A Novel Computational Technique for Impulsive Fractional Differential Equations

Changyou Ma

doi:10.3390/sym11020216

Abstract

A computational technique for impulsive fractional differential equations is proposed in this paper. Adomian decomposition method plays an efficient role for approximate analytical solutions for ordinary or fractional calculus. Semi-analytical method is proposed by use of the Adomian polynomials. The method successively updates the initial values and gives the numerical solutions on different impulsive intervals. As one of the numerical examples, an impulsive fractional logistic differential equation is given to illustrate the method.

Highlights

Fractional calculus appears frequently in various applied topics [1,2,3,4,5,6,7] and pure mathematics [8,9,10,11,12].They are employed to depict the long-interaction of different statues of the systems
Various novel algorithms based on the new Adomian polynomials can be considered. It was successfully used in fractional differential equations [28] where a semi-analytical method was developed
To the best of our knowledge, we did not find any work on semi-analytical solutions for impulsive fractional differential equations

Summary

Introduction

Fractional calculus appears frequently in various applied topics [1,2,3,4,5,6,7] and pure mathematics [8,9,10,11,12]. The impulsive differential equation is not a continuous time system but the one combining both continuous and discrete point information. It depicts the impact of external conditions which may be negative or positive. Various novel algorithms based on the new Adomian polynomials can be considered It was successfully used in fractional differential equations [28] where a semi-analytical method was developed. To the best of our knowledge, we did not find any work on semi-analytical solutions for impulsive fractional differential equations This paper combines both analytical and numerical solutions’ features to develop a semi-analytical method

Definitions and Properties of Fractional Calculus

Adomian Polynomials

Semi-Analytical Method Based on Adomian Polynomials

Numerical Solutions based on Adomian Polynomials

Conclusions