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 cas...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.展开更多
This paper investigates a solution technique for solving a two-dimensional Kuramoto-Sivashinsky equation discretized using a finite difference method. It consists of an order reduction method into a coupled system of ...This paper investigates a solution technique for solving a two-dimensional Kuramoto-Sivashinsky equation discretized using a finite difference method. It consists of an order reduction method into a coupled system of second-order equations, and to formulate the fully discretized, implicit time-marched system as a Lyapunov- Sylvester matrix equation. Convergence and stability is examined using Lyapunov criterion and manipulating generalized Lyapunov-Sylvester operators. Some numerical implementations are provided at the end to validate the theoretical results.展开更多
文摘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.
文摘This paper investigates a solution technique for solving a two-dimensional Kuramoto-Sivashinsky equation discretized using a finite difference method. It consists of an order reduction method into a coupled system of second-order equations, and to formulate the fully discretized, implicit time-marched system as a Lyapunov- Sylvester matrix equation. Convergence and stability is examined using Lyapunov criterion and manipulating generalized Lyapunov-Sylvester operators. Some numerical implementations are provided at the end to validate the theoretical results.
