Opuscula Math. 34, no. 2 (2014), 311-326
Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions
Abstract. We consider the classical solutions of mixed problems for infinite, countable systems of parabolic functional differential equations. Difference methods of two types are constructed and convergence theorems are proved. In the first type, we approximate the exact solutions by solutions of infinite difference systems. Methods of second type are truncation of the infinite difference system, so that the resulting difference problem is finite and practically solvable. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for the given functions. The comparison system is infinite. Parabolic problems with deviated variables and integro-differential problems can be obtained from the general model by specifying the given operators.
Keywords: nonlinear parabolic equations, functional difference equations, infinite systems, Volterra type operators, nonlinear estimates of the Perron type, truncation methods.
Mathematics Subject Classification: 35R10, 35K51, 65M10.