The semi-linear equation -u(xx) - epsilon u(yy) = f(x, y, u) with Dirichlet
boundary conditions is solved by an O(h(4)) finite difference method, whic
h has local truncation error O(h(2)) at the mesh points neighboring the bou
ndary and O(h(4)) at most interior mesh points. It is proved that the finit
e difference method is O(h(4)) uniformly convergent as h --> 0. The method
is considered in the form of a system of algebraic equations with a nine di
agonal sparse matrix. The system of algebraic equations is solved by an imp
licit iterative method combined with Gauss elimination. A Mathematica modul
e is designed for the purpose of testing and using the method. To illustrat
e the method, the equation of twisting a springy rod is solved. (C) 2000 Jo
hn Wiley & Sons, Inc.