# Ulam–Hyers Stability of Linear Differential Equation with General Transform

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

**Definition 1**

**.**Let $g\left(s\right)$ be an integrable function defined for $s\ge 0$, $u\left(\omega \right)\ne 0$ and $v\left(\omega \right)$ are positive real functions; we define the general integral transform $\mathcal{G}\left(\omega \right)$ of $g\left(s\right)$ by the formula

**Definition 2**

**.**Let ${g}_{1}\left(s\right)$ and ${g}_{2}\left(s\right)$ have new integral transforms ${\mathcal{G}}_{1}\left(\omega \right)$ and ${\mathcal{G}}_{2}\left(\omega \right)$, respectively. Then, the new integral transform of their convolution is given as

**Theorem 1**

**.**Let $g\left(s\right)$ is differentiable, and considering the positive real functions $u\left(\omega \right)$ and $v\left(\omega \right)$, then

- $\mathcal{T}\{{g}^{{}^{\prime}}\left(s\right);\omega \}=v\left(\omega \right)\mathcal{G}\left(\omega \right)-u\left(\omega \right)g\left(0\right)$,
- $\mathcal{T}\{{g}^{{}^{\prime \prime}}\left(s\right),\omega \}={v}^{2}\left(\omega \right)\mathcal{T}\{g\left(s\right);\omega \}-v\left(\omega \right)u\left(\omega \right)g\left(0\right)-u\left(\omega \right){g}^{{}^{\prime}}\left(0\right)$,
- $\mathcal{T}\{{g}^{n}\left(s\right);\omega \}={u}^{n}\left(\omega \right)\mathcal{T}\{g\left(s\right);\omega \}-u\left(\omega \right){\sum}_{k=0}^{n-1}{v}^{n-1-k}\left(\omega \right){g}^{\left(k\right)}\left(0\right)$.

## 3. Main Stability Results

**Definition 3**

**.**

- The Equation (1) has Ulam–Hyers stability, if for any $\epsilon >0$ and continuously differentiable function $g\left(s\right)$ satisfies$$\begin{array}{c}\hfill \left|{g}^{{}^{\prime}}\left(s\right)+\gamma g\left(s\right)\right|\le \epsilon ,\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}s\ge 0,\end{array}$$$$\begin{array}{c}\hfill \left|g\left(s\right)-h\left(s\right)\right|\le \mathcal{K}\epsilon ,\forall \phantom{\rule{4pt}{0ex}}s\ge 0.\end{array}$$
- The Equation (2) has Ulam–Hyers stability, if for any $\epsilon >0$ and continuously differentiable function $g\left(s\right)$ satisfies$$\begin{array}{c}\hfill \left|{g}^{{}^{\prime}}\left(s\right)+\gamma g\left(s\right)-r\left(s\right)\right|\le \epsilon ,\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}s\ge 0,\end{array}$$$$\begin{array}{c}\hfill \left|g\left(s\right)-h\left(s\right)\right|\le \mathcal{K}\epsilon ,\forall \phantom{\rule{4pt}{0ex}}s\ge 0,\end{array}$$

**Definition 4**

- The Equation (1) has Ulam–Hyers ϕ-stability, if for every $\epsilon >0$ and continuously differentiable function $g\left(s\right)$ satisfies$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left|{g}^{{}^{\prime}}\left(s\right)+\gamma g\left(s\right)\right|\le \varphi \left(s\right)\epsilon ,\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}s\ge 0,\hfill \end{array}$$$$\left|g\left(s\right)-h\left(s\right)\right|\le \mathcal{K}\varphi \left(s\right)\epsilon ,\forall \phantom{\rule{4pt}{0ex}}s\ge 0.$$
- The Equation (2) has Ulam–Hyers ϕ-stability, if for every $\epsilon >0$ and continuously differentiable function $g\left(s\right)$ satisfies$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left|{g}^{{}^{\prime}}\left(s\right)+\gamma g\left(s\right)-r\left(s\right)\right|\le \varphi \left(s\right)\epsilon ,\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}s\ge 0,\hfill \end{array}$$$$\left|g\left(s\right)-h\left(s\right)\right|\le \mathcal{K}\varphi \left(s\right)\epsilon ,\forall \phantom{\rule{4pt}{0ex}}s\ge 0,$$

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Corollary 1.**

**Proof.**

**Corollary 2.**

**Proof.**

## 4. Discussion on Additional Stability

**Definition 5**

**.**The Mittag–Leffler function denoted by ${E}_{\delta}\left(z\right)$ is defined as

**Remark 1.**

**Remark 2.**

**Remark 3.**

**Remark 4.**

**Remark 5.**

## 5. Examples

**Example 1.**

**Example 2.**

## 6. Applications of General Integral Transform

**Example 3.**

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Function $\mathit{g}\left(\mathit{s}\right)={\mathcal{T}}^{-1}\left\{\mathcal{G}\left(\mathit{\omega}\right)\right\}$ | New Integral Transforms $\mathcal{G}\left(\mathit{\omega}\right)=\mathcal{T}\left\{\mathit{g}\right(\mathit{s});\mathit{\omega}\}$ |
---|---|

1 | $\frac{u\left(\omega \right)}{v\left(\omega \right)}$ |

s | $\frac{u\left(\omega \right)}{v{\left(\omega \right)}^{2}}$ |

${s}^{\beta}$ | $\frac{\mathsf{\Gamma}(\beta +1)u\left(\omega \right)}{u{\left(\omega \right)}^{\beta +1}},\phantom{\rule{4pt}{0ex}}\beta >0$ |

$sins$ | $\frac{u\left(\omega \right)}{v{\left(\omega \right)}^{2}+1}$ |

$sin\left(as\right)$ | $\frac{au\left(\omega \right)}{{a}^{2}+v{\left(\omega \right)}^{2}},\phantom{\rule{4pt}{0ex}}\mathrm{if}\phantom{\rule{4pt}{0ex}}v\left(\omega \right)>\left|\mathcal{J}\left(a\right)\right|$ |

$coss$ | $\frac{v\left(\omega \right)u\left(\omega \right)}{v{\left(\omega \right)}^{2}+1}$ |

${e}^{s}$ | $\frac{u\left(\omega \right)}{v\left(\omega \right)-1},\phantom{\rule{4pt}{0ex}}v\left(s\right)>1$ |

$sH(s-1)$ | $\frac{{e}^{-v\left(\omega \right)}(v\left(\omega \right)+1)u\left(\omega \right)}{v{\left(\omega \right)}^{2}}$ |

${g}^{{}^{\prime}}\left(s\right)$ | $v\left(\omega \right)\mathcal{G}\left(\omega \right)-u\left(\omega \right)g\left(0\right)$ |

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**MDPI and ACS Style**

Pinelas, S.; Selvam, A.; Sabarinathan, S.
Ulam–Hyers Stability of Linear Differential Equation with General Transform. *Symmetry* **2023**, *15*, 2023.
https://doi.org/10.3390/sym15112023

**AMA Style**

Pinelas S, Selvam A, Sabarinathan S.
Ulam–Hyers Stability of Linear Differential Equation with General Transform. *Symmetry*. 2023; 15(11):2023.
https://doi.org/10.3390/sym15112023

**Chicago/Turabian Style**

Pinelas, Sandra, Arunachalam Selvam, and Sriramulu Sabarinathan.
2023. "Ulam–Hyers Stability of Linear Differential Equation with General Transform" *Symmetry* 15, no. 11: 2023.
https://doi.org/10.3390/sym15112023