Abstract The ill-posedness of the forward-backward heat problem arises in boundary layer problems, fluid dynamics, plasma physics, astrophysics and the study of propagation of an electron beam through the solar corona. In this study, a new truly meshless method is developed for the numerical solution of the two dimensional forward-backward heat equation. We propose a novel method based on the domain decomposition scheme and RBF method with an adaptive nodes technique. Specifically, the physical domain is divided into two subdomains each defining a forward or a backward subproblem. The resulting subproblems are dealt with by a radial basis function meshfree method for spatial dimension and a finite difference scheme for the time derivative followed by an adaptive algorithm to achieve a desired accuracy. In addition, we prove that the time discrete scheme is stable and convergent. Some numerical experiments will be presented to show the performance of our collocation scheme.

中文翻译：

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二维前后向热问题的RBF自适应方法的收敛性分析和稳定性

摘要 边界层问题、流体动力学、等离子体物理学、天体物理学和电子束穿过日冕传播的研究中出现了前后热问题的不适定性。在这项研究中，开发了一种新的真正无网格方法，用于二维前后热方程的数值解。我们提出了一种基于域分解方案和具有自适应节点技术的 RBF 方法的新方法。具体来说，物理域被分成两个子域，每个子域定义一个前向或后向子问题。由此产生的子问题由空间维度的径向基函数无网格方法和时间导数的有限差分方案处理，然后是自适应算法，以达到所需的精度。此外，我们证明了时间离散方案是稳定且收敛的。将提供一些数值实验来展示我们的搭配方案的性能。