scholarly journals Nonclassical Problem for Ultraparabolic Equation in Abstract Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-15
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
Initial Conditions ◽  
A Priori ◽  
Nonlocal Problem ◽  
Ultraparabolic Equation ◽  
Nonclassical Problem ◽  
Order Elliptic Operator ◽  
Priori Estimates ◽  
Abstract Spaces

Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.

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.

Three-Level Difference Schemes on Non-Uniform in Time Grids

2001 ◽  
Vol 1 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Piotr Matus ◽  
Elena Zyuzina
Keyword(s):  
Hilbert Spaces ◽  
Wave Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Approximation ◽  
Difference Schemes ◽  
Nonuniform Grids ◽  
Priori Estimates ◽  
Long Time ◽  
The Right

Abstract In this work, a stability of three-level operator-difference schemes on nonuniform in time grids in Hilbert spaces is studied. A priori estimates of a long time stability (for t → ∞) in the sense of the initial data and the right-hand side are obtained in different energy norms without demanding the quasiuniformity of the grid. New difference schemes of the second order of local approximation on nonuniform grids both in time and space on standard stencils for parabolic and wave equations are adduced.

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 On a Class of Multitime Evolution Equations with Nonlocal Initial Conditions

2007 ◽  
Vol 2007 ◽  
pp. 1-26 ◽  
Author(s):  
F. Zouyed ◽  
F. Rebbani ◽  
N. Boussetila
Keyword(s):  
Evolution Equation ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
A Priori Estimates ◽  
Initial Conditions ◽  
A Priori ◽  
Nonlocal Initial Conditions ◽  
Priori Estimates

The existence and uniqueness of the strong solution for a multitime evolution equation with nonlocal initial conditions are proved. The proof is essentially based on a priori estimates and on the density of the range of the operator generated by the considered problem.

On the Solvability of a Nonlocal Problem Arising in Dynamics of Moisture Transfer

2003 ◽  
Vol 10 (4) ◽  
pp. 607-622
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Moisture Transfer ◽  
A Priori Estimates ◽  
A Priori ◽  
Nonlocal Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Evolution Problems ◽  
Priori Estimates ◽  
Basic Source ◽  
Initial Boundary Value

Abstract In the recent years, evolution problems with an integral term in the boundary conditions have received a great deal of attention. Such problems, in general, are nonself-adjoint, and this poses the basic source of difficulty, which can considerably complicate the application of standard functional and numerical techniques. To avoid these complications, we have introduced a nonclassical function space to establish a priori estimates without any additional complication as compared to the classical evolution problems. As an example of the applicability of this way of solving problems of this type, we investigate an initial-boundary value problem for a pseudoparabolic equation which combines Neumann and integral conditions.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

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