Solutions of the Fractional Differential Equations Including Caputo–Fabrizio, Caputo, and Integer Order Derivatives via SMV Polynomials

Abstract

Fractional linear and nonlinear differential equations with the Caputo–Fabrizio, Caputo, and integer order derivatives are considered in this paper. An approximate solution of the problem is written as a truncated series of the shifted Morgan-Voyce (SMV) polynomials with unknown coefficients. Our goal is to compute the numerical values of the unknown coefficients. First, the Caputo–Fabrizio, Caputo, and integer order derivatives of the approximate solution expressed in terms of SMV polynomials are presented in the form of the matrix relations. The main advantage of these matrix relations is that they convert the differential equation, including three different types of derivatives, into a system of algebraic equations, which allows us to easily transfer the problem into computer programming. Furthermore, the convergence of the method is investigated in the Sobolev space. Finally, the application of the method is presented by using numerical examples. In the numerical examples, figures and tables are used to discuss the effect of different values of fractional order on the solution and to show the accuracy of the method by comparing it with existing numerical solutions.