A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions

This paper is devoted to develop a new matrix scheme for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions. We first transform these equations into equivalent integro partial differential equations (PDEs). Such these integro-PDEs contain both of the initial and boundary conditions and can be solved numerically in a more appropriate manner. Subsequently, all the existing known and unknown functions in the latter equations are approximated by Bernoulli polynomials and operational matrices of differentiation and integration together with the completeness of these polynomials can be used to reduce the integro-PDEs into the associated algebraic generalized Sylvester equations. For solving these algebraic equations, an efficient Krylov subspace iterative method (i.e., BICGSTAB) is implemented. Two numerical examples are given to demonstrate the efficiency, accuracy, and versatility of the proposed method.