Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

Abstract

This study presents numerical simulations of generalized two‐dimensional (2D) and three‐dimensional (3D) Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, which are coupled nonlinear partial differential equations. A meshless collocation method based on barycentric rational interpolation is developed for space variable of the KGZ equations. For time discretization, an explicit low storage fourth order Runge Kutta method is proposed after transforming KGZ equations to system of ordinary differential equations by introducing auxiliary variables. L∞ and L2 error norms for some test problems are computed. Obtained numerical results and comparisons with finite element methods indicate that barycentric rational interpolation method is an efficient method for solving multidimensional generalized KGZ system numerically.

Simulation of predator–prey system with two-species, two chemicals and an additional chemotactic influence via direct meshless local Petrov–Galerkin method

“…The linear barycentric interpolation method is developed in Oruc (2022a) to simulate 2D fractional Rayleigh–Stokes problem for a heated generalized second grade fluid. A meshless collocation method based on barycentric rational interpolation is developed in Oruc (2022b) for solving the multidimensional generalized Klein–Gordon–Zakharov equations with power law nonlinearity. A local radial basis function-finite difference (RBF-FD) method is proposed in Oruc (2021) for solving 1D and 2D coupled Schrodinger–Boussinesq equations.…”

Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter–Saxton Equation

“…Non-linear partial differential equations play a vital role in modeling real-life phenomena, especially in the field of applied sciences such as meteorology, aerospace engineering, oceanography, geology, astronomy, and many other disciplines. For the realistic treatment of these non-linear models, several numerical techniques such as the barycentric interpolation collocation method [1], Fourier spectral method [2], and reproducing kernel method [3] have been adopted for the numerical treatment of these non-linear partial differential equations, such as the Klein-Gordon-Zakharov equations, Schrödinger equation, Duffing systems, and many more. The Hunter-Saxton equation is one among those non-linear partial differential equations that describe waves of a nematic liquid crystal in an enormous director field and arise at the short-wave limit of the Camassa-Holm equationan integrable model of shallow-water waves propagating uni-directionally across a flat bottom [4].…”