Browsing by Subject "infinite systems"

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Access status: Open Access ,
Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions
(2014) Czernous, Wojciech; Jaruszewska-Walczak, Danuta
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.
Access status: Open Access ,
Stability of solutions of infinite systems of nonlinear differential-functional equations of parabolic type
(2006) Zabawa, Tomasz
A parabolic initial boundary value problem and an associated elliptic Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations are considered. It is shown that the solutions of the parabolic problem is asymptotically stable and the limit of the solution of the parabolic problem as $t\to\infty$ is the solution of the associated elliptic problem. The result is based on the monotone methods.