Rapid Convergence of Approximate Solutions for Fractional Differential Equations

Xiran Wu,Junyan Bao,Yufeng Sun

doi:10.1155/2020/5370524

Abstract

In this paper, we develop a generalized quasilinearization technique for a class of Caputo’s fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions of order m (m≥0), and we obtain the convergence of the sequences of approximate solutions by establishing the convergence of order k (k≥2).

Highlights

Fractional differential equations have received attention from some researchers because they have extensive application in mechanics, biochemistry, electrical engineering, medicine, and many other fields
There are some results on the monotone sequences of approximate solutions converging uniformly to a solution of fractional differential equations by employing monotone iterative technique and generalized monotone iterative method coupled with the method of upper and lower solutions, which can be found in [11,12,13,14,15,16,17,18,19,20,21]
The quasilinearization method [22] is one of the effective methods to obtain a sequence of approximate solutions with quadratic convergence, and it is extremely useful in scientific computations due to its accelerated rate of convergence as in [23, 24]

Summary

Introduction

Fractional differential equations have received attention from some researchers because they have extensive application in mechanics, biochemistry, electrical engineering, medicine, and many other fields (see [1,2,3,4,5,6]). There are some results on the monotone sequences of approximate solutions converging uniformly to a solution of fractional differential equations by employing monotone iterative technique and generalized monotone iterative method coupled with the method of upper and lower solutions, which can be found in [11,12,13,14,15,16,17,18,19,20,21]. Inspired and motivated by [34, 35], in the present paper, we will discuss the rapid convergence of approximate solutions of fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions with coupled lower and upper solutions, and construct sequences of approximate solutions that converge rapidly to the extremal solutions of (1) by using an improved quasilinearization method (rate of convergence k ≥ 2)

Preliminaries

Main Results

CCCA ð51Þ