Homotopy analysis method for heat radiation equations

Here, the homotopy analysis method (HAM), one of the newest analytical methods which is powerful and easy-to-use, is applied to solve heat transfer problems with high nonlinearity order. Also, the results are compared with the perturbation and numerical Runge–Kutta methods and homotopy perturbation method (HPM). Here, homotopy analysis method is used to solve an unsteady nonlinear convective–radiative equation containing two small parameters of ?₁ and ?₂. The homotopy analysis method contains the auxiliary parameter h¯, which provides us with a simple way to adjust and control the convergence region of solution series.     

Assessment of homotopy–perturbation and perturbation methods in heat radiation equations

One of the newest analytical methods to solve the nonlinear heat transfer equations is using both homotopy and perturbation methods in equations. Here, homotopy–perturbation method is applied to solve heat transfer problems with high nonlinearity order. The origin of using this method is the difficulties and limitations of perturbation or homotopy. It has been attempted to show the capabilities and wide-range applications of the homotopy–perturbation method in comparison with the previous ones in solving heat transfer problems. In this research, homotopy–perturbation method is used to solve an unsteady nonlinear convective-radiative equation and a nonlinear convective-radiative conduction equation containing two small parameters of ε₁ and ε₂.     

Analytical solution of natural convection of Darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by means of HAM

In this paper, fluid flow and heat transfer of vertical full cone embedded in porous media have been studied. Similarity solution for full cone subjected to wall temperature boundary conditions gives us a nonlinear ordinary differential equation which the obtained nonlinear ODE has been solved through homotopy analysis method (HAM). The obtained analytical solution in comparison with the numerical ones represents a remarkable accuracy. The results also indicate that HAM can provide us with a convenient way to control and adjust the convergence region. By the way we also calculate Nusselt number that is an important parameter in heat transfer by obtained analytical solution by HAM.     

He's homotopy perturbation method for two‐dimensional heat conduction equation: Comparison with finite element method

Heat conduction appears in almost all natural and industrial processes. In the current study, a two‐dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy‐based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20292     

An explicit solution for the combined heat and mass transfer by natural convection from a vertical wall in a non-Darcy porous medium

An analytic technique, namely the homotopy analysis method, is applied to solve the combined heat and mass transfer by natural convection adjacent to a vertical wall in a non-Darcy porous medium governed by a set of three fully coupled, highly nonlinear similarity equations. An explicit, totally analytic and uniformly valid solution is derived, which agrees well with numerical results.     

Semi-exact solution of elastic non-uniform thickness and density rotating disks by homotopy perturbation and Adomian's decomposition methods. Part I: Elastic solution

In this work, two powerful analytical methods, namely homotopy perturbation method (HPM) and Adomian's decomposition method (ADM), are introduced to obtain distributions of stresses and displacements in rotating annular elastic disks with uniform and variable thicknesses and densities. The results obtained by these methods are then compared with the verified variational iteration method (VIM) solution. He's homotopy perturbation method which does not require a “small parameter” has been used and a homotopy with an imbedding parameter p ∈ [0,1] is constructed. The method takes the full advantage of the traditional perturbation methods and the homotopy techniques and yields a very rapid convergence of the solution. Adomian's decomposition method is an iterative method which provides analytical approximate solutions in the form of an infinite power series for nonlinear equations without linearization, perturbation or discretization. Variational iteration method, on the other hand, is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. This study demonstrates the ability of the methods for the solution of those complicated rotating disk cases with either no or difficult to find fairly exact solutions without the need to use commercial finite element analysis software. The comparison among these methods shows that although the numerical results are almost the same, HPM is much easier, more convenient and efficient than ADM and VIM.     

A novel effective approach for solving nonlinear heat transfer equations

In this paper, a novel analytic technique, namely the Laplace transform new homotopy perturbation method (LTNHPM), is applied for solving the nonlinear differential equations arising in the field of heat transfer. This approach is a new modification to the homotopy perturbation method based on the Laplace transform. Unlike the previous approach implemented by the present authors for these problems, the present method does not consider the initial approximation as a power series. The nonlinear convective–radiative cooling equation and nonlinear equation of conduction heat transfer with the variable physical properties are chosen as illustrative examples. The exact solution has been found for the first case and for the others; results with remarkable accuracy have been achieved which verify the efficiency as well as accuracy of the presented approach. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20411     

The application of differential transformation method to nonlinear equations arising in heat transfer

In this paper two nonlinear heat transfer problems were solved by considering variable specific heat coefficient. The calculations are carried out by using differential transformation method (DTM) which is a semi-numerical-analytical solution technique. By using DTM, the nonlinear constrained governing equations are reduced to recurrence relations and related initial conditions are transformed into a set of algebraic equations. The principle of differential transformation is briefly introduced, and then applied for the aforementioned problems. The solutions are subsequently solved by a process of inverse transformation. The current results are then compared with those derived from the variational iteration method (VIM), homotopy perturbation method (HPM), perturbation method (PM) and the exact solutions in order to verify the accuracy of the proposed method. The findings reveal that the DTM can achieve more suitable results in predicting the solution of such problems.     

Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer

We consider one of the newest analytical methods, the Optimal Homotopy Asymptotic Method (OHAM), to solve nonlinear equations arising in heat transfer. Two specific applications are considered: cooling of a lumped system with variable specific heat and the temperature distribution equation in a thick rectangular fin radiation to free space. Results obtained by OHAM, which does not need small parameters are compared with numerical results and a very good agreement was found. This method provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. The results reveal that the proposed method is explicit, effective and easy to use.     

Solution of strongly nonlinear ordinary differential equations arising in heat transfer with optimal homotopy asymptotic method

An analytic approximate procedure, called optimal homotopy asymptotic method (OHAM) is considered for the solution of strongly nonlinear differential equations arising in heat transfer. Two particular applications are chosen: convecting-radiating cooling of a lumped system with variable specific heat and fins with temperature dependent surface heat flux. The effectiveness of the method, which is independent of the small parameter, is investigated by comparing the results obtained with the exact and the results already present in the literature.     

Approximate Analytical Solution of Stagnation Point Flow and Heat Transfer over an Exponential Stretching Sheet with Convective Boundary Condition

This study is concerned with the stagnation point flow and heat transfer over an exponential stretching sheet via an approximate analytical method known as optimal homotopy asymptotic method (OHAM). The governing partial differential equations are converted into ordinary nonlinear differential equations using similarity transformations available in the literature. The heat transfer problem is modeled using two‐point convective boundary condition. These equations are then solved using the OHAM approach. The effects of controlling parameters on the dimensionless velocity, temperature, friction factor, and heat transfer rate are analyzed and discussed through graphs and tables. It is found that the OHAM results match well with numerical results obtained by Runge–Kutta Fehlberg fourth‐fifth order method for different assigned values of parameters. The rate of heat transfer increases with the stretching parameter. It is also found that the stretching parameter reduces the hydrodynamic boundary layer thickness whereas the Prandtl number reduces the thermal boundary layer thickness.     

Optimal Homotopy Asymptotic Solutions for Nonlinear Ordinary Differential Equations Arising in Flow and Heat Transfer due to Nonlinear Stretching Sheet

Optimal homotopy asymptotic method (OHAM) is used to obtain solutions for nonlinear ordinary differential equations (ODEs) arising in fluid flow and heat transfer at a nonlinear stretching sheet. The solutions for skin friction and temperature gradient for some special cases are tabulated and compared with the available numerical results in the literature. Moreover, OHAM is found to be very easy to use and the technique could be used for solving coupled nonlinear systems of ordinary differential equations arising in science and engineering.     

Application of the homotopy perturbation method for the solution of inverse heat conduction problem

This paper deals with an application of the homotopy perturbation method for the solution of inverse heat conduction problem. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of functions describing the temperature and heat flux on the boundary, when the temperature measurements in the domain are known. Examples illustrating discussed application and confirming utility of this method in such a type of problem was also presented.     

Investigation of heat transfer in a geometry with variable cross section

In this paper, the thermal conductivity that plays an important role in heat transfer efficiency is investigated. He's homotopy perturbation method (HPM) and Adomian decomposition method (ADM) are applied to the nonlinear heat transfer equations and the comparison between results is illustrated within several plots. Here, thermal conductivity has been considered as a function of temperature. Also the variable heat generation has been taken into account. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20266 Copyright © 2009 Wiley Periodicals, Inc.     

The application of the homotopy perturbation method to one-phase inverse Stefan problem

In this paper, the application of the homotopy perturbation method for solving the inverse Stefan problem is presented. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of the functions describing temperature and heat flux on the boundary, when the position of the moving interface is known.     

Analytical study of heat and mass transfer in unsteady MHD radiant flow of Williamson nanofluid over stretching sheet with heat generation and chemical reaction

This paper presents the analytical study of heat and mass transfer in a two-dimensional time-dependent flow of Williamson nanofluid near a permeable stretching sheet by considering the effects of external magnetic field, viscous dissipation, Joule heating, thermal radiation, heat source, and chemical reaction. Suitable transformations are introduced to reformulate the governing equations and the boundary conditions convenient for computation. The resulting sets of nonlinear differential equations are then solved by the homotopy analysis method. The study on the effects of relevant parameters on fluid velocity, temperature, and concentration profiles is analyzed and presented in graphical and tabular forms. Upon comparison of the present study with respect to some other previous studies, a very good agreement is obtained. The study points out that the transfer of heat can substantially be enhanced by decreasing viscoelasticity of the fluid and the transfer of mass can be facilitated by increasing permeability of the stretching sheet.     

Effect of nonlinear thermal radiation on silver and copper water nanofluid flow due to a rotating disk with variable thickness in the presence of nonuniform heat source/sink using the homotopy analysis method

An analysis of a steady axisymmetric heat transfer nanofluid flow due to a rotating disk having variable thickness in the presence of nonlinear radiation and nonuniform heat source/sink is presented. Water with Copper (Cu) and Silver (Ag) nanoparticles are utilized in the investigation. The governing equations along with boundary conditions are solved using the homotopy analysis method. A parametric study of the physical parameters is done and results are displayed in the form of graphs. The findings indicate that nonlinear radiation has a significant effect on temperature as well as on wall heat transfer when compared with linear case, which is more useful in few engineering processes.     

Effect of thermal radiation on magnetohydrodynamic three‐dimensional motion of a nanofluid past a shrinking surface under the influence of a heat source

An analytical technique known as the homotopy analysis method is used to acquire solutions for magnetohydrodynamic 3‐D motion of a viscous nanofluid over a saturated porous medium with a heat source and thermal radiation. The governing nonlinear partial differential equations are changed to ordinary differential equations employing appropriate transformations. Validation of the present result is done with the help of error analysis for flow and temperature. The influences of pertinent parameters on momentum, energy, and Nusselt number are studied and discussed. The major findings are: the velocity of the nanofluid is affected by the nanoparticle volume fraction and the thickness of the thermal boundary layer becomes thinner and thinner subject to sink, whereas the effect is revered in case of the source.     

Semianalytical method of solution for solid phase diffusion in lithium ion battery electrodes: Variable diffusion coefficient

A semianalytical methodology based on the integral transform technique is proposed to solve the diffusion equation with concentration dependent diffusion coefficient in a spherical intercalation electrode particle. The method makes use of an integral transform pair to transform the nonlinear partial differential equation into a set of ordinary differential equations, which is solved with less computational efforts. A general solution procedure is presented and two illustrative examples are used to demonstrate the usefulness of this method for modeling of diffusion process in lithium ion battery electrode. The solutions obtained using the method presented in this study are compared to the numerical solutions.     

Comparison study of differential transform method with finite difference method for magnetoconvection in a vertical channel

In this paper, coupled nonlinear equations governing the flow for magnetoconvection in a vertical channel for open and short circuits are solved. The calculations are carried out by using differential transformation method (DTM) which is a semi‐numerical–analytical solution technique. By using DTM, the nonlinear constrained governing equations are reduced to recurrence relations and related initial conditions are transformed into a set of algebraic equations. The principle of differential transformation is briefly introduced, and then applied for the aforementioned problems. The current results are then compared with those derived from the finite difference method (FDM) and perturbation method (PM) in order to verify the accuracy of the proposed method. The findings reveal that the DTM can achieve more suitable results in predicting the solution of such problems. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21035