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Zeitschrift für Analysis und ihre Anwendungen


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Volume 32, Issue 3, 2013, pp. 313–337
DOI: 10.4171/ZAA/1487

Published online: 2013-07-03

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

Lucjan Sapa[1]

(1) AGH University of Science and Technology, Krakow, Poland

Classical solutions of nonlinear second-order partial di erential functional equations of parabolic type with Dirichlet's condition are approximated in the paper by solutions of associated implicit di erence 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 di erence 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