Lagrange radial basis function collocation method for boundary value problems in $1$D

This paper introduces the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving 1D partial differential equations. Our method addresses the trade-off principle, which is a key challenge in standard RBF collocation methods, by maintaining the accuracy and convergence of the numerical solution, while improving the stability and efficiency. We prove the existence and uniqueness of the numerical solution for specific differential operators, such as the Laplacian operator, and for positive definite RBFs. Additionally, we introduce a perturbation into the main matrix, thereby developing the perturbed LRBF method (PLRBF); this allows for the application of Cholesky decomposition, which significantly reduces the condition number of the matrix to its square root, resulting in the CPLRBF method. In return, this enables us to choose a large value for the shape parameter without compromising stability and accuracy, provided that the perturbation is carefully selected. By doing so, highly accurate solutions can be achieved at an early level, significantly reducing central processing unit (CPU) time. Furthermore, to overcome stagnation issues in the RBF collocation method, we combine LRBF and CPLRBF with multilevel techniques and obtain the Multilevel PLRBF (MuCPLRBF) technique. We illustrate the stability, accuracy, convergence, and efficiency of the presented methods in numerical experiments with a 1D Poisson equation. Although our approach is presented for 1D, we expect to be able to extend it to higher dimensions in future work.