scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

Source identification problems for hyperbolic differential and difference equations

2019 ◽  
Vol 27 (3) ◽  
pp. 301-315 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Fathi Emharab
Keyword(s):  
Difference Scheme ◽  
Source Identification ◽  
Identification Problem ◽  
Stability Estimates ◽  
Identification Problems ◽  
Order Of Accuracy ◽  
One Dimensional ◽  
First Order ◽  
Accuracy Difference ◽  
The Difference

Abstract In the present study, a source identification problem for a one-dimensional hyperbolic equation is investigated. Stability estimates for the solution of the source identification problem are established. Furthermore, a first-order-of-accuracy difference scheme for the numerical solution of the source identification problem is presented. Stability estimates for the solution of the difference scheme are established. This difference scheme is tested on an example, and some numerical results are presented.

scholarly journals Existence and uniqueness results for an inverse problem for a semilinear equation with final overdetermination

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 847-858 ◽  
Author(s):  
Ali Sazaklioglu ◽  
Abdullah Erdogan ◽  
Allaberen Ashyralyev
Keyword(s):  
Banach Space ◽  
Inverse Problem ◽  
Difference Scheme ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Order Of Accuracy ◽  
First Order ◽  
Existence And Uniqueness Results ◽  
Semilinear Equation ◽  
Uniqueness Results

In the present paper, unique solvability of a source identification inverse problem for a semilinear equation with a final overdetermination in a Banach space is investigated. Moreover, the first order of accuracy Rothe difference scheme is presented for numerically solving this problem. The existence and uniqueness result for this difference scheme is given. The efficiency of the proposed method is evaluated by means of computational experiments.

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.

Propagator Method for Numerical Solution of Convective Fisher Equation

2011 ◽  
Vol 222 ◽  
pp. 387-390
Author(s):  
Daiga Zaime ◽  
Janis S. Rimshans ◽  
Sharif E. Guseynov
Keyword(s):  
Difference Scheme ◽  
Iteration Process ◽  
Fisher Equation ◽  
Implicit Difference Scheme ◽  
Nonlinear Convection ◽  
First Order ◽  
Truncation Errors ◽  
Propagator Method ◽  
Left And Right ◽  
The Difference

Propagator numerical method was developed as an effective tool for modeling of linear advective dispersive reactive (ADR) processes [1]. In this work implicit propagator difference scheme for Fisher equation with nonlinear convection (convective Fisher equation) is elaborated. Our difference scheme has truncation errors of the second order in space and of the first order in time. Iteration process for implicit difference scheme is proposed by introducing forcing terms in the left and right sides of the difference equation. Convergence and stability criterions for the elaborated implicit propagator difference scheme are obtained.

scholarly journals Generation and Analysis of a New Implicit Difference Scheme for the Korteweg-de Vries Equation

2018 ◽  
Vol 173 ◽  
pp. 03006
Author(s):  
Yuri Blinkov ◽  
Vladimir Gerdt ◽  
Konstantin Marinov
Keyword(s):  
Difference Scheme ◽  
Vries Equation ◽  
Implicit Difference Scheme ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
Algorithmic Approach ◽  
De Vries ◽  
Volume Method ◽  
Modified Equation ◽  
Two Parameter

In this paper we apply our computer algebra based algorithmic approach to construct a new finite difference scheme for the two-parameter form of the Korteweg-de Vries equation. The approach combines the finite volume method, numerical integration and difference elimination. Being implicit, the obtained scheme is consistent and unconditionally stable. The modified equation for the scheme shows that its accuracy is of the second order in each of the mesh sizes. Using exact one-soliton solution, we compare the numerical behavior of the scheme with that of the other two schemes known in the literature and having the same order of accuracy. The comparison reveals numerical superiority of our scheme.

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

FINITE-DIFFERENCE METHOD OF ORDER SIX FOR THE TWO-DIMENSIONAL STEADY AND UNSTEADY BOUNDARY-LAYER EQUATIONS

2005 ◽  
Vol 16 (05) ◽  
pp. 757-780 ◽  
Author(s):  
A. A. SALAMA ◽  
A. A. MANSOUR
Keyword(s):  
Boundary Layer ◽  
Numerical Experiments ◽  
Two Dimensional ◽  
Unsteady Boundary Layer ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
High Order Method ◽  
Boundary Layer Equations ◽  
Special Cases ◽  
Blasius Equation

In this article, we propose a high order method for solving steady and unsteady two-dimensional laminar boundary-layer equations. This method is convergent of sixth-order of accuracy. It is shown that this method is unconditionally stable. The unsteady separated stagnation point flow, the Falkner–Skan equation and Blasius equation are considered as special cases of these equations. Numerical experiments are given to illustrate our method and its convergence.

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 An Unconditionally Stable Parallel Difference Scheme for Telegraph Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
A. Borhanifar ◽  
Reza Abazari
Keyword(s):  
Difference Scheme ◽  
Domain Decomposition ◽  
Numerical Experiments ◽  
Telegraph Equation ◽  
Unconditional Stability ◽  
Implicit Method ◽  
Alternating Group ◽  
Unconditionally Stable ◽  
Parallel Difference ◽  
Stability And Accuracy

We use an unconditionally stable parallel difference scheme to solve telegraph equation. This method is based on domain decomposition concept and using asymmetric Saul'yev schemes for internal nodes of each sub-domain and alternating group implicit method for sub-domain's interfacial nodes. This new method has several advantages such as: good parallelism, unconditional stability and better accuracy than original Saul'yev schemes. The details of implementation and proving stability are briefly discussed. Numerical experiments on stability and accuracy are also presented.

scholarly journals Stability Analysis of the Explicit Difference Scheme for Richards Equation

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 352
Author(s):  
Fengnan Liu ◽  
Yasuhide Fukumoto ◽  
Xiaopeng Zhao
Keyword(s):  
Difference Scheme ◽  
Numerical Experiments ◽  
Richards Equation ◽  
Induction Method ◽  
Explicit Difference Scheme ◽  
Time Step ◽  
The Difference ◽  
The Stability ◽  
Forward Euler ◽  
Parabolic Term

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.

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