A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations
Introduction
During the last decades, some major contributions have been made to both the theory and applications of the fractional differential equations motivated by various practical engineering and physical problems. These applications cross diverse disciplines, such as chemical physics [1], viscoelasticity [2], electricity [3], finance [4], control theory [5], biomedical engineering [1], fluid mechanics [6] and other sciences (see [6], [7], [8] and references therein). In fact, it has been found that the fractional-order models are more adequate than the previously used integer-order models [9], [10], [11], because fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different substances.
Space fractional partial differential equations (SFPDEs) are used to model anomalous diffusive and super-diffusive systems, where a particle plume spread faster than the classical Brownian motion model predicts [12]. The left and right space fractional derivatives allow the modeling of flow regime impacts from either side of the domain [13]. The space fractional advection–dispersion equation (SFADE) is used to describe the transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns [14], [15] and it is also proved as a benefit tool to model the transport of passive tracers carried by fluid flow in a porous medium for the groundwater hydrology research [16]. Several numerical methods have been proposed in the last few years for solving SFADE such that finite difference method [17], [18], [19], [20], finite volume method [21], random walk method [22], random walk particle tracking method [23], [24], and spectral methods [25], [26], [27]. The fractional telegraph equation is commonly used in the study of wave phenomena and also in propagation of electric signals in a cable transmission line [9], [28], [29].
On the other hand, it was pointed out that many of the universal electromagnetic, acoustic and mechanical responses can be modeled accurately using the time-fractional diffusion and wave–diffusion equations. The power law wave equation [30] is used to model sound wave propagation in anisotropic media that exhibits frequency dependent attenuation . In human tissue, experimental evidence indicates that the power law is . Saadatmandi et al. [31] gave an efficient numerical algorithm based on the Sinc–Legendre collocation method for the time-fractional convection–diffusion equation with variable coefficients on a finite domain. In [32], [33], Cui presented high-order compact finite difference methods to solve the fractional diffusion and time fractional diffusion equation. Luchko [34] discussed the multi-term time-fractional diffusion equation with the variable coefficients. Liu et al. [35] proposed an implicit numerical method for a class of fractional advection–dispersion models in which they discussed five fractional models. They also proposed some computationally effective numerical methods for simulating the multi-term time-fractional wave–diffusion equations [30].
The spectral method is an efficient tool for computing approximate solutions of differential equations because of its high-order accuracy (see for instance [36], [37], [38], [39], [40], [41], [42], [43]). The use of the spectral method in both temporal and spatial discretizations of fractional partial differential equations may significantly reduce the storage requirement because, as compared to low order methods, much fewer time and space levels are needed to compute a smooth solution. The main idea of spectral methods is to express the solution of the differential equation as a sum of basis functions and then to choose the coefficients in order to minimize the error between the numerical and exact solutions as well as possible.
The main goal of this paper is to construct an efficient spectral algorithm to solve multi-term time–space fractional partial differential equation. We start by expanding the spectral solution in terms of a truncated shifted Jacobi vector, and then the left- and right-sided Caputo fractional derivatives and Riemann–Liouville fractional integral of the solution are expressed in terms of their operational matrices. A shifted Jacobi tau algorithm is investigated for treating both temporal and spatial discretizations. Indeed, this algorithm is basically constructed on the shifted Jacobi tau method in combination with the constructed operational matrices. Consequently, the proposed algorithm converts the multi-term time–space fractional partial differential equation into a system of algebraic equations. A special attention is paid for the error analysis and the convergence of the method. Finally, some applications are given for time and space fractional advection–diffusion equation, telegraph equation and power law wave equation.
The paper is organized as follows. In Section 2, we present some fractional calculus preliminaries and properties of Jacobi polynomials and then we construct the operational matrices of Jacobi polynomials with a particular focus on the Caputo definition. In Section 3, by using tau spectral method, we construct and develop an algorithm for the solution of the multi-term time–space fractional differential equations with Dirichlet conditions. In Section 4, the convergence analysis of the proposed spectral algorithm is investigated. Section 5 reports numerical experiments with four illustrative examples. The last section consists of some obtained conclusions.
Section snippets
Preliminaries and fundamentals
In the first part of this section, we recall some basic properties of fractional calculus theory. Then we present some properties of the Jacobi polynomials (see, e.g. [44], [45]) which sets the stage for the third part, discussing key ideas related to the operational matrices of fractional integral and derivative of Jacobi polynomials.
Multi-term time–space fractional partial differential equations
Multi-term time fractional partial differential equations model several physical phenomena, for example the power law wave equation [60], the Szabo wave equation [61], fractional diffusion equation [34], fractional telegraph equation [28] and neutron telegraph equation [9]. However, studies of these multi-term time–space fractional differential equation still under development.
In this section, we consider the multi-term time and space fractional partial differential equation with nonhomogeneous
Error analysis
In this section, an upper bound of the absolute errors will be given by using Lagrange interpolation polynomials. Also, by using the Tau method with error estimation and the residual correction method [65], [66], an efficient error estimation will be given for the multi-term time fractional partial differential equations.
Numerical results and comparisons
In this section, we present four numerical examples to demonstrate the accuracy and applicability of the proposed method. We also compare the results given from our scheme and those reported in the literature such as difference scheme with spline function [73], implicit difference approximation [18] and fractional predictor–corrector method [30].
Example 1 As a first application, we offer the following one-dimensional Riesz space fractional reaction dispersion equation [73], [74]:
Conclusion
In this paper, we have proposed a space–time spectral algorithm based on shifted Jacobi tau technique combined with the associated operational matrices of left-sided Caputo, right-sided Caputo fractional derivatives and Riemann–Liouville fractional integrals. This algorithm was employed for solving the multi-term time–space fractional differential equation. The space and time fractional derivatives were given in the Caputo sense. Moreover, space fractional derivative includes the left and right
Acknowledgements
The authors are very grateful to the reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.
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