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

Abstract

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...

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

“…The shifted Gegenbauer polynomials G k (t) defined above in Equation (8) are orthogonal [25,28,30] with respect to L 2 -space on the interval [0, 1].…”

“…In another study, the solutions of Bagley-Torvik equation have been presented via GWM which is an application to neural networking [19]. However, the Gegenbauer wavelets method [19,25,[28][29][30] still carries some drawbacks that need to be addressed. A significant deficiency in employing this method for more complex problems is its weakness in handling nonlinearity where this method gets diverged when it is used to compute solutions for nonlinear equations involving strong nonlinearities, such as exponential, cubic, or transcendental, and so on.…”

An extension of the Gegenbauer pseudospectral method for the time fractional Fokker‐Planck equation

“…This section is devoted to the numerical experiments, for demonstrating the effectiveness of the GPS method to solve numerically the TFFPE (Equations [22][23][24]. We implemented the GPS method with MATLAB 8.5 software in a PC laptop COREi3 with 2.13 GHz of CPU and 4 GB of RAM.…”

“…A Legendre pseudospectral method has been developed for the determination of the control parameter in a 3‐dimensional diffusion equation in Shamsi and Dehghan . In Izadkhah and Saberi Nadjafi, for solving the time fractional convection‐diffusion equations with variable coefficients, Gegenbauer spectral method have been proposed. The aim of this work is to present an effective numerical method for the TFFPE along with supplementary conditions by considering the Gegenbauer pseudospectral (GPS) method for the fractional derivative in time, while the spatial derivatives is approximated by pseudospectral method based on Chebyshev‐Gauss‐Lobatto (CGL) nodes.…”

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

“…Fractional PDEs play a key role in modeling some physical phenomena such as particle transport process in anomalous diffusion, which has applications in semiconductors, finance, electrochemistry, etc. () The Caputo temporal fractional derivative of u ( x , t ) is defined as The paper is organized as follows. Section 2 gives some preliminaries of Bernstein and dual Bernstein polynomials.…”