An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation

Abstract

In this paper, an optimization method based on a new class of basis functions, namely generalized polynomials (GPs), is proposed for nonlinear variable-order time fractional diffusion-wave equation. Variable-order time fractional derivative is expressed in the Caputo sense. In the proposed method, solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. In this way, some new operational matrices of the ordinary and fractional derivatives are derived for these basis functions. The residual function and its 2-norm are employed for converting the problem under study to an optimization one and then choosing the unknown free coefficients and control parameters optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters. The validity and effectiveness of the method are demonstrated by solving some numerical examples. The results demonstrate that the proposed method is a powerful algorithm with good accuracy for solving such kind of problems.