Computing the Set of RBF-FD Weights Within the Integrals of a Kernel-Based Function and Its Applications

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This paper offers an approach to computing Radial Basis Function–Finite Difference (RBF-FD) weights by integrating a kernel-based function. We derive new weight sets that effectively approximate both the first and second differentiations of a function, demonstrating their utility in interpolation and the resolution of Partial Differential Equations (PDEs). Particularly, the paper evaluates the theoretical weights in interpolation tasks, highlighting the observed numerical orders, and further applies these weights to solve two distinct time-dependent PDE problems.