# Fractional Partial Differential Equations with Boundary Conditions

## TL;DR

This paper establishes the connection between stochastic processes and fractional PDEs with boundary conditions, proving well-posedness and providing a new numerical approximation method for these equations.

## Contribution

It introduces a novel embedding technique for finite state Markov processes into Feller processes, ensuring well-posedness and convergence for fractional PDEs with boundary conditions.

## Key findings

- Proved well-posedness of fractional PDEs in $C_0(om)$ and $L_1(om)$.
- Developed a new method for embedding Markov processes into Feller processes.
- Provided a numerical approximation approach for solutions to fractional PDEs.

## Abstract

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\Omega)$ and $L_1(\Omega)$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.