scholarly journals Generalization of the Lax–Ryabenky–Philippov theorem to nonlinear problems

2021 ◽  
Vol 65 (4) ◽  
pp. 391-396
Author(s):  
P. P. Matus
Keyword(s):  
Banach Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Finite Difference Methods ◽  
Nonlinear Problems ◽  
Equivalence Theorem ◽  
Convergence Criterion ◽  
Quasilinear Parabolic Equations ◽  
Stability Of Difference Schemes ◽  
The Stability

In this paper, Lax’s equivalence theorem, which states that stability is a necessary and sufficient condition for its convergence in the presence of an approximation of a difference scheme, is generalized to abstract nonlinear difference problems with operators acting in finite dimensional Banach spaces. In contrast to linear finite-difference methods, such a criterion in the nonlinear case can be established only for unconditionally stable computational methods, when the corresponding a priori estimates take place for sufficiently small |h| ≤ h0. In this case, the value of h0 depends both on the consistency of discrete and continuous norms in Banach spaces, and on the magnitude of the perturbation of the input data of the problem. The proven convergence criterion is used to study the stability of difference schemes approximating quasilinear parabolic equations with nonlinearities of unbounded growth with respect to the initial data.

Stability of Difference Schemes for Nonlinear Time-dependent Problems

2003 ◽  
Vol 3 (2) ◽  
pp. 313-329 ◽  
Author(s):  
Piter Matus
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Quasilinear Parabolic Equations ◽  
Basic Point ◽  
Nonlinear Part ◽  
Priori Estimates ◽  
The Difference ◽  
The Stability

AbstractIn the present paper, a priori estimates of the stability in the sense of the initial data of the difference schemes approximating quasilinear parabolic equations and nonlinear transfer equation have been obtained. The basic point is connected with the necessity of estimating all derivatives entering into the nonlinear part of the difference equations. These estimates have been proved without any assumptions about the properties of the differential equations and depend only on the behavior of the initial and boundary conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for the finite instant of time t 6 t0 connected with the fact that the solution of the Riccati equation becomes infinite. is already associated with the behavior of the second derivative of the initial function and coincides with the time of the exact solution destruction (heat localization in the peaking regime). A close relation between the stability and convergence of the difference scheme solution is given. Thus, not only a priori estimates for stability have been established, but it is also shown that the obtained conditions permit exact determination of the time of destruction of the solution of the initial boundary value problem for the original nonlinear differential equation in partial derivatives. In the present paper, concrete examples confirming the theoretical conclusions are given.

On the Stability of Differential-operator Equations and Operator-difference Schemes as t → ∞

2002 ◽  
Vol 2 (2) ◽  
pp. 153-170 ◽  
Author(s):  
Boško Jovanović ◽  
Sergey Lemeshevsky ◽  
Peter Matus
Keyword(s):  
Asymptotic Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Nonlinear Problems ◽  
Abstract Cauchy Problem ◽  
Difference Schemes ◽  
Operator Equations ◽  
Priori Estimates ◽  
The Difference ◽  
The Stability

AbstractFor the abstract Cauchy problem for a parabolic equation a priori estimates of the global and asymptotic stability in various energy norms have been obtained. Similar problems are also considered for the second-order equation. In the latter case, a priori estimates of the asymptotic stability by the initial data have been obtained. The corresponding estimates of the global stability for three-level operator difference schemes have been proved. Estimates of the asymptotic behavior of the solution for quasi-linear multidimensional equations with unbounded nonlinearity have been obtained. The corresponding mathematical apparatus permitting one to prove unconditional monotonicity of the difference schemes approximating nonlinear problems is presented.

scholarly journals MONOTONE ECONOMICAL SCHEMES FOR QUASILINEAR PARABOLIC EQUATIONS

2002 ◽  
Vol 7 (2) ◽  
pp. 207-216
Author(s):  
N. V. Dzenisenko ◽  
A. P. Matus ◽  
P. P. Matus
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Quasilinear Parabolic Equation ◽  
Uniform Norm ◽  
Additive Scheme ◽  
Quasilinear Parabolic Equations ◽  
The Maximum Principle ◽  
Priori Estimates ◽  
The Difference ◽  
Quasilinear Parabolic

In order to approximate a multidimensional quasilinear parabolic equation with unlimited nonlinearity the economical vector‐additive scheme is constructed. It is shown that its solution satisfies the maximum principle and, hence, the scheme is monotone. The proof is based on the equivalence of the vector‐additive scheme and the scheme of summarized approximation (locally one‐dimensional scheme). The a priori estimates of the difference solution in the uniform norm are obtained.

On the Strong Stability of Operator-Difference Schemes in Time-Integral Norms

2001 ◽  
Vol 1 (1) ◽  
pp. 72-85 ◽  
Author(s):  
Boško S. Jovanović ◽  
Piotr P. Matus
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Stability ◽  
Difference Schemes ◽  
Initial Condition ◽  
Time Integral ◽  
Right Hand ◽  
Priori Estimates ◽  
The Stability

Abstract In this paper we investigate the stability of two-level operator-difference schemes in Hilbert spaces under perturbations of operators, the initial condition and right hand side of the equation. A priori estimates of the error are obtained in time- integral norms under some natural assumptions on the perturbations of the operators.

STABILITY AND CONVERGENCE OF TWO-LEVEL DIFFERENCE SCHEMES IN INTEGRAL WITH RESPECT TO TIME NORMS

1998 ◽  
Vol 08 (06) ◽  
pp. 1055-1070 ◽  
Author(s):  
ALEXANDER A. SAMARSKII ◽  
PETR P. MATUS ◽  
PETR N. VABISHCHEVICH
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Mathematical Physics ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Priori Estimates ◽  
Linear Problems ◽  
The Difference ◽  
The Stability

Nowadays the general theory of operator-difference schemes with operators acting in Hilbert spaces has been created for investigating the stability of the difference schemes that approximate linear problems of mathematical physics. In most cases a priori estimates which are uniform with respect to the t norms are usually considered. In the investigation of accuracy for evolutionary problems, special attention should be given to estimation of the difference solution in grid analogs of integral with respect to the time norms. In this paper a priori estimates in such norms have been obtained for two-level operator-difference schemes. Use of that estimates is illustrated by convergence investigation for schemes with weights for parabolic equation with the solution belonging to [Formula: see text].

scholarly journals A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 75
Author(s):  
Hristo Kiskinov ◽  
Magdalena Veselinova ◽  
Ekaterina Madamlieva ◽  
Andrey Zahariev
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Integral Representations ◽  
Distributed Delays ◽  
Finite Time Stability ◽  
Gronwall’S Inequality ◽  
Priori Estimates ◽  
The Stability ◽  
Traditional Approaches

In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.

scholarly journals Study of some mathematical models for nonstationary filtration processes

2020 ◽  
pp. 1-12
Author(s):  
Н.Л. Гольдман
Keyword(s):  
Mathematical Models ◽  
Parabolic Equations ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Nonlinear Problems ◽  
Parabolic Problems ◽  
Smooth Functions ◽  
Nonlinear Parabolic ◽  
Nonstationary Filtration

Рассматриваются математические модели, связанные с изучением нестационарных процессов фильтрации в подземной гидродинамике. Они представляют собой нелинейные задачи для параболических уравнений с неизвестной функцией источника в правой части. Одна из постановок является системой, которая состоит из краевой задачи с граничными условиями первого рода и из уравнения, задающего закон изменения по времени искомой функции источника. В другой постановке соответствующая система включает в себя краевую задачу с граничными условиями второго рода. Указанные постановки существенно отличаются от обычных краевых задач для параболических уравнений. Цель исследования - установить для этих нелинейных параболических задач условия однозначной разрешимости в классе гладких функций на основе априорных оценок метода Ротэ. We consider some mathematical models connected with the study of nonstationary filtration processes in underground hydrodynamics. These models involve nonlinear problems for parabolic equations with unknown source functions. One of the problems is a system consisting of a boundary value problem of the first kind and an equation describing a time dependence of the sought source function. In the other problem, the corresponding system is distinguished from the first one by boundary conditions of the second kind. These problems essentially differ from usual boundary value problems for parabolic equations. The aim of our study is to establish conditions of unique solvability in a class of smooth functions for the considered nonlinear parabolic problems. The proposed approach involves the proof of a priori estimates for the Rothe method.

scholarly journals On a Priori Estimates of Solutions of Systems of Higher Order Nonlinear Functional-Differential Inequalities

2007 ◽  
Vol 14 (3) ◽  
pp. 445-456
Author(s):  
Ivan Kiguradze ◽  
Nino Partsvania
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Higher Order ◽  
Infinite Interval ◽  
Differential Inequalities ◽  
Nonlinear Functional ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Functional Differential ◽  
The Stability

Abstract Systems of higher order nonlinear functional-differential inequalities arising in the theory of boundary value problems as well as in the stability theory are considered on a finite and an infinite interval. On a finite interval, a priori estimates are obtained for solutions of these systems satisfying boundary conditions of periodic type, and on an infinite interval, a priori estimates are established for the derivatives of bounded solutions.

DISCONTINUOUS/CONTINUOUS LEAST-SQUARES FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS

2005 ◽  
Vol 15 (06) ◽  
pp. 825-842 ◽  
Author(s):  
RICKARD E. BENSOW ◽  
MATS G. LARSON
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Finite Element Methods ◽  
A Priori Estimates ◽  
A Priori ◽  
Solution Space ◽  
First Order ◽  
Poisson Problem ◽  
Priori Estimates ◽  
The Stability

Least-squares finite element methods (LSFEM) are useful for first-order systems, where they avoid the stability consideration of mixed methods and problems with constraints, like the div-curl problem. However, LSFEM typically suffer from requirements on the solution to be very regular. This rules out, e.g., applications posed on nonconvex domains. In this paper we study a least-squares formulation where the discrete space is enriched by discontinuous elements in the vicinity of singularities. The weighting on the interelement terms are chosen to give correct regularity of the solution space and thus making computation of less regular problems possible. We apply this technique to the first-order Poisson problem, show coercivity and a priori estimates, and present numerical results in 3D.

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