A Numerical Method for Distributed-Order Time Fractional 2d Sobolev Equation

Abstract

In this work, the distributed-order time fractional 2D Sobolev equation is introduced. The orthonormal Bernoulli polynomials, as a renowned family of basis functions, are employed to solve this problem. To effectively use of these polynomials in constructing a suitable methodology for this equation, some operational matrices regarding the ordinary and fractional derivative of them are derived. In the developed method, by approximating the unknown solution by means of these polynomials and using the mentioned matrices, as well as applying the collocation technique, a system of algebraic equations (in which the unknowns are the expansion coefficients of the solution function) is obtained, which by solving it, a solution for the main problem is obtained. By providing four test problems, the capability and accuracy of the scheme are studied.