Homotopy Analysis Method

Homotopy Analysis Method

DOI: 10.4018/978-1-5225-2713-8.ch007
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Abstract

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.
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Basic Idea Of Ham

Let us assume the following nonlinear differential equation in form of:

. (1) where is a nonlinear operator, is an independent variable and is the solution of equation. We define the function, as follows:
. (2) where, and is the initial guess which satisfies the initial or boundary condition and if:
, (3) and using the generalized homotopy method, Liao’s so-called zero-order deformation equation will be:
. (4) where is the auxiliary parameter which helps us increase the results convergency, is the auxiliary function and is the linear operator. It should be noted that there is a great freedom to choose the auxiliary parameter , the auxiliary function , the initial guess and the auxiliary linear operator . This freedom plays an important role in establishing the keystone of validity and flexibility of HAM as shown in this chapter. However, there are some fundamental rules of solutions which should be regarded in choosing and .

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