A novel meshless method for solving long-term evolution problem on irregular domain

In the context of method of approximate particular solutions (MAPS), we propose a novel meshless computational scheme based on hybrid radial basis function (RBF)-polynomial bases to solve both parabolic and hyperbolic partial differential equations over a large terminal time interval on irregular spatial domain. By using space–time approach, the original time-dependent problem is firstly reformulated into an elliptic boundary value problem. Integrated polyharmonic splines (PS)-type RBF kernels in conjunction with multivariate polynomials are then employed to construct the approximate solution space. This superior combination enables us to stably achieve highly accurate solution. Due to the adoption of the polyharmonic splines, the difficulty of determining a suitable shape parameter of RBF is alleviated. Moreover, employing the recently developed ghost point method, the precision and stability of the approximation can be further enhanced. For numerical verification, four examples are investigated to demonstrate the robustness of the proposed methodology in terms of the aforementioned advantages. © 2024 Elsevier Inc.