An approximation of the analytical solution of some nonlinear heat transfer equations: a survey by using Homotopy analysis method

Abstract In this letter, the approximate solution of nonlinear heat diffusion and heat transfer and also the energy balance for a differential fin element are developed via Homotopy Analysis Method HAM. This method is a strong and easy-to-use analytic tool for investigating nonlinear problems, which does not need small parameters. 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. By suitable choice of the auxiliary parameter h , we can obtain reasonable solutions for large modulus. In this study, we compare obtained results through HAM results, with those of homotopy perturbation method and the exact solutions. The first differential equation to be solved is a straight fin with a temperature-dependent thermal conductivity and the second one is the modeling equation of a cooling Lumped system with variable specific heat

Higher-order approximation of cubic–quintic duffing model

We apply an Artificial Parameter Lindstedt-Poincaré Method (APL-PM) to find improved approximate solutions for strongly nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution which makes it a unique solution. It is demonstrated that this method works very well for the whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreement of the approximate frequencies with the exact one has been observed and discussed. Moreover, it is not limited to the small parameter such as in the classical perturbation method. Interestingly, This study revealed that the relative error percentage in the second-order approximate analytical period is less than 0.042% for the whole parameter values. In addition, we compared this analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations

Higher-order approximation of cubic–quintic duffing model

We apply an Artificial Parameter Lindstedt-Poincaré Method (APL-PM) to find improved approximate solutions for strongly nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution which makes it a unique solution. It is demonstrated that this method works very well for the whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreement of the approximate frequencies with the exact one has been observed and discussed. Moreover, it is not limited to the small parameter such as in the classical perturbation method. Interestingly, This study revealed that the relative error percentage in the second-order approximate analytical period is less than 0.042% for the whole parameter values. In addition, we compared this analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations

Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation

Two new analytical methods to solve nonlinear heat transfer equations are homotopy perturbation method and homotopy analysis method. Here, homotopy analysis method, which gives us a vast freedom to choose the answer type, is applied to solve nonlinear heat transfer differential equations and analytical results are compared with those of HPM and the numerical results. In this study, the procedure of HAM is applied to two cases in different ways according to the physics of the target problem. Comparing the two methods, our attention is focused on the results accuracy; and applicability of different methods in many cases with different limitation is studied. In the two examples of this paper, the effect of small parameter increaser on the accuracy of the analytical results of two methods also has been studied. The first differential equation is the modeling equation of a cooling lumped system with combined convection and radiation. The second one is the modeling equation of heat transfer with conduction in a slab of thermal dependent conductivity. © 2007 Elsevier B.V. All rights reserved

Analytical study of natural convection in high Prandtl number

In case of natural convection modeling, when Boussinesq assumption is used, we encounter coupled nonlinear differential equations. In this work, the authors have modeled natural heat convection by implementing one of the newest analytical methods of solving nonlinear differential equations called homotopy analysis method (HAM), which gives us a vast freedom to choose the answer type. We have used an iterating analytical method in order that cope with nonlinearity. Also, we apply some provisions because of particular difficulties that are caused by coupling problem. A new adapting boundary condition is proposed in this work that is based on an initial guess and then it is developed to the solution expression. We must notice that HAM is applied to our case study according to the physics of the target problem. © 2008 Elsevier Ltd. All rights reserved

Approximate Analysis of MHD Squeeze Flow between Two Parallel Disks with Suction or Injection by Homotopy Perturbation Method

An analysis has been performed to study magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid. We investigate the combined effect of inertia, electromagnetic forces, and suction or injection. With the introduction of a similarity transformation, the continuity and momentum equations governing the squeeze flow are reduced to a single, nonlinear, ordinary differential equation. An approximate solution of the equation subject to the appropriate boundary conditions is derived using the homotopy perturbation method (HPM) and compared with the direct numerical solution (NS). Results showing the effect of squeeze Reynolds number, Hartmann number and the suction/injection parameter on the axial and radial velocity distributions are presented and discussed. The approximate solution is found to be highly accurate for the ranges of parameters investigated. Because of its simplicity, versatility and high accuracy, the method can be applied to study linear and nonlinear boundary value problems arising in other engineering applications