An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

Abstract

The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, the equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method.

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

“…Comparison between numerical results of four numerical methods are listed in Table for Example case II with various choices of n and l = 1. From this comparison, we obtain the introduced method in this article that is applicable and gives highly accurate results comparing with the results given in El‐Mesiry, Liu et al, and Maleknejad et al While in Table , we bring results of the maximum absolute errors of the method given in Liu et al and our suggested method for Example case II with disjoint values of l and n . From the aforementioned results in Table , the values of E max using the proposed method are smaller than that given in Liu et al with the same size of Chebyshev polynomials of the second kind.…”

“…Definition Variable‐order Caputo fractional derivative () The variable‐order fractional derivative definition of order α ( t ) for the function u ( t ) ∈ C m [0, b ] can be defined in the Caputo sense as follows: …”

“… where λ and μ are constants. Furthermore, according to the Caputo operator presented in Equation , we can claim that() …”

“…Table of the numerical results will not be presented in this case because the exact solution is obtained. To show that our presented method is more accurate than the one introduced in Liu et al, in Table , comparisons between the proposed method results and those obtained in Liu et al are introduced for various choices of l and n . Also, as seen in Table , the vector C T = [ c 0 , c 1 ,…, c n ] obtained is mainly composed of three terms, namely, [ c 0 , c 1 , c 2 ].…”

“…Consequently, researchers are interested to obtain the numerical solutions of the models described by the variable‐order fractional differential equations. Some of these methods are Legendre wavelets, second‐kind Chebyshev polynomials, Bernstein polynomials, Legendre polynomials, and the operational matrix of fractional integration founded by the shifted second‐kind Chebyshev polynomials . In addition, some other numerical techniques for solving the variable‐order fractional differential equations are given in Chen et al, Shen et al, Tavares et al, and Nagy et al…”