In this paper, we propose an effective method to solve partial differential equations dependent on time with Neumann boundary condition, by examining its effectivity on direct and inverse reaction–diffusion equation. This method merges the radial point interpolation and the Hermite-type interpolation techniques to provide us suitable tools to impose the boundary condition. This technique is called meshless radial point Hermite interpolation “MRPHI” which utilizes the radial basis function and its derivative to prepare suitable shape functions that are the key for expanding the high-order derivative. This procedure is tested on some types of two-dimensional diffusion equations to show stability through the time in different arbitrary domains.

中文翻译：

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通过直接和逆二维反应-扩散方程对无网格径向点 Hermite 插值的新见解

在本文中，我们通过检验其对正反应扩散方程和逆反应扩散方程的有效性，提出了一种求解具有诺依曼边界条件的时间依赖偏微分方程的有效方法。这种方法结合了径向点插值和 Hermite 类型的插值技术，为我们提供了合适的工具来施加边界条件。这种技术被称为无网格径向点 Hermite 插值“MRPHI”，它利用径向基函数及其导数来准备合适的形状函数，这是扩展高阶导数的关键。此过程在某些类型的二维扩散方程上进行了测试，以显示不同任意域中随时间变化的稳定性。