Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation

Mallory, Kristina; Gorder, Robert A. Van

doi:10.1007/s11075-013-9765-0

Cited by 15 publications

(15 citation statements)

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“…Here q ∈ [0, 1] is an embedding parameter, such that upon setting q = 0 we have Note that an entire literature exists for the case where h = −1, in which case the method is often referred to as the homotopy perturbation method (HPM); see [17][18][19]. On the other hand, when h is picked in a specific way to minimize the error in the approximate solution formed through truncation, the method is often referred to as the optimal homotopy analysis method (OHAM); see [9][10][11][12]. We shall not go into details of the HAM solution procedure for the sake of brevity; see [1,3,4,13] for details and many worked examples.…”

Section: Mathematical Underpinnings Of the Generalized Taylor Seriesmentioning

confidence: 99%

“…The HAM has proven useful for a variety of such problems [4][5][6][7][8], owing to the fact that it is unique among analytical or perturbation methods in that it gives a way to minimize the error of approximations by way of an auxiliary parameter, commonly referred to as a convergence control parameter. For instance, one may minimize the error or residual error of approximate solutions over all possible choices of this parameter, and this process is referred to as the optimal HAM (or, OHAM); see [9][10][11][12]. The HAM also gives one great freedom in selecting the form of the solutions via representation of solutions [1,13], since one has control over the type of basis functions employed in such a representation.…”

Section: Introductionmentioning

confidence: 99%

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On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

Gorder

2016

Numer Algor

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In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.

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Section: Mathematical Underpinnings Of the Generalized Taylor Seriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

Gorder

2016

Numer Algor

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“…This is essentially the essence of the so-called optimal HAM: the convergence control parameter h is optimally selected in order to minimize some measure of the error of solutions. For some examples of where this technique of minimizing error with respect to the convergence control parameter was successful, see [21,22,23,24,25,26,27].…”

Section: Derivation Of Vims From the Hammentioning

confidence: 99%

The variational iteration method is a special case of the homotopy analysis method

Gorder

2015

Applied Mathematics Letters

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The variational iteration method is a special case of the homotopy analysis method Robert A. Van Gorder PII: S0893-9659(15)00030-0 DOI: http://dx.Please cite this article as: R.A. Van Gorder, The variational iteration method is a special case of the homotopy analysis method, Appl. Math. Lett. (2015), http://dx. AbstractIn the present paper, we demonstrate that the variational iteration method (and all of its optimal analogues) are specific cases of the more general homotopy analysis method. To do so, we derive the variational iteration method starting with the homotopy analysis method. The optimal variational iteration method, which also appears in the literature, can be described completely within the context of the optimal homotopy analysis method. Alternately, the optimal homotopy analysis method can be used to construct more general iterative methods of the same ilk, which is potentially useful for solving partial differential equations.

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“…In minimum error method the optimal values of the convergence-control parameters are obtained by the minimum of average squared residual error. The choice of the optimal values of the convergence-control parameters by using minimum error method as defined by Liao [25] is often referred to as OHAM and has been applied in the literature (see [26][27][28][29][30]). Here, the optimal values for the convergence-control parameters are obtained through the minimum of total average squared residual error.…”

Section: Introductionmentioning

confidence: 99%

Series solutions of non-similarity boundary layer flows of nano-fluids over stretching surfaces

et al. 2014

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In this paper the convergent series solutions for the non-similarity flow of viscous fluid with nanoparticles are given. Fundamental equations employed in the mathematical modelling include the novel aspects of Brownian motion and thermophoresis. Non-similarity flow is induced by a stretching sheet with arbitrary velocity. The so-called homotopy analysis method (HAM) is applied to gain the convergent series solutions of the nonlinear partial differential equation. It is noticed that flow field, temperature and nanoparticle volume fraction profile are greatly influenced by the physical parameters such as Prandtl number, Brownian motion parameter, thermophoresis parameter and Lewis number. To the best of our knowledge, the present analysis seems to be a first attempt to non-similarity boundary layer flows of viscous fluids with nanoparticles.

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Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation

Cited by 15 publications

References 44 publications

On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

The variational iteration method is a special case of the homotopy analysis method

Series solutions of non-similarity boundary layer flows of nano-fluids over stretching surfaces

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