ALBERT

Description: The class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo, has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl. Math. , 48 , 279–319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58 , 309–317; Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond. A , 463 , 357–367). The new classes include fully implicit constitutive relations for the stress and the deformation gradient, and the interesting sub-class wherein the Cauchy–Green tensor or the linearized strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the Green–St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A , 463 , 357–367), other possible implicit relationships between other stress and kinematic measures have not been investigated. This paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic bodies, when they are expressed through implicit relations for different stress and stretch/strain measures.

Description: This work investigates the existence and non-existence of travelling wave solutions for Kolmogorov-type delayed lattice reaction–diffusion systems. Employing the cross iterative technique coupled with the explicit construction of upper and lower solutions in the theory of quasimonotone dynamical systems, we can find two threshold speeds $c^* $ and $c_* $ with $c^* \geq c_* 〉 0.$ If the wave speed is greater than $c^* ,$ then we establish the existence of travelling wave solutions connecting two different equilibria. On the other hand, if the wave speed is smaller than $c_* ,$ we further prove the non-existence result of travelling wave solutions. Finally, several ecological examples including one-species, two-species and three-species models with various functional responses and time delays are presented to illustrate the analytical results.

Description: In this paper, sufficient criteria for global asymptotic stability of a general stochastic non-autonomous Lotka–Volterra system with infinite delays are established. Then these criteria are applied to a competitive system, a predator–prey system and a cooperative system to demonstrate their applicability and effectiveness. Some recent results are extended and improved.

Description: In the paper, the problem of existence of almost periodic solutions to non-linear impulsive systems of Caputo fractional order is addressed. Sufficient conditions for stability of almost periodic solutions are also established. The investigations are carried out by using of a fractional comparison principle and the Lyapunov's function method. We apply our results to an impulsive cellular neural network model of fractional order.

Description: In this paper we propose a limited memory method, called the Broyden rank reduction (BRR) method, to efficiently compute fixed points of high-dimensional non-linear maps. The method is based on the singular value decomposition and has good convergence properties. We discuss some basic properties of the BRR method and prove convergence for linear systems. The method is developed for a concrete problem in engineering. In chemical engineering, periodically forced processes in packed bed reactors are generally simulated using pseudo-homogeneous one-dimensional models on a coarse grid discretization. In these simulations, one typically uses Broyden's method because of its simplicity and the fact that appropriate initial conditions can be chosen in a natural way. A disadvantage of Broyden's method is the memory usage to store the Broyden matrix. The BRR method resolves this issue and allows us to consider more complicated and accurate models. We show that the BRR method developed to simulate periodically forced processes is in certain situations a good option to compute fixed points of high-dimensional non-linear maps and illustrate this using examples from integral equations and parabolic partial differential equations. To benchmark our method, we conclude the paper with a comparison of the BRR method with the generally preferred Newton–GMRES( m ) method.

Description: In this paper, the relationship between representation formulas for unique regularized solutions of inverse source, as well as backward problems with final overdetermination for evolution equations and singular value decomposition (SVD) of corresponding input–output operators, is analysed. For each considered inverse problem this representation formula is derived via the solution of appropriate adjoint problems and the parameter of regularization. The Lagrange multiplier method is then used to show that each regularized inverse problem is equivalent to a coupled system of two (direct and adjoint) parabolic (or hyperbolic) problems, and the Lagrange multiplier is the weak solution of the corresponding adjoint problem. In the constant coefficient linear parabolic and hyperbolic equations cases, it is proved that the coupled problem is equivalent to the normal equation corresponding to each inverse/backward problem. As a result, an equivalence of the representation formula for unique regularized solution and the SVD of the input–output operator is obtained. This approach, in particular, allows one to construct regularizing filters for a considered class of inverse problems. Some numerical examples are presented to illustrate the usefulness and effectiveness of the proposed approach.

Description: This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier–Stokes equations, complexified in time. Specifically, for periodic boundary conditions on [0, L ] 2 , and a force g D( A (α–1)/2 ), we show that there is a fixed strip about the real-time axis on which a uniform bound $|A^{\alpha}u| 〈 m_{\alpha}\nu\kappa_{0}^{\alpha}$ holds for each $\alpha \in {\mathbb {N}}$ . Here is viscosity, $\kappa_{0}=2\pi/L$ , and m α is explicitly given in terms of g and α. We show that if any element in A is in $\mathcal{D}(A^{\alpha})$ , then all of A is in $\mathcal{D}(A^{\alpha})$ , and likewise with $\mathcal{D}(A^{\alpha})$ replaced by $C^{\infty}(\Omega)$ . We demonstrate the universality of this ‘all for one, one for all’ law is also shown to hold for the union of a hierarchical set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a non-zero force by showing that if this is so, the force must be in a particular function class.

Description: An exact solution is obtained for the steady plane magnetohydrodynamics oblique stagnation-point flow of a homogeneous, incompressible, electrically conducting micropolar fluid over a rigid uncharged dielectric at rest. The space is permeated by a non-uniform external magnetic field and by a suitable uniform external electric field. The total magnetic field in the fluid is parallel to the velocity at infinity. The problem is reduced to a non-linear ordinary differential boundary value problem by using similarity transformations. The effects of the magnetic field on the velocity and on the microrotation profiles are presented graphically and discussed.

Description: In this paper, we analyse the connection between approximate non-linear self-adjointness of perturbed partial differential equations (PDEs) and non-linear self-adjointness of the corresponding unperturbed PDEs, and consequently provide a simple approach to discriminate approximate non-linear self-adjointness of perturbed PDEs. Moreover, a succinct approximate conservation law formula by virtue of the known conservation laws of the corresponding unperturbed PDEs is given in an explicit form. As an application, we classify a class of perturbed wave equations to be approximately non-linearly self-adjoint and construct its general approximate conservation law formula. The specific examples demonstrate that approximate non-linear self-adjointness can be used to construct new approximate conservation laws of perturbed PDEs.

Description: Practical physical applications of mathematical nature are frequently met in engineering technology and involve the low-Reynolds number flow of micropolar conducting fluids under the effect of magnetic fields. Here, we consider the three-dimensional (3D) creeping motion (Stokes flow), in steady state, of a non-conductive colloidal suspension of ferromagnetic material embedded within an electrically conductive, viscous and incompressible, carrier liquid. In such cases, the ferromagnetic particles behave as rigid magnetic dipoles and react in the presence of an externally applied magnetic field, which is of general form and arbitrarily orientated in the 3D space. Therein, an induced magnetic field of minor significance is created, while the effective viscosity of the fluid is increasing and an additional magnetic pressure is appeared. The consistency of the governing set of partial differential equations with the principles of both ferrohydrodynamics and magnetohydrodynamics is established by taking into account magnetization and electrical conductivity of the fluid, respectively. Our main intension is to use the potential representation theory to improve previous models and construct a new complete and unique integro-differential representation of the magnetic Stokes flow of conducting liquids, valid for any non-axisymmetric geometry, which provides the velocity and total pressure fields in a closed form and in terms of easy-to-find potentials, via a semi-analytical formalism. To demonstrate the usefulness of our analytical approach, we assume a degenerate case of the aforementioned method to simulate the creeping flow of a micropolar fluid with conductive properties inside a circular duct.