Notes on the homotopy analysis method: Some definitions and theorems

Liao, Shijun

doi:10.1016/j.cnsns.2008.04.013

Cited by 753 publications

(404 citation statements)

References 52 publications

(82 reference statements)

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“…The first is based on the Homotopy Analysis Method (HAM) [33,30,31] applied to the discretized version of the integral equation proposed by Dang [21]. The method starts from an initial guess and deforms it continuously in order to find better and better approximations of the solution of the integral equation.…”

Section: Introductionmentioning

confidence: 99%

Optimal execution with non-linear transient market impact

Curato

Gatheral

Lillo

2016

Quantitative Finance

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We study the problem of the optimal execution of a large trade in the presence of nonlinear transient impact. We propose an approach based on homotopy analysis, whereby a well behaved initial strategy is continuously deformed to lower the expected execution cost. We find that the optimal solution is front loaded for concave impact and that its expected cost is significantly lower than that of conventional strategies. We then consider brute force numerical optimization of the cost functional; we find that the optimal solution for a buy program typically features a few short intense buying periods separated by long periods of weak selling. Indeed, in some cases we find negative expected cost. We show that this undesirable characteristic of the nonlinear transient impact model may be mitigated either by introducing a bid-ask spread cost or by imposing convexity of the instantaneous market impact function for large trading rates.

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Section: Introductionmentioning

confidence: 99%

Optimal execution with non-linear transient market impact

Curato

Gatheral

Lillo

2016

Quantitative Finance

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“…Generally speaking, all traditional non-perturbation methods as we mentions above, can not guarantee the convergence of approximation series. In this section we introduce the basic idea of the HAM as presented in the book of Shijun Liao (Liao, 2003(Liao, , 2009(Liao, , 2012.…”

Section: Homotopy Analysis Methods (Ham)mentioning

confidence: 99%

Mathematical Modelling of Spray Combustion-Numerical and Analytical Analysis with application to Engineering Science

Nave¹,

Ajadi²,

Gol’dshtein³

2016

APR

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In the present paper we applied two well-known analytical method to the problem of thermal explosion of monodisperse and polydisperse fuel spray. The methods are the method of integral manifold (MIM) and the homotopy analysis method (HAM). The MIM method used as a basic tools for the analysis of SPS system of ordinary differential equations which means that the physical/ mathematical model should contain a small parameter in the governing equations. The HAM is always valid no matter whether there exist small physical parameters or not in contrast to the classical perturbation methods which requires the existence of a small parameter in the system (in general this is not the case). According to the theory of HAM, the convergence and the rate of solution series are dependent on the convergent control parameter ℏ. This means that this parameter gives one a convenient way to adjust and to control the convergent region of the solutions. γ (conventional parameter of Semenov's theory of thermal explosion), the final dimensionless adiabatic temperature of the thermally insulated system after explosion. This parameter is small compared with unity for most gaseous mixture due to the high exothermicity and activation energy of the chemical reactionwhere ϵ d is the emissivity of the droplet surface ϵ 1,2 dimensionless parameters, introduced for the first time (Gol'dshtein, Goldfarb, Shreiber, & Zinoviev, 1996) and describe the relations between the thermo physical properties of the gas and liquid phases

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“…To determine the optimal values of f , θ and φ , we use the so-called averaged residual error defined by. 35,36 …”

Section: Optimal Convergence-control Parametersmentioning

confidence: 99%

On the double diffusive convection flow of Eyring-Powell fluid due to cone through a porous medium with Soret and Dufour effects

Khan

Sultan

2015

AIP Advances

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This paper devotedly study the double diffusive Darcian convection flow of Eyring-Powell fluid from a cone embedded in a homogeneous porous medium with the effects of Soret and Dufour. Arising set of non-linear partial differential equations are transformed through a suitable self-similar transformation into a set of nonlinear ordinary differential equations. Further, the numerical and the analytical solutions of the governing equations are elucidated by using numerical method as well as non-perturbation scheme. Numerical values are presented through tables for the skin friction coefficients, Nusselt number and Sherwood number. The obtained results are validated by comparing the analytical results with previously published results obtained by bvp4c for the numerical values of physical quantities. The effect of various parameters on the velocity, temperature and concentration profiles is discussed and also shown graphically.

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Notes on the homotopy analysis method: Some definitions and theorems

Cited by 753 publications

References 52 publications

Optimal execution with non-linear transient market impact

Optimal execution with non-linear transient market impact

Mathematical Modelling of Spray Combustion-Numerical and Analytical Analysis with application to Engineering Science

On the double diffusive convection flow of Eyring-Powell fluid due to cone through a porous medium with Soret and Dufour effects

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