Numerical study of modified Adomian's method applied to Burgers equation

Basto, Mário; Semião, Viriato; Calheiros, Francisco

doi:10.1016/j.cam.2006.09.001

Cited by 31 publications

(21 citation statements)

References 58 publications

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“…Adomian's decomposition method [1][2][3][4]8,10,11,16,24,26,27,40] defines the solutions by the infinite series…”

Section: Adomian's Decomposition Methodsmentioning

confidence: 99%

“…Various powerful methods for obtaining the exact solutions for the nonlinear partial differential equations have been proposed (see, for example ). Among of these methods are Bäcklund transformation method [12,[31][32][33], the inverse scattering transform method , extended tanh method [9][10][11][12][13], Adomian Pade approximation [28], variational iteration method [5,6,9,17,22,30], homotopy perturbation method [18][19][20]23,40,41], various Lindstedt-Poincare methods [21,22], Adomian decomposition method [1][2][3][4]8,10,11,16,24,26,27,40], F-expansion method [34], exp-function method [35], the Jacobi elliptic function expansion method [37], the complex hyperbolic function method [39] and so on. The objective of this paper is to use the modified variational iteration method (MVIM) to solve the generalized nonlinear Hirota Satsuma coupled KdV equations [14,15,37,38] …”

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confidence: 99%

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On using the modified variational iteration method for solving the nonlinear coupled equations in the mathematical physics

Zayed

Rahman

2010

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This paper applied the modified variational iteration method to the nonlinear coupled partial differential equations via the generalized nonlinear Hirota Satsuma coupled KdV equations, the nonlinear coupled Kortewge-de Vries KdV equations and the nonlinear shallow water equations together with the initial conditions. The proposed modification is made by introducing Adomian's polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discritization, liberalization, perturbation, or restrictive assumptions.

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“…Adomian's decomposition method [1][2][3][4]8,10,11,16,24,26,27,40] defines the solutions by the infinite series…”

Section: Adomian's Decomposition Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

On using the modified variational iteration method for solving the nonlinear coupled equations in the mathematical physics

Zayed

Rahman

2010

Ricerche mat.

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“…Following [2], the variant with µ = 0.1 is calculated. Two versions of the scheme (35) are tested : with σ = 1 (implicit) and with σ = 0.5 (Crank-Nicolson).…”

Section: Numerical Testmentioning

confidence: 99%

Discrete-analytical methods for the implementation of variational principles in environmental applications

Penenko

Tsvetova

2009

Journal of Computational and Applied Mathematics

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a r t i c l e i n f o MSC: 58E30 76R99 49Q12 81T80 65M06 65M32 65N99 65K10 Keywords: Variational principle Convection-diffusion Monotonic scheme Discrete-analytical approximation Numerical environment modelling Adjoint sensitivity problem Direct and inverse modelling a b s t r a c tA new method of constructing numerical schemes on the base of a variational principle for models including convection-diffusion operators is proposed. An original element is the use of analytical solutions of local adjoint problems formulated for the operators of convectiondiffusion within the framework of the splitting technique. This results in numerical schemes which are absolutely stable, monotonic, transportive, and differentiable with respect to the state functions and parameters of the model. Artificial numerical diffusion is avoided due to the analytical solutions. The variational technique provides strong consistency between the numerical schemes of the main and adjoint problems. A theoretical study of the new class of schemes is given. The quality of the numerical approximations is demonstrated by an example of the non-linear Burgers equation. These new schemes enhance our variational methodology of environmental modelling. As one of the environmental applications, an inverse problem of risk assessment for Lake Baikal is presented.

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“…The function 0 ( ) gives the function satisfies the Burgers equation at time = 0. Some previous studies on the Burgers equation have been conducted, for example, see [4][5][6][7]. Staggered grid numerical methods have been active research areas for solving conservation law problems.…”

Section: Introductionmentioning

confidence: 99%