Homotopy Analysis Method

In this chapter, the analytic solution of nonlinear partial differential equations arising in heat transfer is obtained using the newly developed analytic method, namely the Homotopy Analysis Method (HAM). The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.

Homotopy Perturbation Method

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. 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. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

Assessment of Homotopy Perturbation and Variational Iteration Methods in Heat Transfer Equations

We explained VIM and HPM methods in Chapter 1 and Chapter 2. In this chapter, we compare these methods for nonlinear heat transfer problems.

Akbari-Ganji's Method

The nonlinear partial differential equation governing on the mentioned system has been investigated by a simple and innovative method which we have named it Akbari-Ganji's Method or AGM. It is notable that this method has been compounded by Laplace transform theorem in order to covert the partial differential equation governing on the afore-mentioned system to an ODE and then the yielded equation has been solved conveniently by this new approach (AGM). One of the most important reasons of selecting the mentioned method for solving differential equations in a wide variety of fields not only in heat transfer science but also in different fields of study such as solid mechanics, fluid mechanics, chemical engineering, etc. in comparison with the other methods is as follows: Obviously, according to the order of differential equations, we need boundary conditions so in the case of the number of boundary conditions is less than the order of the differential equation, this method can create additional new boundary conditions in regard to the own differential equation and its derivatives. Therefore, a solution with high precision will be acquired. With regard to the afore-mentioned explanations, the process of solving nonlinear equation(s) will be very easy and convenient in comparison with the other methods.

Optimal Homotopy Analysis Method

Optimal homotopy analysis method is a powerful tool for nonlinear differential equations. In this method, the convergence of the series solutions is controlled by one or more parameters which can be determined by minimizing a certain function. There are several approaches to determine the optimal values of these parameters, which can be divided into two categories, i.e., global optimization approach and step-by-step optimization approach. In the global optimization approach, all the parameters are optimized simultaneously at the last order of approximation. However, this process leads to a system of coupled, nonlinear algebraic equations in multiple variables which are very difficult to solve. In the step-by-step approach, the optimal values of these parameters are determined sequentially, that is, they are determined one by one at different orders of approximation. In this way, the computational efficiency is significantly improved, especially when high order of approximation is needed. In this chapter, we provide extensive examples in heat transfer equations.

Adomian's Decomposition Method

In this chapter, we apply Adomian's Decomposition Method to find appropriate solutions of heat and mass transfer in the two-dimensional and axisymmetric unsteady flow between parallel plates and squeezing flow and heat transfer between two parallel disks with velocity slip, temperature jump which are of utmost importance in applied and engineering sciences.

Differential Transformation Method

In this chapter, a new linearization procedure based on Differential Transformation Method (DTM) will be presented. The procedure begins with solving nonlinear differential equation by DTM. The effectiveness of the procedure is verified using a heat transfer nonlinear equation. The simulation result shows the significance of the proposed technique.

Introduction to Heat Transfer

In this chapter, we offer general definition of heat transfer. In another chapter, we offer the solution of heat transfer's problems. It is our target in this book.

He's Variational Iteration Method

In this chapter, a variational iteration method (VIM) has been applied to nonlinear heat transfer equation. The concept of the variational iteration method is introduced briefly for applying this method for problem solving. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. The results reveal that the VIM is very effective and convenient in predicting the solution of such problems.