Solving Nonlinear Variable-Order Time Fractional Convection-Diffusion Equation with Generalized Polynomials

Abstract

AbstractThis chapter presents a new nonlinear variable-order (VO) time fractional convection-diffusion equation (NV-TFCDE). The model generalizes the standard fixed-order nonlinear time fractional convection-diffusion equation. The VO time fractional derivative is described in the Caputo type leading to an optimization method for the NV-TFCDE. The proposed approach is based on a new class of basis functions, namely the generalized polynomials (GP). The solution of the problem under consideration is expanded in terms of the GP with unknown free coefficients (FC) and control parameters (CP). A new VO time fractional operational matrix (V-TFOM) for the GP transforms the problem into a system of nonlinear algebraic equations with unknown FC and CP leading to the approximate solution. Several numerical examples show that the proposed method is efficient.KeywordsVariable-order time fractional convection-diffusion equationGeneralized polynomialsVariable-order time fractional operational matrixOptimization methodFree coefficientsControl parameters