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Abstract
In the present paper, we introduce the fractional Bernstein series solution (FBSS) to solve the fractional diffusion equation, which is a generalization of the classical diffusion equation. The Bernstein polynomial method is a promising one and can be generalized to more complicated problems in fractional partial differential equations. To get the FBSS, we first convert all terms in the problem to matrix forms. Then, the fundamental matrix equation is obtained and thus, the solution is obtained. Two error estimation methods based on a residual correction procedure and the consecutive approximations are incorporated to find the estimate and bound of the absolute error. The perturbation and stability analysis of the method is given. We apply the method to some illustrative examples. The numerical results are compared with the exact solutions and known second-order methods. The outcomes of the numerical examples are very encouraging and show that the FBSS is highly useful in solving fractional partial problems. The results show the accuracy and effectiveness of the method.
Highlights
Fractional derivatives have been used to model many problems in science, e.g., physics [1,2,3], medicine [4], hydrology [5], biomedical problems [6], dynamics of particles [7] and applied sciences [8]
The linear fractional diffusion equation is considered for scientists and engineers [9]
Some linear and non-linear systems of ordinary differential equations have been solved by Bernstein polynomials method (BPM) and the accuracy has been improved by a residual correction procedure [34]
Summary
Fractional derivatives have been used to model many problems in science, e.g., physics [1,2,3], medicine [4], hydrology [5], biomedical problems [6], dynamics of particles [7] and applied sciences [8]. We consider the following one-dimensional fractional diffusion equation (FDE),. The operational matrices of the BPM have been used to get the numerical solutions of a class of third-order ordinary differential equations [27]. The BPM with new modifications was employed to solve fractional differential equations [29]. The same method was used to solve some types of ordinary differential equations [30,31,32]. Some linear and non-linear systems of ordinary differential equations have been solved by BPM and the accuracy has been improved by a residual correction procedure [34]. The fractional Bernstein series solution (FBSS) method is introduced and applied to solve Equation (1) numerically.
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