# Lyapunov-Sylvester operators for Generalized Nonlinear Euler-Poisson-Darboux System

## TL;DR

This paper develops a new numerical method using Lyapunov-Sylvester operators for solving a nonlinear Euler-Poisson-Darboux system, demonstrating improved accuracy and efficiency over existing methods.

## Contribution

It introduces a novel finite difference scheme transforming the system into a Lyapunov-Sylvester operator framework, proving its stability, convergence, and efficiency.

## Key findings

- The scheme is uniquely solvable, stable, and convergent.
- Numerical results show higher accuracy than existing methods.
- The method leads to faster computational algorithms.

## Abstract

In this paper a nonlinear Euler-Poisson-Darboux system is considered. In a first part, we proved the genericity of the hypergeometric functions in the development of exact solutions for such a system in some special cases leading to Bessel type differential equations. Next, a finite difference schemle in two-dimensional case has been developed. The continuous system is transformed into an algebraic quasi linear discrete one leading to generalized Lyapunov-Sylvester operators. The discrete algebraic system is proved to be uniquely solvable, stable and convergent based on Lyapunov criterion of stability and Lax-Richtmyer equivalence theorem for the convergence. A numerical example has been provided at the end to illustrate the efficiency of the theoretical results. The present method is thus proved to be more accurate than existing ones and lead to faster algorithms.