• KSCE Journal of Civil and Environmental Engineering Research
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• v.13 no.3
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• pp.129-138
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• 1993

In this study, the applicability of finite analytic method (FAM) is studied by selecting linearized-Burgers equation and Burgers equation which have convective and diffusive behaviors as the model equation of Navier-Stokes equations and by comparing numerical solution of finite difference method (FDM) and finite analytic method. The results are as follows. It is shown that the convergence of FAM for steady-state analytic solution of linearized-Burgers equation and Burgers equation is better than that of FDM under the same criteria. Also the accuracy of FAM for transient solution of Burgers equation is excellent. Especially, it is shown that oscillation phenomenon due to dispersion errors which occur according to the choice of grid size in FDM does not occur in FAM at all. So, it can be thought that FAM is numerically very stable scheme, which is free from dispersion errors.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.191-205
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• 2008

In this paper, we construct the analytic solutions and numerical solutions for a two-dimensional Riemann problem for Burgers' equation. In order to construct the analytic solution, we use the characteristic analysis with the shock and rarefaction base points. We apply the composite scheme suggested by Liska and Wendroff to compute numerical solutions. The result is coincident with our analytic solution. This demonstrates that the composite scheme works pretty well for Burgers' equation despite of its simplicity.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.81-108
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• 2003

We consider the Korteweg-de Vries-Burgers (KdVB) equation on the domain [0,11]. We derive a control law which guarantees $L^2$-global exponential stability, $H^3$-global asymptotic stability, and $H^3$-semiglobal exponential stability.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.685-695
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• 2020

In this paper, we used modified tanh-coth method, combined with Riccati equation and secant hyperbolic ansatz to construct abundantly many real and complex exact travelling wave solutions to KdV-Burgers (KdVB) equation with forcing term. The real part is the sum of the shock wave solution of a Burgers equation and the solitary wave solution of a KdV equation with forcing term, while the imaginary part is the product of a shock wave solution of Burgers with a solitary wave travelling solution of KdV equation. The method gives more solutions than the previous methods.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.4
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• pp.293-305
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• 2009

In this article, we study a reduced-order modelling for distributed feedback control problem of the Burgers equations. Brief review of the centroidal Voronoi tessellation (CVT) are provided. A weighted (nonuniform density) CVT is introduced and low-order approximate solution and compensator-based control design of Burgers equation is discussed. Through weighted CVT (or CVT-nonuniform) method, obtained low-order basis is applied to low-order functional gains to design a low-order controller, and by using the low-order basis order of control modelling was reduced. Numerical experiments show that a solution of reduced-order controlled Burgers equation performs well in comparison with a solution of full order controlled Burgers equation.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.637-645
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• 2016

In this paper, we study dynamical Bifurcation of the Burgers-Fisher equation. We show that the equation bifurcates an invariant set ${\mathcal{A}}_n({\beta})$ as the control parameter ${\beta}$ crosses over $n^2$ with $n{\in}{\mathbb{N}}$. It turns out that ${\mathcal{A}}_n({\beta})$ is homeomorphic to $S^1$, the unit circle.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.573-581
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• 2008

In this paper, a finite difference method for a Cauchy problem of RLW-Burgers equation was considered. Although the equation is not energy conservation, we have given its the energy conservative finite difference scheme with condition. Convergence and stability of the difference solution were proved. Numerical results demonstrate that the method is efficient and reliable.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.43-55
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• 2007

In this paper, an $H^1-Galerkin$ mixed finite element method is used to approximate the solution as well as the flux of Burgers' equation. Error estimates have been derived. The results of the numerical experiment show the efficacy of the mixed method and justifies the theoretical results obtained in the paper.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.193-213
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• 2017

We investigate an efficient approximation of solution to stochastic Burgers equation driven by an additive space-time noise. We discuss existence and uniqueness of a solution through the Orstein-Uhlenbeck (OU) process. To approximate the OU process, we introduce the Karhunen-$Lo{\grave{e}}ve$ expansion, and sparse grid stochastic collocation method. About spatial discretization of Burgers equation, two separate finite element approximations are presented: the conventional Galerkin method and Galerkin-conservation method. Numerical experiments are provided to demonstrate the efficacy of schemes mentioned above.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.29-51
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• 2004

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers' equation. This method is based upon a space-time variational form of Burgers' equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates in $L^2(\Omega)$ are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.