Quenching and blow-up problems are the non-linear singular problems which are undoubtedly very important mathematical problems in the literature. In this paper, we propose to extend the method of particular solutions (MPS) for solving various quenching and blow-up problems defined on different geometries using polynomial basis functions. We successfully solved these non-linear singular problems and compared our results with the results obtained from well established numerical methods such as finite element method (FEM), finite difference method (FDM), boundary element method (BEM) and method of fundamental solutions (MFS). Different numerical experiments presented in this paper clearly verified the effectiveness of our proposed method.

polynomial basis functions, method of particular solutions, non-linear partial differential equations, Picard iteration, quenching and blow up problems.