# An Integro-Differential Equation of the Fractional Form: Cauchy Problem and Solution

## TL;DR

This paper solves a fractional integro-differential Cauchy problem involving a pseudo-differential Riesz operator, providing explicit solutions for Gaussian initial conditions and the Dirac delta case, advancing understanding of fractional PDEs.

## Contribution

It introduces explicit solutions to a fractional PDE with Gaussian and delta initial conditions, expanding analytical methods for such equations.

## Key findings

- Explicit solution for Gaussian initial condition.
- Solution for Dirac delta initial condition.
- Enhanced understanding of fractional PDE behavior.

## Abstract

We solve the Cauchy problem defined by the fractional partial differential equation $[\partial_{tt}-\kappa\mathbb{D}]u=0$, with $\mathbb{D}$ the pseudo-differential Riesz operator of first order, and the initial conditions $u(x,0)=\mu(\sqrt{\pi}x_0)^{-1}e^{-(x/x_0)^2}$, $u_t(x,0)=0$. The solution of the Cauchy problem resulting from the substitution of the Gaussian pulse $u(x,0)$ by the Dirac delta distribution $\varphi(x)=\mu\delta(x)$ is obtained as corollary.