In this thesis, we consider two collocation methods, namely, Haar Wavelet Collocation Method (HWCM) and Meshless Collocation Method (MCM) with Radial Basis Functions (RBFs). Two new algorithms based on HWCM and MCM are developed for one-dimensional linear elliptic problems with single as well as double interfaces. The proposed methods are extended to include two- and three-dimensional elliptic problems. We also consider parabolic interface problems in which time derivative is involved. The time derivative is approximated bynite di erence and Crank-Nicolson schemes. Both the proposed methods are developed for linear parabolic problems as well. The new numerical methods are also applied to nonlinear elliptic and parabolic problems. The algorithms are designed for all the new developed methods. The MATLAB software is used for implementations and testing of all the new methods. Several numerical experiments are taken into account in order to check the performance of the new developed methods. The numerical results are compared with exact solution as well as existing methods in literature which show the superiority of the proposed methods. A comparative study between HWCM and MCM in terms of e ciency, accuracy and applicability is also performed.