Novel operational matrix method for the numerical solution of nonlinear reaction–advection–diffusion equation of fractional order

Abstract

In this work, a new scheme has been developed for the numerical solution of the fractional order reaction–advection–diffusion equation. To approximate the problem the authors have used Vieta–Fibonacci polynomials as basis functions and derived for the first time the operational matrices with the said polynomials for integer and fractional order Caputo differential operator. Using these operational matrices and collocating the residual together with initial and boundary conditions at certain collocation points, the problem is reduced to a system of algebraic equations. An approximate solution to the problem can be obtained by solving this system of equations. The efficiency and accuracy of the proposed method are validated through error analysis between the obtained numerical results and the existing analytical results for the particular forms of the considered model. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.