scholarly journals A Note on the Stability of the Integral-Differential Equation of the Parabolic Type in a Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Unique Solvability ◽  
Parabolic Type ◽  
Stability Estimates ◽  
The Difference ◽  
The Stability ◽  
Integral Differential Equation

The integral-differential equation of the parabolic type in a Banach space is considered. The unique solvability of this equation is established. The stability estimates for the solution of this equation are obtained. The difference scheme approximately solving this equation is presented. The stability estimates for the solution of this difference scheme are obtained.

A Note on the Stability of the Integral-Differential Equation of the Parabolic Type

2011 ◽  
Author(s):  
Maksat Ashyraliyev ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Differential Equation ◽  
Parabolic Type ◽  
The Stability ◽  
Integral Differential Equation

scholarly journals On Gronwall’s type integral inequalities with singular kernels

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1041-1049 ◽  
Author(s):  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Fractional Differential Equations ◽  
Parabolic Type ◽  
Integral Inequalities ◽  
Stability Estimates ◽  
Type Integral ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Integral Differential Equation

In this paper, the generalizations of Gronwall?s type integral inequalities with singular kernels are established. In applications, theorems on stability estimates for the solutions of the nonliner integral equation and the integral-differential equation of the parabolic type are presented. Moreover, these inequalities can be used in the theory of fractional differential equations.

scholarly journals A note on the parabolic identification problem with involution and Dirichlet condition

2020 ◽  
Vol 99 (3) ◽  
pp. 130-139
Author(s):  
A. Ashyralyev ◽  
◽  
A.S. Erdogan ◽  
A. Sarsenbi ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Difference Scheme ◽  
Source Identification ◽  
Parabolic Problem ◽  
Identification Problem ◽  
Dirichlet Condition ◽  
Stability Estimates ◽  
Well Posedness ◽  
The Difference

A space source of identification problem for parabolic equation with involution and Dirichlet condition is studied. The well-posedness theorem on the differential equation of the source identification parabolic problem is established. The stable difference scheme for the approximate solution of this problem is presented. Furthermore, stability estimates for the difference scheme of the source identification parabolic problem are presented. Numerical results are given.

Well-posedness of a nonlocal boundary value difference elliptic problem

2019 ◽  
Vol 14 (5) ◽  
pp. 507
Author(s):  
Allaberen Ashyralyev ◽  
Ayman Hamad
Keyword(s):  
Difference Scheme ◽  
Numerical Solution ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Well Posedness ◽  
The Difference ◽  
The Stability ◽  
Coercive Stability ◽  
Second Order Of Approximation

The second order of approximation two-step difference scheme for the numerical solution of a nonlocal boundary value problem for the elliptic differential equation [see formula in PDF] in an arbitrary Banach space E with the positive operator A is presented. The well-posedness of the difference scheme in Banach spaces is established. In applications, the stability, almost coercive stability and coercive stability estimates in maximum norm in one variable for the solutions of difference schemes for numerical solution of two type elliptic problems are obtained.

scholarly journals On Third Order Stable Difference Scheme for Hyperbolic Multipoint Nonlocal Boundary Value Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Boundary Value Problem ◽  
Difference Scheme ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
The Difference

This paper presents a third order of accuracy stable difference scheme for the approximate solution of multipoint nonlocal boundary value problem of the hyperbolic type in a Hilbert space with self-adjoint positive definite operator. Stability estimates for solution of the difference scheme are obtained. Some results of numerical experiments that support theoretical statements are presented.

scholarly journals Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskiĭ
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Continuous Functions ◽  
High Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Smooth Functions ◽  
Order Of Accuracy ◽  
High Order Of Accuracy ◽  
The Difference

It is well known the differential equation−u″(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the positive operatorAis ill-posed in the Banach spaceC(E)=C((−∞,∞),E)of the bounded continuous functionsϕ(t)defined on the whole real line with norm‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions inC(τ,E)of these difference schemes is established.

scholarly journals A Note on the Parabolic Differential and Difference Equations

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Yasar Sozen ◽  
Pavel E. Sobolevskii
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Positive Operator ◽  
Difference Equations ◽  
Well Posedness ◽  
Holder Space ◽  
Strongly Positive Operator ◽  
Ill Posed ◽  
General Banach Space

The differential equationu'(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the strongly positive operatorAis ill-posed in the Banach spaceC(E)=C(ℝ,E)with norm‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder spaceCα(E)=Cα(ℝ,E)with norm‖ϕ‖Cα(E)=sup−∞<t<∞‖ϕ(t)‖E+sup−∞<t<t+s<∞(‖ϕ(t+s)−ϕ(t)‖E/sα),0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme inC(ℝτ,E)spaces is proved. The well-posedness of this difference scheme inCα(ℝτ,E)spaces is obtained.

scholarly journals Stability of the Difference Schemes for the Equations of Weakly Compressible Liquid

2007 ◽  
Vol 7 (3) ◽  
pp. 208-220 ◽  
Author(s):  
P. Matus ◽  
O. Korolyova ◽  
M. Chuiko
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Initial Conditions ◽  
Compressible Liquid ◽  
Stability Conditions ◽  
Differential Problem ◽  
Weakly Compressible ◽  
Linear Problems ◽  
The Difference ◽  
The Stability

Abstract A priory estimates of the stability in the sense of the initial data of the difference scheme approximating weakly compressible liquid equations in the Riemann invariants have been obtained. These estimates have been proved without any assumptions about the properties of the solution of the differential problem and depend only on the behavior of the initial conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for a finite instant of time t≤t_0. In particular, this is confirmed by the fact, that nonfulfilment of these stability conditions lead to the appearance of supersonic flows or domains with large gradients. The questions of uniqueness and convergence of the difference solution are considered also. The results of the computating experiment confirming the theoretical conclusions are given.

scholarly journals On fourth order accuracy stable difference scheme for a multi-point overdetermined elliptic problem

2021 ◽  
Vol 102 (2) ◽  
pp. 45-53
Author(s):  
C. Ashyralyyev ◽  
◽  
G. Akyuz ◽  
◽  
Keyword(s):  
Approximate Solution ◽  
Difference Scheme ◽  
Fourth Order ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Uniqueness Of The Solution ◽  
Accuracy Difference ◽  
The Difference ◽  
Overdetermined Boundary Value Problem ◽  
Coercive Stability

In this paper fourth order of accuracy difference scheme for approximate solution of a multi-point elliptic overdetermined problem in a Hilbert space is proposed. The existence and uniqueness of the solution of the difference scheme are obtained by using the functional operator approach. Stability, almost coercive stability, and coercive stability estimates for the solution of difference scheme are established. These theoretical results can be applied to construct a stable highly accurate difference scheme for approximate solution of multi-point overdetermined boundary value problem for multidimensional elliptic partial differential equations.

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

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