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1
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.
2
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.
3
Content available remote Stability of difference equations generated by parabolic differential functional equations
Ciarski R.
Demonstratio Mathematica
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2005
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Vol. 38, nr 1
101--117
EN
The aim of this paper is to present a numerical approximation for the initial boundary value problem for quasilinear parabolic differential functional equations. The convergence result is proved for the difference scheme with the property that the difference operators approximating mixed derivatives depend on local properties of coefficients of the differential equation. A numerical example is given.
4
A sample time assign for a discrete interval parabolic system with the two-dimensional uncertain parameter space
Mitkowski W., Oprzędkiewicz K.
Systems Science
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2004
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Vol. 30, no 1
43-50
EN
In this paper, the sample time assign problem for a class of uncertain parameter discrete parabolic systems is presented. The system under consideration is described by the discrete abstract state space equation in the Hubert space. The system's uncertainty is described by the two-dimensional vector containing uncertain parameters of a heat equation. The type of the system uncertainty implies that the state operator and the control operator for the discussed system are interval operators. As a measure of the system's uncertainty the weighted mean of widths of interval state equation coefficients and the Euclidean norm of the both widths were accepted. The proposed performance indices allows us to calculate a range of sample time that guarantees that the required maximal permissible system's uncertainty is not exceeded after a system's discretization. The results are depicted by an example.
5
Content available remote Different variants of the boundary element method for parabolic equations
Majchrzak E., Ładyga E., Mendakiewicz J., Piasecka Belkhayat A.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2004
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Vol. 3, nr 1
127--132
EN
In the paper the different algorithms of boundary element method for parabolic equation are presented, this means the 1st and 2nd scheme of the BEM, the BEM using discretization in time and the BEM using Laplace transform.
6
Content available remote Dynamically equivalent perturbations of linear parabolic equations
Czaja R.
Demonstratio Mathematica
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2004
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Vol. 37, nr 2
327--348
EN
A family of abstract parabolic equations with sectorial operator is studied in this paper. The conditions are provided to show that the global attractors for each equation exist and coincide. Although the common dynamics is simple, the examples presented in the final part of the paper indicate that the considered family may contain a linear equation together with a large number of its nonlinear perturbations. The mentioned examples include both scalar second order equations and the celebrated Cahn-Hilliard system.
7
The existence and uniqueness of solution of one coupled plate thermomechanics problem
Krysko V. A., Awrejcewicz J., Bruk V. M.
Journal of Applied Analysis
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2002
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Vol. 8, nr 1
129-139
EN
The thermo-elastic plate system of equations is analysed. The sufficient conditions of existence, uniqueness and continuity dependence on initial data of the Cauchy problem solutions for differential-operational equation of mixed type (a part of the equation of hyperbolic type, and a part of parabolic type) are given in this paper. If the operational coefficients are suitably chosen, the investigated equation can be used to obtain a differential equation describing vibrations of a plate — the modified Germain-Lagrange equation of hyperbolic type. Moreover, in order to define the temperature field, one can use a three-dimensional equation of thermal conductivity (a parabolic equation).
8
Content available remote Periodic solutions for evolution inclusions with time-dependent subdifferentials
Matzakos N., Papageorgiu N. S.
Demonstratio Mathematica
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2002
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Vol. 35, nr 3
521-538
EN
In this paper we examine a periodic evolution equation driven by a time-dependent subdifferential and with a multivalued forcing term. Using a fixed point theorem for pseudo-acyclic multifunctions we prove the existence of periodic trajectories. This approach requires a study of the structure of the solution set of the Cauchy problem, which is also conducted in this paper.
9
Content available remote Oscillation of parabolic delay differential equations
Kubiaczyk I., Saker S. H.
Demonstratio Mathematica
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2002
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Vol. 35, nr 4
791-802
EN
Some new sufficient conditions for oscillation of all solutions of parabolic delay differential equations with several positive and negative coefficients are obtained. Our results extend and improve the well known results in the literature.
10
Content available remote Solving Parabolic Equations by Using the Method of Fast Convergent Iterations
Bondarenko V., Bidyuk P., Bernatska J.
International Journal of Applied Mathematics and Computer Science
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2000
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Vol. 10, no 2
333-344
EN
The paper describes an approach to solving parabolic partial differential equations that generalizes the well-known parametrix method. The iteration technique proposed exhibits faster convergence than the classical parametrix approach. A solution is constructed on a manifold with the application of the Laplace-Beltrami operator. A theorem is formulated and proved to provide a basis for finding a unique solution. Simulation results illustrate the superiority of the proposed approach in comparison with the classical parametrix method.
11
Oscillation criteria for a class of functional parabolic equations
Kusano T., Yoshida N.
Journal of Applied Analysis
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1999
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Vol. 5, nr 1
1-16
EN
Oscillations of parabolic equations with functional arguments are studied, and sufficient conditions are derived for all solutions of certain boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multidimensional problems to one-dimensional problems for functional differential inequalities.
12
Content available Discrete relaxed method for semilinear parabolic optimal control problem
Chryssoverghi I., Coletsos J., Kokkinis B.
Control and Cybernetics
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1999
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Vol. 28, no 2
157-176
EN
We consider an optimal control problem for systems governed by semilinear parabolic partial differential equations with control and state constraints, without any convexity assumptions. A discrete optimization method is proposed to solve this problem in its relaxed form which combines a penalized Armijo type method with a finite element discretization and constructs sequences of discrete Gamkrelidze relaxed controls. Under appropriate assumptions, we prove that accumulation points of these sequences satisfy the relaxed Pontryagin necessary conditions for optimality. Moreover, we show that the Gamkrelidze controls thus generated can be replaced by simulating piecewise constant classical controls.
13
Content available remote Phragmen-Lindelof type theorems for some semilinear elliptic and parabolic equations
Celebi A., Kalantarov V., Tahamtani F.
Demonstratio Mathematica
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1998
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Vol. 31, nr 1
43-54
EN
Phragmen-Lindelof type theorems for some classes of fourth order semilinear elliptic and se
14
Content available remote Oscillations of systems of neutral delay parabolic equations
Li W., Cui B.
Demonstratio Mathematica
|
1998
|
Vol. 31, nr 4
813-824
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