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A finite difference method for quasi-linear and nonlinear differential functional parabolic equations with Neumann's condition

Sapa S.

Commentationes Mathematicae

2009

Vol. 49, [Z] 1

83-106

Classical solutions of nonlinear second-order partial dierential functional equations of parabolic type with Neumann's condition are approximated in the paper by solutions of associated explicit dierence 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 the comparison technique. These theorems in particular cover quasi-linear equations, but such equations are also treated separately. The known results on similar dierence methods can be obtained as particular cases of our simple result.