Computational method for solving space fractional Fisher's nonlinear equation

El-Danaf, Talaat S.; Hadhoud, Adel R.

doi:10.1002/mma.2822

Cited by 13 publications

(2 citation statements)

References 16 publications

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“…In recent years, fractional calculus has played an increasingly important role in various fields and has attracted much interest from scholars due to its extensive applications in modeling many complex problems [1][2][3][4][5][6][7][8]. The fractional integro-differential equation is one of the most active fields in fractional calculus [9][10][11][12][13][14][15], which can be seen as the extension of classical integral equations by replacing integer-order derivatives with fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

Li,

Chen,

Zhou

2023

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In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method is applied for discretizing the spatial derivative.The exponential convergence of our proposed method is demonstrated in detail. Finally, numerical evidence is employed to verify the theoretical results and confirm the expected convergence rate.

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Section: Introductionmentioning

confidence: 99%

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

Li,

Chen,

Zhou

2023

Axioms

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“…Authors of [38] investigated the high-order and unconditionally stable difference scheme for the solution of modified anomalous fractional sub-diffusion equation by the inclusion of a secondary fractional time derivative acting on a diffusion operator. In [39], a numerical method developed for solving the fractional Fisher's equation by the quadratic spline functions. A truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations is introduced in [40].…”

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confidence: 99%

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Izadkhah

Saberi‐Nadjafi

2014

Math Methods in App Sciences

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In this paper, we study the numerical solution to time-fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time-fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. IntroductionRecently, fractional differential operators are indisputably found to play a fundamental role in the modeling of a considerable number of phenomena. Because of the nonlocal property of fractional derivative, they can utilize for modeling of memory-dependent phenomena and complex media such as porous media and anomalous diffusion [1][2][3][4]. They have been used in modeling turbulent flow [5,6], chaotic dynamics of classical conservation systems [7], and even finance [8,9] (see [1] for more information).Also, fractional calculus emerged as an important and efficient tool for the study of dynamical systems where classical methods reveal strong limitations. In [10], a fractional advection-dispersion equation is derived by extending Fick's first law from isotropic media to heterogeneous media and is particularly suitable for description of the highly skewed and heavy-tailed dispersion processes observed in rivers and other natural media.In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11][12][13], finite element methods [14,15], and spectral methods [16]. In [17][18][19], the authors used their proposed numerical schemes to solve Bagley-Torvik equation and other ordinary fractional differential equations. Gegenbauer polynomials have received much attention for their fundamental properties as well as for their use in applied mathematics. The described functions are a key ingredient to the implementation of spectral and pseudo-spectral methods to solve certain types of differential equations. Gegenbauer polynomials are a convenient basis for polynomial approximations because they are eigenfunctions of corresponding differential operators. For numerical methods, it is usually as convenient as efficient to convert between representations of a polynomial by expansion coefficients or by function values, respectively.Spectral approximations, such as the Fourier approximation based upon trigonometric polynomials for periodic problems, and the Chebyshev, Legendre, or the general Gegenbauer approximation based upon polynomials for nonperiodic problems are exponentially accurate for analytic functions [20][21][22][23]. In [24], the authors provided collocation method for natural convection heat transfer equations embedded in porous medium by using the rational Gegenbauer polynomials. Gottlieb and Shu in [25] used the Geg...

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Legendre Collocation Spectral Method for Solving Space Fractional Nonlinear Fisher’s Equation

Liu

2016

Theory, Methodology, Tools and Applications for Modeling and Simulation of Complex Systems

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Computational method for solving space fractional Fisher's nonlinear equation

Cited by 13 publications

References 16 publications

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Legendre Collocation Spectral Method for Solving Space Fractional Nonlinear Fisher’s Equation

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