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  • Hindawi  (12)
  • 1
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    Newtonian and Non-Newtonian Fluids through Permeable Boundaries (2014)
    Hindawi
    Details
    Publication Date: 2014-10-02
    Description: We considered the situation where a container with a permeable boundary is immersed in a larger body of fluid of the same kind. In this paper, we found mathematical expressions at the permeable interface of a domain , where . is defined as a smooth two-dimensional (at least class ) manifold in . The Sennet-Frenet formulas for curves without torsion were employed to find the expressions on the interface . We modelled the flow of Newtonian as well as non-Newtonian fluids through permeable boundaries which results in nonhomogeneous dynamic and kinematic boundary conditions. The flow is assumed to flow through the boundary only in the direction of the outer normal n, where the tangential components are assumed to be zero. These conditions take into account certain assumptions made on the curvature of the boundary regarding the surface density and the shape of ; namely, that the curvature is constrained in a certain way. Stability of the rest state and uniqueness are proved for a special case where a “shear flow” is assumed.
    Print ISSN: 1024-123X
    Electronic ISSN: 1563-5147
    Topics: Mathematics , Technology
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  • 2
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    Extension of Matched Asymptotic Method to Fractional Boundary Layers Problems (2014)
    Hindawi
    Details
    Publication Date: 2014-11-21
    Description: We were concerned with the description of the boundary layers problems within the scope of fractional calculus. However, we will note that one of the main methods used to solve these problems is the matched asymptotic method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the matched asymptotic method to the scope of fractional derivative, we proposed a relatively new derivative called the beta-derivative. We presented some useful information for this operator. With the reward of this operator, we presented the idea of matched asymptotic method in finding solutions of the fractional boundary layers problems. The method was illustrated with an example.
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  • 3
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    On Analysis of Fractional Navier-Stokes Equations via Nonsingular Solutions and Approximation (2015)
    Hindawi
    Details
    Publication Date: 2015-07-02
    Description: Until now, all the investigations on fractional or generalized Navier-Stokes equations have been done under some restrictions on the different values that can take the fractional order derivative parameter . In this paper, we analyze the existence and stability of nonsingular solutions to fractional Navier-Stokes equations of type () defined below. In the case where , we show that the stability of the (quadratic) convergence, when exploiting Newton’s method, can only be ensured when the first guess is sufficiently near the solution . We provide interesting well-posedness and existence results for the fractional model in two other cases, namely, when and
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  • 4
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    Existence Results for a Michaud Fractional, Nonlocal, and Randomly Position Structured Fragmentation Model (2014)
    Hindawi
    Details
    Publication Date: 2014-06-13
    Description: Until now, classical models of clusters’ fission remain unable to fully explain strange phenomena like the phenomenon of shattering (Ziff and McGrady, 1987) and the sudden appearance of infinitely many particles in some systems having initial finite number of particles. That is why there is a need to extend classical models to models with fractional derivative order and use new and various techniques to analyze them. In this paper, we prove the existence of strongly continuous solution operators for nonlocal fragmentation models with Michaud time derivative of fractional order (Samko et al., 1993). We focus on the case where the splitting rate is dependent on size and position and where new particles generating from fragmentation are distributed in space randomly according to some probability density. In the analysis, we make use of the substochastic semigroup theory, the subordination principle for differential equations of fractional order (Prüss, 1993, Bazhlekova, 2000), the analogy of Hille-Yosida theorem for fractional model (Prüss, 1993), and useful properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005). We are then able to show that the solution operator to the full model is positive and contractive.
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  • 5
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    Mathematical Solvability of a Caputo Fractional Polymer Degradation Model Using Further Generalized Functions (2014)
    Hindawi
    Details
    Publication Date: 2014-06-26
    Description: The continuous fission equation with derivative of fractional order , describing the polymer chain degradation, is solved explicitly. We prove that, whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalized -function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.
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  • 6
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    Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities (2017)
    Hindawi
    Details
    Publication Date: 2017-01-01
    Description: After the recent introduction of the Caputo-Fabrizio derivative by authors of the same names, the question was raised about an eventual comparison with the old version, namely, the Caputo derivative. Unlike Caputo derivative, the newly introduced Caputo-Fabrizio derivative has no singular kernel and the concern was about the real impact of this nonsingularity on real life nonlinear phenomena like those found in shallow water waves. In this paper, a nonlinear Sawada-Kotera equation, suitable in describing the behavior of shallow water waves, is comprehensively analyzed with both types of derivative. In the investigations, various fixed-point theories are exploited together with the concept of Piccard K-stability. We are then able to obtain the existence and uniqueness results for the models with both versions of derivatives. We conclude the analysis by performing some numerical approximations with both derivatives and graphical simulations being presented for some values of the derivative order γ. Similar behaviors are pointed out and they concur with the expected multisoliton solutions well known for the Sawada-Kotera equation. This great observation means either of both derivatives is suitable to describe the motion of shallow water waves.
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  • 7
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    Extension of Matched Asymptotic Method to Fractional Boundary Layers Problems (2014)
    Hindawi
    Details
    Publication Date: 2014-01-01
    Description: We were concerned with the description of the boundary layers problems within the scope of fractional calculus. However, we will note that one of the main methods used to solve these problems is the matched asymptotic method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the matched asymptotic method to the scope of fractional derivative, we proposed a relatively new derivative called thebeta-derivative. We presented some useful information for this operator. With the reward of this operator, we presented the idea of matched asymptotic method in finding solutions of the fractional boundary layers problems. The method was illustrated with an example.
    Print ISSN: 1024-123X
    Electronic ISSN: 1563-5147
    Topics: Mathematics , Technology
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  • 8
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    Newtonian and Non-Newtonian Fluids through Permeable Boundaries (2014)
    Hindawi
    Details
    Publication Date: 2014-01-01
    Description: We considered the situation where a container with a permeable boundary is immersed in a larger body of fluid of the same kind. In this paper, we found mathematical expressions at the permeable interfaceΓof a domainΩ, whereΩ⊂R3.Γis defined as a smooth two-dimensional (at least classC2) manifold inΩ. The Sennet-Frenet formulas for curves without torsion were employed to find the expressions on the interfaceΓ. We modelled the flow of Newtonian as well as non-Newtonian fluids through permeable boundaries which results in nonhomogeneous dynamic and kinematic boundary conditions. The flow is assumed to flow through the boundary only in the direction of the outer normaln, where the tangential components are assumed to be zero. These conditions take into account certain assumptions made on the curvature of the boundary regarding the surface density and the shape ofΩ; namely, that the curvature is constrained in a certain way. Stability of the rest state and uniqueness are proved for a special case where a “shear flow” is assumed.
    Print ISSN: 1024-123X
    Electronic ISSN: 1563-5147
    Topics: Mathematics , Technology
    Published by Hindawi
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  • 9
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    Mathematical Solvability of a Caputo Fractional Polymer Degradation Model Using Further Generalized Functions (2014)
    Hindawi
    Details
    Publication Date: 2014-01-01
    Description: The continuous fission equation with derivative of fractional orderα, describing the polymer chain degradation, is solved explicitly. We prove that, whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalizedG-function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.
    Print ISSN: 1024-123X
    Electronic ISSN: 1563-5147
    Topics: Mathematics , Technology
    Published by Hindawi
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  • 10
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    On Analysis of Fractional Navier-Stokes Equations via Nonsingular Solutions and Approximation (2015)
    Hindawi
    Details
    Publication Date: 2015-01-01
    Description: Until now, all the investigations on fractional or generalized Navier-Stokes equations have been done under some restrictions on the different values that can take the fractional order derivative parameterβ. In this paper, we analyze the existence and stability of nonsingular solutions to fractional Navier-Stokes equations of type (ut+u·∇u+∇p-Re-1(-∇)βu=f  in  Ω×(0,T]) defined below. In the case whereβ=2, we show that the stability of the (quadratic) convergence, when exploiting Newton’s method, can only be ensured when the first guessU0is sufficiently near the solutionU. We provide interesting well-posedness and existence results for the fractional model in two other cases, namely, when1/2
    Print ISSN: 1024-123X
    Electronic ISSN: 1563-5147
    Topics: Mathematics , Technology
    Published by Hindawi
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