scholarly journals Numerical Solution of Stochastic Hyperbolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Necmettin Aggez ◽  
Maral Ashyralyyewa
Keyword(s):  
Difference Scheme ◽  
Numerical Solution ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergence Estimate ◽  
Numerical Examples ◽  
Convergence Estimates ◽  
The Difference ◽  
Initial Boundary Value

A two-step difference scheme for the numerical solution of the initial-boundary value problem for stochastic hyperbolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of the difference scheme are obtained for different initialboundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

scholarly journals Conservative Linear Difference Scheme for Rosenau-KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinsong Hu ◽  
Youcai Xu ◽  
Bing Hu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Difference ◽  
Second Order Convergence ◽  
Initial Boundary Value ◽  
Theoretical Results

A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed. The difference scheme simulates two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

scholarly journals An Average Linear Difference Scheme for the Generalized Rosenau-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Maobo Zheng ◽  
Jun Zhou
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Discrete Energy ◽  
The Difference ◽  
Initial Boundary Value ◽  
Theoretical Results

An average linear finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-KdV equation is proposed. The existence, uniqueness, and conservation for energy of the difference solution are proved by the discrete energy norm method. It is shown that the finite difference scheme is 2nd-order convergent and unconditionally stable. Numerical experiments verify that the theoretical results are right and the numerical method is efficient and reliable.

scholarly journals An Approximation of Semigroups Method for Stochastic Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Mehmet Emin San
Keyword(s):  
Difference Scheme ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Single Step ◽  
Nonlocal Boundary Value Problem ◽  
Convergence Estimate ◽  
Numerical Examples ◽  
Convergence Estimates ◽  
The Difference

A single-step difference scheme for the numerical solution of the nonlocal-boundary value problem for stochastic parabolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In application, the convergence estimates for the solution of the difference scheme are obtained for two nonlocal-boundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

Exact Difference Schemes for Hyperbolic Equations

2007 ◽  
Vol 7 (4) ◽  
pp. 341-364 ◽  
Author(s):  
P. Matus ◽  
A. Kołodyńska
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Difference Scheme ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Cauchy Problem ◽  
Exact Difference ◽  
Constant Coefficients ◽  
Initial Boundary Value

AbstractIn the present paper, an exact difference scheme for the initial boundary- value problem of the third kind for an inhomogeneous hyperbolic equation of the second order with constant coefficients has been constructed on ordinary rectangular grids with constant space and time steps, where the Courant number γ=1. Later we proved a priori estimates of the stability in energy norm. For a quasi-linear wave equation on the moving characteristic grid a difference scheme has been constructed, which has the second order of approximation for the initial boundary-value problem and is exact for the Cauchy problem. The computational results for smooth functions and for a weak solution confirm the high accuracy of the introduced algorithm. We have also constructed exact difference schemes for the Cauchy problem for a system of two hyperbolic equations of the first order with constant coefficients on grids with constant space and time steps. Stability in energy norm for one of the constructed schemes has been proved. Using a method analogous to that used for the nonlinear wave equation a difference scheme for a nonlinear gas dynamic system has been constructed.

scholarly journals Conservative Difference Scheme for Generalized Rosenau-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yan Luo ◽  
Youcai Xu ◽  
Minfu Feng
Keyword(s):  
Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Discrete Analogue ◽  
The Difference ◽  
Second Order Convergence ◽  
Initial Boundary Value ◽  
Conservative Difference ◽  
Theoretical Results

A conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of generalized Rosenau-KdV equation is proposed. The difference scheme shows a discrete analogue of the main conservation law associated to the equation. On the other hand the scheme is implicit and stable with second order convergence. Numerical experiments verify the theoretical results.

scholarly journals Stability with respect to coefficients of solution of difference schemes approximating initial boundary-value problems for semi-linear hyperbolic equations

2020 ◽  
Vol 64 (2) ◽  
pp. 135-143
Author(s):  
P. P. Matus ◽  
S. V. Lemeshevsky
Keyword(s):  
Difference Scheme ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Differential Problem ◽  
Domain Of Existence ◽  
The Stability ◽  
The One ◽  
Initial Boundary Value

The stability with respect to coefficients of solution of a difference scheme approximating the initial boundary-value problem for the one-dimensional semi-linear hyperbolic equation is studied. The estimates of the solutions of both differential and difference problems are obtained. In the domain of existence of the solution, the estimates for perturbation of the solution of a difference scheme with respect to perturbation of the coefficients of the equation are obtained. These estimates are consistent with the estimates for the differential problem. In all cases, the method of energy inequalities, the Bihari inequality and its mesh analogue are used.

scholarly journals On the stable difference scheme for the time delay telegraph equation

2020 ◽  
Vol 99 (3) ◽  
pp. 105-119
Author(s):  
A. Ashyralyev ◽  
◽  
K. Turk ◽  
D. Agirseven ◽  
◽  
...  
Keyword(s):  
Time Delay ◽  
Difference Scheme ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Telegraph Equation ◽  
Stability Estimates ◽  
The Difference ◽  
Initial Boundary Value

The stable difference scheme for the approximate solution of the initial boundary value problem for the telegraph equation with time delay in a Hilbert space is presented. The main theorem on stability of the difference scheme is established. In applications, stability estimates for the solution of difference schemes for the two type of the time delay telegraph equations are obtained. As a test problem, one-dimensional delay telegraph equation with nonlocal boundary conditions is considered. Numerical results are provided.

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