Adaptive Radial Basis Function Interpolation for Time-Dependent Partial Differential Equations

In this thesis we have proposed the meshless adaptive method by radial basis functions (RBFs) for the solution of the time-dependent partial differential equations (PDEs) where the approximate solution is obtained by the multiquadrics (MQ) and the local scattered data reconstruction has been done by polyharmonic splines. We choose MQ because of its exponential convergence for sufficiently smooth functions. The solution of partial differential equations arising in science and engineering, frequently have large variations occurring over small portion of the physical domain, the challenge then is to resolve the solution behaviour there. For the sake of efficiency we require a finer grid in those parts of the physical domain whereas a much coarser grid can be used otherwise. During our journey, we come up with different ideas and have found many interesting results but the main motivation for the one-dimensional case was the Korteweg-de Vries (KdV) equation rather than the common test problems. The KdV equation is a nonlinear hyperbolic equation with smooth solutions at all times. Furthermore the methods available in the literature for solving this problem are rather fully implicit or limited literature can be found using explicit and semi-explicit methods. Our approach is to adaptively select the nodes, using the radial basis function interpolation. We aimed in, the extension of our method in solving two-dimensional partial differential equations, however to get an insight of the method we developed the algorithms for one-dimensional PDEs and two-dimensional interpolation problem. The experiments show that the method is able to track the developing features of the profile of the solution. Furthermore this work is based on computations and not on proofs.