Implicit Difference Methods for Differential Functional Parabolic Equations with Dirichlet's Condition

Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition are approximated in the paper by solutions of associated implicit difference functional equations. The functional dependence is of the Volterra type. Nonlinear estimates of the generalized Perron type for given functions are assumed. The convergence and stability results are proved with the use of discrete functional inequalities and the comparison technique. In particular, these theorems cover quasi-linear equations. However, such equations are also treated separately. The known results on similar difference methods can be obtained as particular cases of our simple result.