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.
TopBasic 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
.