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

Abstract This paper introduces the Lagrange collocation method with radial basis functions (LRBF) as a novel approach for solving 1D partial differential equations. Our method addresses the trade-off principle, a key challenge in standard RBF collocation methods, by maintaining the accuracy and convergence of the numerical solution while improving 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 to the main matrix, developing the perturbed LRBF method (PLRBF) which allows for the application of Cholesky decomposition, significantly reducing 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 the perturbation is carefully selected. By doing so, highly accurate solutions can be achieved at an early level, significantly reducing CPU time. Furthermore, to overcome stagnation issues in the RBF collocation method, we combine LRBF and CPLRBF with multilevel techniques and get the Multilevel PLRBF (MuC-PLRBF). 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 extend it to higher dimensions in future work.