Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaces

The nonlinear wave phenomenon constitutes a significant research field and is a capable mathematical model for representing the transmission of energy in physical processes. This paper concentrates on the numerical solution for the nonlinear regularized long-wave and nonlinear extended Fisher–Kolmogorov models in two-space dimension. The solutions of these models are approximated via the finite difference technique and the localized radial basis function partition of unity. The association of the technique results in a meshfree method that does not require linearizing the nonlinear terms. In the first step, the partial differential equation (PDE) is transformed to a system of nonlinear ordinary differential equations (ODEs) by means of radial kernels. Subsequently, the nonlinear ODE system is approximated using a high-order ODE solver. The global collocation methods pose a considerable computational burden due to the calculation of the dense algebraic system. The proposed approach is based on a decomposition of the original domain into several subdomains via a kernel approximation on every subdomain. Numerical results such as the motion of single solitary and two solitary waves confirm the efficiency and accuracy of the method and are good agreement with the results of existing works in literature.