# A Novel Computational Technique for Impulsive Fractional Differential Equations

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Definitions and Properties of Fractional Calculus

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Remarks**:

**Property**

**1.**

**Lemma**

**1**

**[14].**The impulsive fractional differential Equation (1) is equivalent to the following integral equation of fractional order

#### 2.2. Adomian Polynomials

## 3. Semi-Analytical Method Based on Adomian Polynomials

- Assume the solution in a series form as$$x(t)={\displaystyle \sum _{i=0}^{\infty}{c}_{i}{(t-{t}_{0})}^{i\mathsf{\alpha}}}$$
- Substituting (12) into (8), with Adomian polynomials, the coefficients of ${c}_{i}$ are obtained as$$\{\begin{array}{l}{c}_{n+1}=\frac{\mathsf{\Gamma}(1+n\mathsf{\alpha})}{\mathsf{\Gamma}(1+(n+1)\mathsf{\alpha})}{A}_{n}[{c}_{0},{c}_{1},\dots ,{c}_{n}],0\le n,\\ {c}_{0}={x}_{0}+{\displaystyle \sum _{j=1}^{k}{y}_{j}}.\end{array}$$
- ${x}_{n}$ can be obtained as$${x}_{n}=\psi ({c}_{0},{t}_{0},{\displaystyle \sum _{j=1}^{k}{y}_{j},t}),t\in ({t}_{{N}_{k}},{t}_{{N}_{k+1}}].$$
- Set $t\in [{t}_{0},T]$, $t=ih$, $H=\frac{T}{N}$, $h=\frac{H}{K}$, $i=0,1,\dots ,NK$ and let ${x}_{i}^{*}=\psi ({x}_{i-1}^{*},{t}_{i-1},{\displaystyle \sum _{j=1}^{k}{y}_{j},\hspace{0.17em}}{t}_{i}),$ where ${x}_{0}^{*}={c}_{0}$. We can obtain the numerical solutions ${x}_{0}^{*},\dots ,{x}_{i}^{*}$.

## 4. Numerical Solutions based on Adomian Polynomials

## 5. Conclusions

- It is still challenging work to do error analysis. For many nonlinear cases, the exact solution is unknown and numerical errors cannot be obtained. We will pay attention to this topic in the near future;
- In this method, we generally adopt a fractional series expansion which is a fractional analogy of the Taylor series. What about other expansions which satisfy the features of the new polynomials? For example, how can series solutions be found for boundary value problems? Hence, it is very important to develop new ideas for this topic.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Ma, C.
A Novel Computational Technique for Impulsive Fractional Differential Equations. *Symmetry* **2019**, *11*, 216.
https://doi.org/10.3390/sym11020216

**AMA Style**

Ma C.
A Novel Computational Technique for Impulsive Fractional Differential Equations. *Symmetry*. 2019; 11(2):216.
https://doi.org/10.3390/sym11020216

**Chicago/Turabian Style**

Ma, Changyou.
2019. "A Novel Computational Technique for Impulsive Fractional Differential Equations" *Symmetry* 11, no. 2: 216.
https://doi.org/10.3390/sym11020216