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.

中文翻译：

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一种求解不规则域长期演化问题的新型无网格方法


在近似特定解法 （MAPS） 的背景下，我们提出了一种基于混合径向基函数 （RBF） -多项式基的新型无网格计算方案，以在不规则空间域上求解大终端时间间隔上的抛物线和双曲偏微分方程。首先，利用时空方法将原来的时相关问题重新表述为椭圆边值问题。然后采用积分多谐波样条函数 （PS） 型 RBF 核与多元多项式相结合来构建近似解空间。这种卓越的组合使我们能够稳定地实现高精度的解决方案。由于采用了多谐波样条，减少了确定 RBF 合适形状参数的难度。此外，采用最近开发的鬼点方法，可以进一步提高近似的精度和稳定性。对于数值验证，研究了四个示例，以证明所提出的方法在上述优势方面的稳健性。