Two-point Taylor approximations of the solutions of two-dimensional boundary value problems

We consider two-dimensional elliptic boundary value problems of the form L(U)=F in Ω⊂R2,Ω bounded and open, with a Dirichlet boundary condition U|∂Ω=H, where L is a second order linear differential operator whose coefficients, as well as the functions F and H are differentiable up to a certain degree. In a recent paper [C. Kesan, Taylor polynomial solutions of second order linear partial differential equations, Appl. Math. Comput. 152 (2004) 29-41], a matrix algorithm is introduced to compute the standard Taylor polynomial of the solution U at a certain point in Ω. We propose an alternative formulation of the problem based on a redefinition of the unknown U and the use of the standard Frobenius method that simplifies the computation of the Taylor coefficients of U. We also consider the use of a two-point Taylor representation of the solution, instead of the classical one-point Taylor representation, which gives a more uniform approximation of the solution.