# On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Auxiliary Results

#### 2.1. Problem Formulation

#### 2.2. Mellin-Barnes Representations of the Fundamental Solution

#### 2.3. Special Functions of the Wright Type

## 3. New Integral Representations of the Fundamental Solution

**Theorem**

**1.**

**Proof.**

## 4. New Closed-Form Formulas for Particular Cases of the Fundamental Solution

**Theorem**

**2.**

- (a)
- For $\beta =\alpha $ and $n=2$ under the condition $1<\alpha \le 2$:$${G}_{\alpha ,\alpha ,2}(\mathrm{x},t)=\left(\right)open="\{"\; close>\begin{array}{c}\frac{{\left|\mathrm{x}\right|}^{\alpha -2}}{\sqrt{\pi}{t}^{\alpha}}\phantom{\rule{0.166667em}{0ex}}{W}_{\left(\right)open="("\; close=")">\frac{1}{2}-\frac{\alpha}{2},-\frac{\alpha}{2}}\hfill & \left(\right)open="("\; close=")">-{\left(\right)}^{\frac{\left|\mathrm{x}\right|}{t}}\alpha \end{array}\phantom{\rule{4.pt}{0ex}}if\phantom{\rule{4.pt}{0ex}}\left|\mathrm{x}\right|t,\hfill $$
- (b)
- For $\beta =\frac{3}{2}\alpha $ and $n=2$ under the condition $1<\alpha \le \frac{4}{3}$:$${G}_{\alpha ,\frac{3}{2}\alpha ,2}(\mathrm{x},t)=\frac{\sqrt{3}}{2{\pi}^{2}{\left|\mathrm{x}\right|}^{2}}{}_{1}{\mathsf{\Psi}}_{3}\left(\right)open="["\; close="]">\genfrac{}{}{0pt}{}{(1,1)}{\left(\right)open="("\; close=")">\frac{1}{3},\frac{\alpha}{2}},\left(\right)open="("\; close=")">0,-\frac{\alpha}{2}$$

**Proof.**

- (i)
- $\left|\mathrm{x}\right|<t$;
- (ii)
- $\left|\mathrm{x}\right|>t$.

## 5. Discussion

## Conflicts of Interest

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**MDPI and ACS Style**

Luchko, Y.
On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation. *Mathematics* **2017**, *5*, 76.
https://doi.org/10.3390/math5040076

**AMA Style**

Luchko Y.
On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation. *Mathematics*. 2017; 5(4):76.
https://doi.org/10.3390/math5040076

**Chicago/Turabian Style**

Luchko, Yuri.
2017. "On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation" *Mathematics* 5, no. 4: 76.
https://doi.org/10.3390/math5040076