Shifted ultraspherical pseudo-Galerkin method for approximating the solutions of some types of ordinary fractional problems

Mohamed Abdelhakem,Mamdouh El-Kady,Doha Mahmoud,Dumitru Baleanu

doi:10.1186/s13662-021-03247-6

Abstract

In this work, a technique for finding approximate solutions for ordinary fraction differential equations (OFDEs) of any order has been proposed. The method is a hybrid between Galerkin and collocation methods. Also, this method can be extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The spatial approximations with their derivatives are based on shifted ultraspherical polynomials (SUPs). Modified Galerkin spectral method has been used to create direct approximate solutions of linear/nonlinear ordinary fractional differential equations, a system of ordinary fraction differential equations, fractional integro-differential equations, or fractional optimal control problems. The aim is to transform those problems into a system of algebraic equations. That system will be efficiently solved by any solver. Three spaces of collocation nodes have been used through that transformation. Finally, numerical examples show the accuracy and efficiency of the investigated method.

Highlights

ordinary fraction differential equations (OFDEs) play a role in many branches and applications
The importance of the Galerkin method (GM) is that, for the presented technique, it can be used for the solution of a wide class of ordinary fractional problems (OFPs)
The work is coordinated as follows: in Sect. 2, we introduce Caputo’s fractional derivative

Summary

Introduction

OFDEs play a role in many branches and applications. These applications can be found in risk theory [1], physics [2], biological phenomena, and diseases [3,4,5,6]. For OFDEs, in [12] the authors presented a spectral method using shifted Chebyshev polynomials (SCHPs) of the second kind. The importance of the Galerkin method (GM) is that, for the presented technique, it can be used for the solution of a wide class of ordinary fractional problems (OFPs). The lemma will be used to introduce a general form for the fractional derivative and the integer order integration of SUPs. Proof Straightforward using Eq (3) and the binomial theorem. The following corollary proves the fractional derivatives of SUPs in the Caputo sense. The theorem will establish the general form of the SUPs approximation of a function φ(x).

Solving OFDEs by SU-pseudo-GM Consider OFDEs as follows:

Method

Findings

Conclusion