scholarly journals Stability of delay parabolic difference equations

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 995-1006 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Deniz Agirseven
Keyword(s):  
Parabolic Equations ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Unbounded Operators ◽  
Stability Of Difference Schemes ◽  
Fractional Spaces ◽  
The Stability ◽  
Delay Differential ◽  
Parabolic Difference

In the present paper, the stability of difference schemes for the approximate solution of the initial value problem for delay differential equations with unbounded operators acting on delay terms in an arbitrary Banach space is studied. Theorems on stability of these difference schemes in fractional spaces are established. In practice, the stability estimates in H?lder norms for the solutions of difference schemes for the approximate solutions of the mixed problems for delay parabolic equations are obtained.

scholarly journals Economical factorized schemes for third-order pseudoparabolic equations

2021 ◽  
pp. 44-57
Author(s):  
Мурат Хамидбиевич Бештоков
Keyword(s):  
General Theory ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Third Order ◽  
The Third ◽  
Stability Of Difference Schemes ◽  
The Stability ◽  
Pseudoparabolic Equations

Изучены экономичные факторизованные схемы для псевдопараболических уравнений третьего порядка. На основе общей теории устойчивости разностных схем доказаны устойчивость и сходимость разностных схем. Economical factorized schemes for pseudo-parabolic equations of the third order are studied. On the basis of the general theory of stability of difference schemes, the stability and convergence of difference schemes are proved.

On the Stable Difference Schemes for the Schrödinger Equation with Time Delay

2020 ◽  
Vol 20 (1) ◽  
pp. 27-38 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Deniz Agirseven
Keyword(s):  
Time Delay ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Stability Of Difference Schemes ◽  
Accuracy Difference ◽  
Initial Boundary Value

AbstractIn the present paper, the first and second order of accuracy difference schemes for the approximate solutions of the initial value problem for Schrödinger equation with time delay in a Hilbert space are presented. The theorem on stability estimates for the solutions of these difference schemes is established. The application of theorems on stability of difference schemes for the approximate solutions of the initial boundary value problems for Schrödinger partial differential equation is provided. Additionally, some illustrative numerical results are presented.

scholarly journals Stability and Monotonicity of Difference Schemes for Nonlinear Scalar Conservation Laws and Multidimensional Quasi-linear Parabolic Equations

2009 ◽  
Vol 9 (3) ◽  
pp. 253-280 ◽  
Author(s):  
P. Matus ◽  
S. Lemeshevsky
Keyword(s):  
Conservation Laws ◽  
Parabolic Equations ◽  
Input Data ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Linear Parabolic Equations ◽  
The Difference ◽  
The Stability

Abstract We have proved the difference analogue of a Bihari-type inequality. Using this inequality, we study the stability in C and monotonicity of the difference schemes approximating initial-boundary value problems for nonlinear conservation laws and multi-dimensional parabolic equations. It has been shown that in the nonlinear case the stability and monotonicity are determined not only by the behavior of the approximate solution but also by its difference derivatives appearing in the nonlinear terms of the equation. The stability estimates are obtained without any assumptions about the properties of the solution and nonlinear coefficients of the differential problem. Here we use restrictions only on input data (initial and boundary conditions and the right-hand side). The sufficient conditions of the shock wave generation is formulated for input data. For the Riemann problem two exact and stable difference schemes are analyzed.

scholarly journals A note on the difference schemes for hyperbolic-elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
A. Ashyralyev ◽  
G. Judakova ◽  
P. E. Sobolevskii
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Accuracy Difference ◽  
The Difference ◽  
The Stability

The nonlocal boundary value problem for hyperbolic-elliptic equationd2u(t)/dt2+Au(t)=f(t),(0≤t≤1),−d2u(t)/dt2+Au(t)=g(t),(−1≤t≤0),u(0)=ϕ,u(1)=u(−1)in a Hilbert spaceHis considered. The second order of accuracy difference schemes for approximate solutions of this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established.

scholarly journals Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Dumitru Baleanu
Keyword(s):  
Parabolic Equation ◽  
Fractional Calculus ◽  
Numerical Solution ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Implicit Schemes ◽  
The Stability

A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.

Stability of Finite-difference Schemes for Semilinear Multidimensional Parabolic Equations

2012 ◽  
Vol 12 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Bosko Jovanovic ◽  
Magdalena Lapinska-Chrzczonowicz ◽  
Aleh Matus ◽  
Piotr Matus
Keyword(s):  
Finite Difference ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Nonlinear Odes ◽  
The Stability ◽  
Initial Boundary Value

Abstract Abstract — We have studied the stability of finite-difference schemes approximating initial-boundary value problem (IBVP) for multidimensional parabolic equations with a nonlinear source of a power type. We have obtained simple sufficient input data conditions, in which the solutions of differential and difference problems are globally bounded for all t. It is shown that if these conditions are not satisfied, then the solution can blow-up (go to infinity) in finite time. The lower bound of the blow-up time has been determined. The stability of the difference solution has been proven. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear ODEs, Bihari-type inequalities and their discrete analogs.

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