An iterative process based on homotopy and perturbation techniques

Mallil, El Hassan; Lahmam, Hassane; Damil, N.; Potier‐Ferry, Michel

doi:10.1016/s0045-7825(00)00198-5

Cited by 41 publications

(37 citation statements)

References 13 publications

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“…Let V 0 be a given trial solution of this non-linear problem. An approximated solution, in the form (2), has been deÿned by using a recently proposed high-order Newton algorithm [3,4]. The aim of this numerical test is to compare the convergence acceleration of sequence (2) using either MMPE or VPA.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…However, VPA seems to be more sensible to the choice of scalar product than MMPE. It seems that one can accelerate the convergence of the high order iterative algorithms proposed in Reference [3,4] without Gram-Schmidt orthonormalization by using the MMPE.…”

Section: Resultsmentioning

confidence: 99%

“…Introducing ( where P k ( ) is a polynomial of degree k. Second, the N − 1 polynoms P N −n ( ) (16n6N − 1) are replaced by N − 1 rational fractions having the same denominator N −1 ( ), as detailed in [3,4,10]. The ÿnal expression is given by…”

Section: Vectorial Padã E Approximantsmentioning

confidence: 99%

“…A class of high-order iterative algorithms has been introduced to solve linear [1,2] or nonlinear [3][4][5][6] problems. In this class of method, the basic idea is ÿrst to deÿne a homotopy transformation, second to search the solution of the problem as integro-power series with respect to the homotopy parameter :…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Damil

Cadou

Potier‐Ferry³

2004

Commun. Numer. Meth. Engng.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Vectorial Padã E Approximantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Damil

Cadou

Potier‐Ferry³

2004

Commun. Numer. Meth. Engng.

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“…Eventually, many analytical and numerical methods have been established and used to reach this goal. Some of the methods are; the perturbation method [1][2][3][4][5], the homotopy perturbation method [3][4][5][6][7], the delta perturbative method [8], the Modified Decomposition method [9][10][11][12][13][14][15][16], the Adomian Decomposition method [15][16][17][18][19][20][21][22], the Laplace Decomposition method [15,16,[22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Applications of the decomposition methods to some nonlinear partial differential equations

Gündoğdu¹,

Gözükızıl²

2018

ntmsci

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Abstract:In this paper, we consider the inhomogeneous Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation with its initial condition. The decomposition methods such as Adomian and Laplace are applied to this equation. Then, the solution satisfying the given initial condition is gained by using each method. To demonstrate these decomposition methods are how effective and convenient for solving the nonlinear partial differential equations(NLPDEs), solutions of some NLPDEs subjected to initial conditions are attained by these method. In the application of the method, it is clearly noticed that there is no need to convert the nonlinear terms into the linear ones due to the Adomian polynomials. In addition to this simplicity, it is seen that each method gives the same solution with a fast convergence by taking the advantage of the noise terms.

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A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

Cadou

Potier‐Ferry²

2009

Numer Methods Biomed Eng

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SUMMARYAn iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non-linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub-steps. In the first sub-step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub-step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well-known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non-linear solvers with few matrix factorizations and few iterations.

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An iterative process based on homotopy and perturbation techniques

Cited by 41 publications

References 13 publications

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Applications of the decomposition methods to some nonlinear partial differential equations

A solver combining reduced basis and convergence acceleration with applications to non‐linear elasticity

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