TY - JOUR
T1 - Study of a Generalized Nonlinear Euler-Poisson-Darboux System: Numerical and Bessel Based Solutions
AU - Riadh , Chteoui
AU - Sabrine , Arfaoui
AU - Anouar , Ben Mabrouk
JO - Journal of Partial Differential Equations
VL - 4
SP - 313
EP - 340
PY - 2020
DA - 2020/08
SN - 33
DO - http://doi.org/10.4208/jpde.v33.n4.2
UR - https://global-sci.org/intro/article_detail/jpde/17862.html
KW - Finite difference method, Lyapunov-Sylvester operators, generalized Euler-Poisson-Darboux equation, hyperbolic equation, Lauricella hypergeometric functions.
AB - <p style="text-align: justify;">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 scheme 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 numerical scheme developed in section 3. The present method is thus proved to be more accurate than existing ones and lead to faster algorithms.</p>