Numerical Solution for the Variable Order Time Fractional Diffusion Equation with Bernstein Polynomials

In this paper, Bernstein polynomials method is proposed for the numerical solution of a class of variable order time fractional diffusion equation. Coimbra variable order fractional operator is adopted, as it is the most appropriate and desirable definition for physical modeling. The Coimbra variable order fractional operator can also be regarded as a Caputo-type definition. The main characteristic behind this approach in this paper is that we derive two kinds of operational matrixes of Bernstein polynomials. With the operational matrixes, the equation is transformed into the products of several dependent matrixes which can also be viewed as the system of linear equations after dispersing the variable. By solving the linear equations; the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.