A numerically efficient and conservative model for a Riesz space-fractional Klein–Gordon–Zakharov system

Hendy, Ahmed S.; Macías‐Díaz, Jorge E.

doi:10.1016/j.cnsns.2018.10.025

Cited by 35 publications

(13 citation statements)

References 42 publications

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“…The nature of the solution v(x, t) obtained by FNDM and q-HATM for Example 1 are, respectively, presented in Figures 1A and 1B. The surface of the exact solution for Equation (33) is illustrated in Figure 1C and the behavior of absolute error for corresponding equation obtained both techniques is plotted in Figures 1D and 1E. Figures 2A and 2B are the response of the obtained solution by the two cases of Example 1 with distinct fractional Brownian motion and standard motion ( = 1).…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…[27][28][29][30][31][32][33] Since most of the complex phenomena are modeled mathematically by nonlinear fractional differential equations, there are many methods in the literature to solve these equations. [27][28][29][30][31][32][33] Since most of the complex phenomena are modeled mathematically by nonlinear fractional differential equations, there are many methods in the literature to solve these equations.…”

Section: Introductionmentioning

confidence: 99%

“…The analytical and numerical solutions for the nonlinear fractional differential equations have fundamental importance. [27][28][29][30][31][32][33] Since most of the complex phenomena are modeled mathematically by nonlinear fractional differential equations, there are many methods in the literature to solve these equations. The Gardner and Cahn-Hilliard equations are studied through distinct techniques such as reduced differential transform method, 34 the modified Kudryashov technique, 35 Adomian decomposition method (ADM), 36 improved (G ′ /G) − expansion method, 37 homotopy perturbation method (HPM), 26 residual power series method (RPSM), 22 and many others.…”

Section: Introductionmentioning

confidence: 99%

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Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

Prakasha

Veeresha

Baskonus

2019

Comp and Math Methods

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The numerical solutions for nonlinear fractional Gardner and Cahn-Hilliard equations arising in fluids flow are obtained with the aid of two novel techniques, namely, fractional natural decomposition method (FNDM) and q-homotopy analysis transform method (q-HATM). Both featured techniques are different from each other since FNDM is algorithmic by the aid of Adomian polynomial and q-HATM is defined by the help of homotopy polynomial. The numerical simulations have been conducted to verify that the proposed schemes are reliable and accurate. The outcomes are revealed through the plots and tables. The comparison of solution obtained by proposed schemes with the available solutions exhibits that both the featured schemes are methodical, efficient, and very exact in solving the nonlinear complex phenomena. KEYWORDS fractional Cahn-Hilliard equation, fractional Gardner equation, fractional natural decomposition method, q-homotopy analysis transform methodwhere is real constant. Here, v (x, t) is the wave function with scaling variables space (x) and time (t), the terms vv x and v 2 v x are symbolize by the nonlinear wave steepening, and v xxx denotes the dispersive wave effects.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

Prakasha

Veeresha

Baskonus

2019

Comp and Math Methods

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“…Naturally, one would wonder which are the effects of considering a fractional diffusion in such HIV-1 systems. At this point, it is important to mention that one of the authors of the present manuscript has devoted part of his efforts to develop numerical methods for Riesz space-fractional partial differential equations [29,30]. In that context, the differentiation order of the diffusion terms affect the speed of propagation of the spread of effects into the medium.…”

Section: Discussionmentioning

confidence: 98%

A finite-difference discretization preserving the structure of solutions of a diffusive model of type-1 human immunodeficiency virus

Alba-Pérez

Macías‐Díaz

2021

Adv Differ Equ

Self Cite

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We investigate a model of spatio-temporal spreading of human immunodeficiency virus HIV-1. The mathematical model considers the presence of various components in a human tissue, including the uninfected CD4+T cells density, the density of infected CD4+T cells, and the density of free HIV infection particles in the blood. These three components are nonnegative and bounded variables. By expressing the original model in an equivalent exponential form, we propose a positive and bounded discrete model to estimate the solutions of the continuous system. We establish conditions under which the nonnegative and bounded features of the initial-boundary data are preserved under the scheme. Moreover, we show rigorously that the method is a consistent scheme for the differential model under study, with first and second orders of consistency in time and space, respectively. The scheme is an unconditionally stable and convergent technique which has first and second orders of convergence in time and space, respectively. An application to the spatio-temporal dynamics of HIV-1 is presented in this manuscript. For the sake of reproducibility, we provide a computer implementation of our method at the end of this work.

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“…2, we introduce the fractional diffusive problem on interest and provide a Crank-Nicolson discretization based on the use of fractional-order centered differences. Here, it is worthwhile to recall that fractional-order centered differences have been used successfully by some of the authors in the discretization of various fractional problems [31][32][33]. Suitable discrete nomenclature is introduced to that end, including some helpful properties of the fractional centered differences and a convenient vector representation of the numerical model.…”

Section: Introductionmentioning

confidence: 99%

Discrete monotone method for space-fractional nonlinear reaction–diffusion equations

2019

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A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion-reaction equation. More precisely, we propose a Crank-Nicolson discretization of a reaction-diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher's equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method. MSC: Primary 65M06; secondary 35K15; 35K55; 35K57

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A numerically efficient and conservative model for a Riesz space-fractional Klein–Gordon–Zakharov system

Cited by 35 publications

References 42 publications

Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

A finite-difference discretization preserving the structure of solutions of a diffusive model of type-1 human immunodeficiency virus

Discrete monotone method for space-fractional nonlinear reaction–diffusion equations

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