Legendre-finite difference method for solving fractional nonlinear Sobolev equation with Caputo derivative

Abstract

The Sobolev equation is known as a partial differential equation with second-order partial derivatives. Initially, the Sobolev equation was used to solve the problem of turbulence in physics. However, due to its extensive use in mathematics and geometry, it soon became one of the most important mathematical equations. Despite the significance of this equation, its solution in fractional and nonlinear cases has received less attention in research. This paper presents a numerical algorithm for solving fractional nonlinear Sobolev model using Legendre polynomials and a finite difference scheme. Firstly, the finite difference plan is employed to discretize the time variable, where stability and convergence of the method are demonstrated. The stability of the method is dependent on the nonlinear part of the problem. The two-dimensional Legendre polynomials are utilized at collocation points to discretize the spatial variables. Finally, several illustrative examples are conducted to evaluate the accuracy of the proposed approach.