Novel method of particular solutions for second-order variable coefficient differential equations on a square

Abstract

In this work, a novel method of particular solutions (MPS, for short) is proposed with Legendre polynomials to solve second-order partial differential equations (PDEs) with a variable coefficient on a square. Specially, we prove the uniqueness of the solution for the given model, which is equipped with a sufficiently smooth variable coefficient excluding elliptic properties. Based on the characteristics of Legendre polynomials, particular solutions, which satisfy the natural boundary condition, are constructed for the variable coefficient within the PDEs. Meanwhile we investigate the a-priori error estimates in $ H^1 $-norm and $ L^2 $-norm of the MPS approximations, which depict the asymptotic super-exponential convergence rates of the numerical approximations for sufficiently smooth solutions. Some numerical examples and convergence rates are listed to depict the highlights of this proposed meshless method.