A reproducing kernel Hilbert space method for nonlinear partial differential equations: applications to physical equations

Abstract

The partial differential equations (PDEs) describe several phenomena in wide fields of engineering and physics. The purpose of this paper is to employ the reproducing kernel Hilbert space method (RKHSM) in obtaining effective numerical solutions to nonlinear PDEs, which are arising in acoustic problems for a fluid flow. In this paper, the RKHSM is used to construct numerical solutions for PDEs which are found in physical problems such as sediment waves in plasma, sediment transport in rivers, shock waves, electric signals' transmission along a cable, acoustic problems for a fluid flow, vibrating membrane, and vibrating string. The RKHSM systematically produces analytic and approximate solutions in the form of series. The convergence analysis and error estimations are discussed to prove the applicability theoretically. Three applications are tested to show the performance and efficiency of the used method. Computational results indicated a good agreement between the exact and numerical solutions.