#
Applications of (h,q)-Time Scale Calculus to the Solution of Partial Differential Equations † ^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Operator of the q-Delta Differential

- (a)
- Jackson [10] discussed the modified version of $q$-derivative and its several consequences in the twentieth century.
- (b)
- For functions that do not have 0 in their definition domain, the ${\u2206}_{q}$-derivative can be calculated. As q is at 1, it decreases to a common derivative.
- (c)
- It can be checked easily that the ${\u2206}_{q}$-operator is a linear operator, i.e.,$$\left(\mathrm{i}\right){\u2206}_{q}\left(f+g\right)\hspace{1em}{\u2206}_{q}f+{\u2206}_{q}g,\phantom{\rule{0ex}{0ex}}\left(\mathrm{ii}\right){\u2206}_{q}\left(\lambda f\right)=\lambda {\u2206}_{q}f$$

## 3. The ${\u2206}_{q}$ Derivative of Few Transcendental Mappings and the Non-Commutative Concept

## 4. q-Time Scale Factorials and q-Timescale Numbers

^{p}; then,

#### 4.1. Remarks

#### 4.2. Partial ${\u2206}_{q}$ Derivative of Multivariable Function in Time Scale

_{1}, t

_{2}, …, t

_{n}) with respect to a variable ${t}_{i}$by

^{th}$\mathrm{order},{\u2206}_{q}$order derivative, we subsequently adopt the identity with respect to ${t}^{j}$.

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Ali, H.; Muhammad, G.; Abbas, M.A.
Applications of (*h*,*q*)-Time Scale Calculus to the Solution of Partial Differential Equations *Comput. Sci. Math. Forum* **2023**, *7*, 56.
https://doi.org/10.3390/IOCMA2023-14388

**AMA Style**

Ali H, Muhammad G, Abbas MA.
Applications of (*h*,*q*)-Time Scale Calculus to the Solution of Partial Differential Equations *Computer Sciences & Mathematics Forum*. 2023; 7(1):56.
https://doi.org/10.3390/IOCMA2023-14388

**Chicago/Turabian Style**

Ali, Hussain, Ghulam Muhammad, and Munawwar Ali Abbas.
2023. "Applications of (*h*,*q*)-Time Scale Calculus to the Solution of Partial Differential Equations *Computer Sciences & Mathematics Forum* 7, no. 1: 56.
https://doi.org/10.3390/IOCMA2023-14388