A new optimized method for solving variable-order fractional differential equations

Abstract

Variable-order fractional derivatives are an extension of constant-order fractional derivatives and
have been introduced in several physical elds. Since the equations described by the variable-order
derivatives are highly complex and also dicult to handle analytically, it is advisable to consider
their numerical solutions. In this paper, a new optimized method based on polynomials is proposed
for solving variable-order fractional dierential equations (VOFDEs) and systems of variable-order
fractional dierential equations (SVOFDEs). To do this, a general polynomial of degree k with
unknown coecients is considered as an approximate solution for the problem under study. By
using the initial conditions some of these unknown coecients are obtained. Finally the rest of these
unknown coecients are obtained optimally by minimizing error of 2-norm of the approximate solution
in a desired interval. In order to demonstrate the eciency of the proposed method some numerical
examples are given. The obtained results show that, the proposed method is very accurate for such
problems.