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

Heat transfer study of convective fin with temperature-dependent internal heat generation by hybrid block method

Purpose: In this study, an implicit one‐step hybrid block method using two off‐step points involving the presence of a third derivative for solving second‐order boundary value problems are subjected to Dirichlet‐mixed conditions. Methodology: To derive this method, the approximate power series solution is interpolated at urn:x-wiley:10992871:media:htj21428:htj21428-math-0001 while its second and third derivatives are collocated at all points urn:x-wiley:10992871:media:htj21428:htj21428-math-0002 on the integrated interval of approximation. Findings: The new derived method not only performs better compared with the existing methods when solving the same problems but also obtains better properties of the numerical method. Afterward, the proposed method is applied to solve the problem of a convective fin with temperature‐dependent internal heat generation. The effects of various physical parameters on temperature distribution are also examined

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