# Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental Concepts

**Definition**

**1**

**Definition**

**2**

**Lemma**

**1**

**Definition**

**3**

**Definition**

**4.**

**Remark**

**1.**

**Theorem**

**1**

## 3. Study of Problem I

#### 3.1. Qualitative Theory of Problem I

**Theorem**

**2.**

**Proof.**

**Lemma**

**2.**

**Proof.**

- $\left({C}_{1}\right)$
- For any $u,v\in \mathrm{R}$, ∃ a constant ${L}_{f}>0$ with$$|f(t,u)-f(t,v)|\le {L}_{f}|u-v|,\phantom{\rule{4pt}{0ex}}t\in [0,b].$$
- $\left({C}_{2}\right)$
- Let ∃ constants ${M}_{f}>0$ and ${C}_{f}>0$, then the given growth condition hold$$\left|f(t,u\left(t\right))\right|\le {M}_{f}+{C}_{f}\left|u\right|,\phantom{\rule{4pt}{0ex}}t\in [0,b].$$

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.2. Numerical Procedure for Problem I

#### 3.3. Numerical Problems to Verify the Establish Analysis

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 4. Study of Problem II

#### 4.1. Qualitative Theory of (2)

- $\left({C}_{3}\right)$
- For any $u,v,w,\phantom{\rule{4pt}{0ex}}\overline{u},\overline{v},\overline{w}\in \mathrm{R}$, there exist a constant ${L}_{{f}_{i}}>0$ such that$$|{f}_{i}(t,u,v,w)-f(t,\overline{u},\overline{v},\overline{w})|\le {L}_{{f}_{i}}\left[\right|u-\overline{u}|+|v-\overline{v}|+|w-\overline{w}\left|\right],\phantom{\rule{4pt}{0ex}}t\in [0,b],\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}i=1,2,3.$$
- $\left({C}_{4}\right)$
- Let there exists constant ${M}_{{f}_{i}}>0$ and ${C}_{{f}_{i}}>0$; such that the given growth condition hold:$$|{f}_{i}(t,u\left(t\right),v\left(t\right),w\left(t\right))|\le {M}_{{f}_{i}}+{C}_{{f}_{i}}\left[\right|u|+\left|v\right|+\left|w\right|],\phantom{\rule{4pt}{0ex}}t\in [0,b],$$

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

#### 4.2. Numerical Procedure to Problem II

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Plots of solution upto first 100 terms at various values of fractional order $\alpha $ of Example (1).

**Figure 2.**Plots of solution upto first three terms at various values of fractional order $\alpha $ of Example (2).

**Figure 3.**Plots of solution upto first three terms at various values of fractional order $\alpha $ of Example (3).

**Figure 4.**Plots of approximate solutions for $u\left(t\right)$ of the system (3) against various fractional order.

**Figure 5.**Plots of approximate solutions for $v\left(t\right)$ of the system (3) against various fractional order.

**Figure 6.**Plots of approximate solutions for $w\left(t\right)$ of the system (3) against various fractional order.

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**MDPI and ACS Style**

Neelma; Eiman; Shah, K.
Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method. *Foundations* **2022**, *2*, 6-19.
https://doi.org/10.3390/foundations2010002

**AMA Style**

Neelma, Eiman, Shah K.
Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method. *Foundations*. 2022; 2(1):6-19.
https://doi.org/10.3390/foundations2010002

**Chicago/Turabian Style**

Neelma, Eiman, and Kamal Shah.
2022. "Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method" *Foundations* 2, no. 1: 6-19.
https://doi.org/10.3390/foundations2010002