# Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction: Formulation of the Problem

**Problem G.**It is required to find a function $u(x,y)\in C\left(\overline{\Omega}\right)$ satisfying Equation (1) and boundary conditions

**Definition**

**1**

## 2. Erdélyi—Kober Transmutation Operator

**Theorem**

**1.**

## 3. Application of the Erdélyi-Kober Operator to the Solution of the Problem

**Problem ${\mathit{G}}_{\mathbf{0}}.$**It is required to find a function ${u}_{1}(x,y)\in C\left(\overline{\Omega}\right)$ satisfying Equation (2) and boundary conditions

- 1.
- $p+q<l+m+1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p+k<l+n+1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left|x\right|<\infty ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left|y\right|<\infty $ or
- 2.
- $p+q=l+m+1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p+k=l+n+1$ and

**Theorem**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Sitnik, S.M.; Karimov, S.T.
Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators. *Mathematics* **2023**, *11*, 951.
https://doi.org/10.3390/math11040951

**AMA Style**

Sitnik SM, Karimov ST.
Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators. *Mathematics*. 2023; 11(4):951.
https://doi.org/10.3390/math11040951

**Chicago/Turabian Style**

Sitnik, Sergei M., and Shakhobiddin T. Karimov.
2023. "Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators" *Mathematics* 11, no. 4: 951.
https://doi.org/10.3390/math11040951