A kernel‐based method for solving the time‐fractional diffusion equation

Abstract

In this paper, we focus on the development and study of a numerical method based on the idea of kernel‐based approximation and finite difference discretization to obtain the solution for the time‐fractional diffusion equation. Using the theory of reproducing kernel, reproducing kernel functions with a polynomial form will be established in polynomial reproducing kernel spaces spanned by the Chebychev basis polynomials. In the numerical method, first the time‐fractional derivative term in the aforementioned equation is approximated by using the finite difference scheme. Then, by the help of collocation method based on reproducing kernel approximation, we will illustrate how to derive the numerical solution in polynomial reproducing kernel space. Finally, to support the accuracy and efficiency of the numerical method, we provide several numerical examples. In numerical experiments, the quality of approximation is calculated by absolute error and discrete error norms.

A Highly Accurate Computational Approach to Solving the Diffusion Equation of a Fractional Order

“…The convergence of the proposed scheme is also verified via the presented results. To provide more evidence for the capability of the presented method, the results of this work are compared with those of a kernel-based method [33] in Table 2. The approximate solution and related absolute errors for µ = 0.2 are plotted in Figure 1.…”

“…Moreover, Figure 3 is depicted to showcase the method's convergence. To compare the obtained results via the present scheme and the kernel-based method [33], Table 4 is tabulated. The L 2 -error at different times versus the number of bases N is reported in Table 5.…”

“…Example 3. To illustrate the accuracy of the method, the following equation from [33] is considered:…”

“…Chen et al [32] proposed the Kansa method for solving the considered Equation (1). A paper focused on the kernel-based scheme to solve (1) is introduced in [33] by Fardi. The subsequent sections of this paper are structured in the following manner: Chebyshev cardinal polynomials and their properties are reviewed and introduced in Section 2. The pseudospectral method is applied to solve the TFDE (1) in Section 3.…”

Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

“…In the future, we plan to extend our numerical approaches for solving generalized fractional models, including the timefractional diffusion equation [61], the time-fractional mobileimmobile advection-dispersion equation [62], the timefractional mobile/immobile transport model [63], and multiterm boundary value problem of variable order [64], etc.…”