the Schwartzian equation of (2+1)-dimensional generalized variable coefficient shallow water wave (SWW) equation is obtained. With the help of Lax pairs, nonlocal symmetries of the SWW equation are constructed which be localized by a complicated calculation process. Furthermore, using the Lie point symmetries of the closed system and Schwartzian equation, some exact interaction solutions are obtained, such as soliton–cnoidal wave solutions. Corresponding 2D and 3D figures are placed to illustrate dynamic behavior of the generalized variable coefficient SWW equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 26 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Meina Sun〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In the frame of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉α〈/mi〉〈mo〉−〈/mo〉〈/math〉solutions defined in the setting of distributional products, the discontinuous solutions to the Riemann problem for a nonlinear chromatography system are constructed. All the discontinuous solutions are obtained within a convenient space of distributions including discontinuous functions and Dirac delta measures. The constructed 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉α〈/mi〉〈mo〉−〈/mo〉〈/math〉solutions are reasonable in comparison with the known results by using other techniques.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 26 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Seong-Ho Son, Kwang-Jae Lee, Won-Kwang Park〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, a direct sampling method (DSM) is designed for a real-time detection of small anomalies from scattering parameters measured by a small number of dipole antennas. Applicability of the DSM is theoretically demonstrated by proving that its indicator function can be represented in terms of an infinite series of Bessel functions of integer order, Hankel function of order zero, and the antenna configurations. Experiments using real-data then demonstrate both the effectiveness and limitations of this method.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Alexander G. Ramm〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉D〈/mi〉〈/math〉 be a bounded domain in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 with a smooth boundary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉S〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉N〈/mi〉〈/math〉 be the unit normal to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉S〈/mi〉〈/math〉 pointing out of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉D〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈mi〉k〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 is a constant.〈/p〉 〈p〉The class of over-determined problems of the type: 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si8.gif"〉〈msup〉〈mrow〉〈mo〉∇〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mspace width="1em"〉〈/mspace〉〈mi〉i〈/mi〉〈mi〉n〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mi〉D〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉u〈/mi〉〈msub〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈/math〉 is studied. Here 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si9.gif"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is the normal derivative of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si10.gif"〉〈mi〉u〈/mi〉〈/math〉 on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉S〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si12.gif"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si13.gif"〉〈mi〉j〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈/math〉 are constants.〈/p〉 〈p〉This problem was not studied as far as the author knows. Our result is the following theorem.〈/p〉 〈p〉〈strong〉Theorem.〈/strong〉 〈em〉If〈/em〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si14.gif"〉〈mrow〉〈mo〉|〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉+〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 〈em〉and the above problem has a solution then〈/em〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉D〈/mi〉〈/math〉 〈em〉is a ball.〈/em〉〈/p〉 〈p〉This Theorem contains several earlier results, proved by the author, including the refined Schiffer’s conjecture and the refined Pompeiu problem.〈/p〉 〈p〉This paper continues the author’s investigations of symmetry problems for partial differential equations.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 17 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): Michele V. Bartuccelli〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work we have analysed the nature of space fluctuations in dissipative Partial Differential Equations (PDEs). By taking a well known and much investigated dissipative PDE as our representative, namely the Swift-Hohenberg Equation, we estimated in an explicit manner the values of the crest factor of its solutions. We believe that the crest factor, namely the ratio between the sup-norm and the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 norm of solutions, is a suitable and proper measure of space fluctuations in solutions of dissipative PDEs. In particular it gives some information on the nature of “soft” and “hard” fluctuations regimes in the flows of dissipative PDEs.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Lin Luo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this letter, we discuss a variable-coefficient Boiti–Leon–Manna–Pempinelli equation. We present its soliton solution and derive its new bilinear Bäcklund transformation through Bell polynomial technique and bilinear method. Finally, we show the variable-coefficient Boiti–Leon–Manna–Pempinelli equation is completely integrable.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters, Volume 94〈/p〉 〈p〉Author(s): Hantaek Bae, Kyungkeun Kang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we provide a new regularity criterion of smooth solutions via the one component of the velocity field in various scaling invariant spaces with a natural growth condition of the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈/math〉 norm near a possible blow-up time.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 27 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Applied Mathematics Letters〈/p〉 〈p〉Author(s): M.A. Jorge Silva, S.B. Pinheiro〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In the present work our main goal is to improve the polynomial decay obtained recently by Santos and Almeida (2017) for a Timoshenko system with type III thermoelasticity. More specifically, in the generic case of different wave speeds of propagation, it is proved by the authors that problem is polynomially stable with decay rate 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉4〈/mn〉〈/mrow〉〈/msup〉〈/math〉 for the Dirichlet boundary condition. Here, our objective is to consider the same problem and prove, still in the general situation of different wave speeds, that the decay rate for the Dirichlet boundary condition is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉, which consists in a faster decay than the previous one.〈/p〉〈/div〉