# Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

## Abstract

**:**

## 1. Introduction

“The theorems above have given us a large amount of information about the smoothness properties of the solutions of fractional differential equations, and in particular about the exact behaviour of the solution as $x\to 0$, most notably the formal asymptotic expansion. […] An aspect of special significance, for example in view of the development of numerical methods, is the question for the precise values of the constants in this expansion, and most importantly the question whether certain coefficients vanish. A suitable generalisation of the Taylor expansion technique for ordinary differential equations described in [88, Chapter I.8] could be useful in this context. Precise results in this connection seem to be unknown at the moment though.”

## 2. Results

#### 2.1. Problem Statement

**Definition**

**1.**

#### 2.2. Algorithm Description

#### 2.2.1. Multi-Order System

**Theorem**

**1.**

**Proof of Theorem 1.**

#### 2.2.2. Implementation of the Differential Transform

**Theorem**

**2.**

**Proof of Theorem 2.**

#### 2.2.3. Applications

**Example**

**1.**

## 3. Discussion

- Transform the multi-term equation into a multi-order differential system.
- Describe an equivalent system of Volterra integral equations.
- Find the general form of the asymptotic expansion of the solution.
- Transform the multi-order differential system to a system of recurrence relations (formal application of FDT).
- Choose convenient orders of FDT.
- Describe the relation between different orders of FDT.

## 4. Methods

#### 4.1. Equivalence and Smoothness Theorems

**Theorem**

**3**

**.**Subject to the conditions specified in Section 2.1, the multi-term Equation (2) with initial conditions in Equation (3) is equivalent to the system of Equation (6) with the initial conditions in Equation (7) in the following sense:

- 1.
- Whenever the function $y\in {C}^{\lceil {\lambda}_{k}\rceil}[0,T]$ is a solution of the IVP given by Equations (2) and (3), the vector-valued function $Y:={({y}_{1},\dots ,{y}_{k})}^{T}$ with$${y}_{j}\left(t\right):=\left\{\begin{array}{cc}y\left(t\right)\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}j=1,\hfill \\ {}_{0}^{C}\phantom{\rule{-0.166667em}{0ex}}{D}_{t}^{{\lambda}_{j-1}}y\left(t\right)\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}j=2,\dots ,k,\hfill \end{array}\right.$$
- 2.

**Theorem**

**4**

**.**Let $0<n$ and $m=\lceil n\rceil $. Moreover, let ${y}_{0}^{\left(0\right)},\cdots ,{y}_{0}^{(m-1)}\in \mathbb{R}$, $K>0$ and ${h}^{\ast}>0$. Define $G:=\{(x,y):x\in [0,{h}^{\ast}],|y-\sum _{k=0}^{m-1}{x}^{k}{y}_{0}^{\left(k\right)}/k!|\le K\}$, and let the function $f:G\to \mathbb{R}$ be continuous. The function $y\in C[0,h]$ for some $0<h<{h}^{\ast}$ is a solution of the initial value problem

**Theorem**

**5**

**.**Let n be a positive irrational number. Consider the initial value problem defined by Equations (22) and (23) and assume that f can be written in the form $f(x,y)=\overline{f}(x,{x}^{n},y)$ where $\overline{f}$ is analytic in a neighborhood of $(0,0,{y}_{0}^{\left(0\right)})$. Then, there exists a uniquely determined analytic function $\overline{y}:(-r,r)\times (-{r}^{n},{r}^{n})\to \mathbb{R}$ with some $r>0$ such that $y\left(x\right)=\overline{y}(x,{x}^{n})$ for $x\in [0,r)$.

**Corollary**

**1**

**.**Under the assumptions of Theorem 5, y is of the form

#### 4.2. Fractional Differential Transformation

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**6.**

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IVP | Initial Value Problem |

DT | Differential Transformation |

FDT | Fractional Differential Transformation |

IFDT | Inverse Fractional Differential Transformation |

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**MDPI and ACS Style**

Rebenda, J. Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders. *Symmetry* **2019**, *11*, 1390.
https://doi.org/10.3390/sym11111390

**AMA Style**

Rebenda J. Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders. *Symmetry*. 2019; 11(11):1390.
https://doi.org/10.3390/sym11111390

**Chicago/Turabian Style**

Rebenda, Josef. 2019. "Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders" *Symmetry* 11, no. 11: 1390.
https://doi.org/10.3390/sym11111390