A lumped Galerkin method for solving the Burgers equation

Kutluay, S.; Esen, Alaattin

doi:10.1080/00207160412331286833

Cited by 25 publications

(22 citation statements)

References 4 publications

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“…The illustrations show the HAM is numerically more accurate than the conventional numerical method in Refs. [1][2][3], also it is easier to use. Results obtained were given in Tables 1-3 and compared with results of the B-spline finite element method [1], the lumped Galerkin method [2] and the modified Adomian de- composition method [3].…”

Section: Convergence Of the Series Solutionmentioning

confidence: 96%

“…Nonlinear phenomena play a crucial role in applied mathematics and physics. The importance of obtaining the exact or approximate solutions of PDEs in physics and mathematics, it is still a hot spot to seek new methods to obtain new exact or approximate solutions [1][2][3][4][5]. For that different methods have been put forward to seek various exact solutions of multifarious physical models described by nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

“…(1) u t + uu x − υu xx = 0, 0 < x < 1, t > 0, subject to the boundary conditions (2) u(0, t) = u(1, t) = 0, t >0, and the initial condition (3) u(x, 0) = sin πx, is known as Burgers' equation. Burgers' model of turbulence is a very important fluid dynamic model and the study of this model and the theory of shock waves has been considered by many authors both for conceptual understanding of a class of physical flows and for testing various numerical methods.…”

Section: Introductionmentioning

confidence: 99%

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On numerical solution of Burgers' equation by homotopy analysis method

İnç

2008

Physics Letters A

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In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions.

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Section: Convergence Of the Series Solutionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On numerical solution of Burgers' equation by homotopy analysis method

İnç

2008

Physics Letters A

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“…Nonlinear phenomena play a crucial role in applied mathematics and physics. The importance of obtaining the exact or approximate solutions of PDEs in physics and mathematics is still a hot topic as regards seeking new methods for obtaining new exact or approximate solutions [2][3][4][5]. For that purpose, different methods have been put forward for seeking various exact solutions of multifarious physical models described using nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method

Abazari

Borhanifar

2010

Computers & Mathematics with Applications

106

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a b s t r a c tIn this paper, the Differential Transformation Method (DTM) is employed to obtain the numerical/analytical solutions of the Burgers and coupled Burgers equations. We begin by showing how the differential transformation method applies to the linear and nonlinear parts of any PDE and give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations. We also compare it against three famous methods, namely the homotopy perturbation method, the homotopy analysis method and the variational iteration method. These results show that the technique introduced here is accurate and easy to apply.

show abstract

“…Nonlinear phenomena play a crucial role in applied mathematics and physics. The importance of obtaining the exact or approximate solutions of PDEs in physics and mathematics is still a hot topic as regards seeking new methods for obtaining new exact or approximate solutions [2][3][4][5]. For that purpose, different methods have been put forward for seeking various exact solutions of multifarious physical models described using nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%