Abstract
In this paper, we solve a time-space fractional diffusion equation. Our methods are based on normalized Bernstein polynomials. For the space domain, we use a set of normalized Bernstein polynomials and for the time domain, which is a semi-infinite domain, we offer an algebraic map to make the rational normalized Bernstein functions. This study uses Galerkin and collocation methods. The integrals in the Galerkin method are established with Chebyshev interpolation. We implemented the proposed methods for some examples that are presented to demonstrate the theoretical results. To confirm the accuracy, error analysis is carried out.
Highlights
Fractional calculus allows mathematicians and engineers better modeling of a wide class of systems with anomalous dynamic behavior and better understanding of the facets of both physical phenomena and artificial processes
The maximum errors L∞ for different values of T and n are listed in Tables and for collocation and Galerkin methods. Note that in these tables we provide CPU time consumed in the algorithms for obtaining the numerical solution, and when we compare these together for one problem, we see the collocation method acting in a shorter time compared with the Galerkin method
For these problems defined in the unbounded time domain, we use the rational normalized Bernstein functions as basis functions to approximate the exact solution
Summary
Fractional calculus allows mathematicians and engineers better modeling of a wide class of systems with anomalous dynamic behavior and better understanding of the facets of both physical phenomena and artificial processes. The normalized Bernstein polynomials with collocation and Galerkin methods are applied to turn the problem into an algebraic system. In Section , Galerkin and collocation methods to approximate the unknown function u(x, t) are applied. The Caputo time and space fractional derivatives of the function u are given as follows.
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