Homotopy Analysis Method

Homotopy Analysis Method

DOI: 10.4018/978-1-5225-2713-8.ch007
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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 978-1-5225-2713-8.ch007.m02is a nonlinear operator, 978-1-5225-2713-8.ch007.m03is an independent variable and 978-1-5225-2713-8.ch007.m04is the solution of equation. We define the function, 978-1-5225-2713-8.ch007.m05 as follows:
. (2) where, 978-1-5225-2713-8.ch007.m07 and 978-1-5225-2713-8.ch007.m08 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 978-1-5225-2713-8.ch007.m11 is the auxiliary parameter which helps us increase the results convergency, 978-1-5225-2713-8.ch007.m12 is the auxiliary function and 978-1-5225-2713-8.ch007.m13 is the linear operator. It should be noted that there is a great freedom to choose the auxiliary parameter 978-1-5225-2713-8.ch007.m14, the auxiliary function 978-1-5225-2713-8.ch007.m15, the initial guess 978-1-5225-2713-8.ch007.m16 and the auxiliary linear operator 978-1-5225-2713-8.ch007.m17. 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 978-1-5225-2713-8.ch007.m18 and 978-1-5225-2713-8.ch007.m19.

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