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1
Content available Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions
Czernous W., Jaruszewska-Walczak D.
Opuscula Mathematica
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2014
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Vol. 34, no. 2
311--326
EN
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.
2
Content available remote Strong maximum principles for infinite systems of parabolic differential-functional inequalities with nonstandard initial inequalities (uj(x,t0)-Kj) + ∑hi(x)(uj(x,Ti-Kj≤0
Brandys J.
Fasciculi Mathematici
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2008
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Nr 39
17-26
EN
The aim of this paper is to present strong maximum principles for infinite systems of parabolic differential-functional inequalities with nonstandard initial inequalities with sums in relatively arbitrary (n+l)-dimensional time-space sets more general than the cylindrical domain.
3
Content available remote Strong maximum principles for infinite systems of parabolic differential-functional inequalities with initial constant estimates
Brandys J.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2007
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Vol. 47, [Z] 2
141-148
EN
The purpose of this paper is to present strong maximum principles for infinite systems of parabolic differential-functional inequalities with initial inequalities in relatively arbitrary (n+1)-dimensional time-space sets more general than the cylindrical domain.
4
Content available Stability of solutions of infinite systems of nonlinear differential-functional equations of parabolic type
Zabawa T.S.
Opuscula Mathematica
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2006
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Vol. 26, no. 1
173-183
EN
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 → ∞ is the solution of the associated elliptic problem. The result is based on the monotone methods.
5
Content available Monotone iterative methods for infinite systems of reaction-diffusion-convection equations with functional dependence
Brzychczy S.
Opuscula Mathematica
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2005
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Vol. 25, no. 1
29-99
EN
We consider the Fourier first initial-boundary value problem for an infinite system of semilinear parabolic differential-functional equations of reaction-diffusion-convection type of the form [formula] where [formula] in a bounded cylindrical domain (0, T] x G := D rcup Rm+1. The right-hand sides of the system are Volterra type functionals of the unknown function z. In the paper, we give methods of the construction of the monotone iterative sequences converging to the unique classical solution of the problem considered in partially ordered Banach spaces with various convergence rates of iterations. We also give remarks on monotone iterative methods in connection with numerical methods, remarks on methods for the construction of lower and upper solutions and remarks concerning the possibility of extending these methods to more general parabolic equations. All monotone iterative methods are based on differential inequalities and, in this paper, we use the theorem on weak partial differential-functional inequalities for infinite systems of parabolic equations, the comparison theorem and the maximum principle. A part of the paper is based on the results of our previous papers. These results generalize the results obtained by several authors in numerous papers for finite systems of semilinear parabolic differential equations to encompass the case of infinite systems of semilinear parabolic differential-functional equations. The monotone iterative schemes can be used for the computation of numerical solutions.
6
Content available Monotone iteration for infinite systems of parabolic equations
Pudełko A.
Opuscula Mathematica
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2005
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Vol. 25, no. 2
307-318
EN
In the paper the Cauchy problem for and infinite system of parabolic type equations is studied. The general operators of the parabolic type of second order with variable coefficients are considered and the system is weakly coupled. Among the obtained results there is a theorem on differential inequality as well as the existence and uniqueness theorem in the class of continuous-bounded functions obtained by monotone iterative method.
7
Content available Existence of solutions of the Dirichlet problem for an infinite system of nonlinear differential-functional equations of elliptic type
Zabawa T.S.
Opuscula Mathematica
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2005
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Vol. 25, no. 2
333-343
EN
The Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations of elliptic type is considered. It is shown the existence of solutions to this problem. The result is based on Chaplygin's method of lower and uper functions.
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