Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2013-07-03
Implicit Difference Methods for Differential Functional Parabolic Equations with Dirichlet's ConditionLucjan Sapa (1) AGH University of Science and Technology, Krakow, Poland
Classical solutions of nonlinear second-order partial dierential functional equations of parabolic type with Dirichlet's condition are approximated in the paper by solutions of associated implicit 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 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 dierence methods can be obtained as particular cases of our simple result.
Keywords: Parabolic differential functional equations, difference methods, stability and convergence, nonlinear estimates of the generalized Perron type
Sapa Lucjan: Implicit Difference Methods for Differential Functional Parabolic Equations with Dirichlet's Condition. Z. Anal. Anwend. 32 (2013), 313-337. doi: 10.4171/ZAA/1487