Application of Variational Iteration and Homotopy-Perturbation Methods to Nonlinear Heat Transfer Equations with Variable Coefficients

Khaleghi, Hassan; Ganji, D.D.; Sadighi, Ali

doi:10.1080/10407780601112878

Cited by 33 publications

(15 citation statements)

References 27 publications

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“…There exist several approaches to solve Eq. (1a), among them: orthogonal collocation method [6], Green function method [9], perturbation method [8,10,11], variational iteration method [8], homotopy-perturbation method [8], direct variational method [12], the least squares method [13], Networks models [14], iterative solutions with the solution of the linear problem as initial approximation [15], Finite difference solutions [5], Lattice Boltzmann method [16], numerical solutions [17], etc.…”

Section: Problem Formulationmentioning

confidence: 99%

On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity

Fabre

Hristov

2016

Heat Mass Transfer

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relationship (Eqs. 9, 10) (m 2 /s) bCoefficient in Eq. (24b) which should be defined trough the initial condition δ (t = 0) = 0 C p Specific heat capacity (J/kg) E L (n, β, t) Squared-error function in accordance with the Langford criterion (Eq. 34) E LT (n, β, t) Squared-error function in accordance with the Langford criterion (Eq. 35) E Mq (p, β, t) Squared-error function in accordance with the Langford criterion and fixed flux BC problem (Eq. 76) e LT (n, β, t) Squared-error sub-function in accordance with the Langford criterion (Eq. 35) e Lq (p, β) Squared-error sub-function in accordance with the Langford criterion and the fixed flux BC problem k Thermal conductivity (W/mK) k (T) Temperature-dependent thermal conductivity (W/mK) k 0 Thermal conductivity of the linear problem (β = 0) (W/mK) m Dimensionless parameter of the nonlinearity (power-law diffusivity) n Dimensionless exponent of the parabolic profile p Dimensionless exponent of the parabolic profile of the assumed profile used to solve Eq. (48) (m) Abstract Closed form approximate solutions to nonlinear transient heat conduction with linearly temperaturedependent thermal diffusivity have been developed by the integral-balance integral method under transient conditions. The solutions uses improved direct approaches of the integral method and avoid the commonly used linearization by the Kirchhoff transformation. The main steps in the new solutions are improvements in the integration technique of the double-integration technique and the optimization of the exponent of the approximate parabolic profile with unspecified exponent. Solutions to Dirichlet and Neumann boundary condition problems have been developed as examples by the classical Heat-balance integral method (HBIM) and the Double-integration method (DIM). Additional examples with HBIM and DIM solutions to cases when the Kirchhoff transform is initially applied have been developed. List of symbolsThermal diffusivity (m 2 /s) a 0 Thermal diffusivity of the linear problem (β = 0) (m 2 /s) a p Thermal diffusivity coefficient in the case of power-law non-linear

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Section: Problem Formulationmentioning

confidence: 99%

On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity

Fabre

Hristov

2016

Heat Mass Transfer

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“…In this letter, we apply the homotopy perturbation method [26][27][28][29][30][31][32][33] to the discussed problem. To illustrate the basic ideas of the new method, we consider the following nonlinear differential equation.…”

Section: Basic Idea Of Homotopy Perturbation Methodsmentioning

confidence: 99%

Application of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction Equations

Tolou¹,

Ganji²,

Hosseini³

et al. 2007

TOMECHJ

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Abstract:In this Paper, Homotopy-Perturbation method (HPM) which is one of the most recent approximate analytical solutions is implemented to solve diffusion-convection-reaction equations (DCRE). The calculations are carried out for two different types of DCRE's such as the Black-Scholes equation used in financial market option pricing and FokkerPlanck equation from plasma physics. The behavior of the approximate solutions of the distribution functions are shown graphically and are compared with those obtained by other theories (e.g. the Adomian decomposition method, ADM). It shows that the numerical results of these methods are the same; while HPM is much easier, more convenient and more efficient than ADM.

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“…[7][8][9]19]) were successfully employed to find approximate analytical solutions for some well-known nonlinear heat transfer problems modeled by using both ordinary differential equations ( [8][9][10][11][12][13][14][15][16]) and partial differential equations ( [17][18][19]). In the present paper we will study several problems from the first category, namely: -1 The cooling of a lumped system involving combined modes of convection and radiation heat transfer ( [8,[10][11][12][13][14][15]) -2 The heat transfer with conduction in a slab of a material with temperature dependent thermal conductivity ( [14,15,9]) -3 The temperature distribution equation in a thick rectangular fin radiating to free space ( [12,13,8])…”

Section: Introductionmentioning

confidence: 99%

Approximate polynomial solutions for nonlinear heat transfer problems using the squared remainder minimization method

Căruntu

Bota

2012

International Communications in Heat and Mass Transfer

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Application of Variational Iteration and Homotopy-Perturbation Methods to Nonlinear Heat Transfer Equations with Variable Coefficients

Cited by 33 publications

References 27 publications

On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity

On the integral-balance approach to the transient heat conduction with linearly temperature-dependent thermal diffusivity

Application of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction Equations

Approximate polynomial solutions for nonlinear heat transfer problems using the squared remainder minimization method

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