In 1992, Liao [1] proposed the Homotopy Analysis Method (HAM) to solve nonlinear differential equations analytically. Since then, HAM has been used to investigate a variety of mathematical and physical problems [2]. As is well known, HAM has the advantage of independence on small physical parameters and adjusting the convergence region and convergence rate of the series solution over perturbation method. However, for some type of auxiliary operator (i.e. base functions), it is usually time-consuming to get high-order approximation, and the number of terms appearing in high order approximation is very huge.

To improve the efficiency of HAM, many scholars have proposed different techniques. Yabushita [3] suggested an optimal HAM approach by minimizing the residual of governing equations. Marinca [4], Niu [5], Liao [6] developed this kind of approach. Lin [7] suggested an iterative technique, in which the initial guess is continuously replaced by intermediate approximation to proceed the computation. Recently, we use orthonormal polynomials/ functions to approximate the right-hand side of the highorder deformation equations to prohibit the rapid growth of the terms appearing in approximate solutions. For differential equations defined on a finite interval, trigonometric functions (or polynomial functions) are usually selected to express solutions. In this case, orthonormal trigonometric functions (or Chebyshev polynomials) are used to approximate right-hand side of high-order deformation equations. In this paper, we generalize this kind of approach for nonlinear problems defined on semi-infinite intervals. The main idea is that orthonormal functions derived from Schmidt-Gram process are used to approximate the right-hand side of high-order deformation equations during computation. For different types of problems, the derived orthonormal functions are different, which are closely related to the solution expression.

In the following section, the modified HAM (MHAM) is presented for boundary layer problems. In Section 3, examples are given to demonstrate it. Conclusions and some discussions are given in the last section.

For convenience, a brief description of the standard HAM will be present first. Then the proposed truncation technique will be followed.

Without loss of generality, consider the differential equation

where N is a nonlinear operator, t denotes the independent variable, is an unknown function. Suppose could be expressed by a set of functions

such that

is uniformly valid, where is a coefficient. In HAM, the zeroth-order deformation equation is constructed as

where

is an auxiliary linear operator, and the auxiliary function. Applying the homotopy-derivative [8]

to both sides of Equation (4), we get the corresponding mth-order deformation equation

where

and

Note that can be obtained by solving linear Equation (7) one after the other. The mth-order approximation of is given by

To measure the accuracy of, the squared residual error for Equation (1) is defined as

where is the domain.

If a successful homotopy analysis solution is obtained, the difficulty to get better approximation is that with the growth of order, the number of terms in higher-order approximation will grow rapidly, resulting in an enormous amount of computing time. To address this problem, we propose a truncation technique. The basic idea is that the right-hand side of Equation (7) is approximated by a set of orthonormal functions.

Suppose that can be expressed by a finite linear combination of linearly independent functions,. Note that may be slightly different from. Define a proper inner product in the linear space spanned by as

where is a weight function. In the framework of HAM, two typical kinds of base functions are usually used for boundary layer problems.

Case 1: suppose that can be expressed by finite linear combination of linearly independent functions

Note that (12) is dependent on three parameters, m and n. To implement the orthonormalization, an order is given to (12) as follows (called triangular order):

.

The inner product is defined as

Case 2: suppose that can be expressed by finite linear combination of linearly independent functions

The inner product is defined as

Applying the Schmidt-Gram process to the first functions, , , , we obtain orthonormal functions. Every time when is got, we approximate it by, to ensure that the number of terms in the right-hand side of Equation (7) will be no more than. That is to say, we replace with its approximation

to proceed the computation in HAM.

To illustrate the efficiency of the truncation technique, two typical examples are considered. The codes are written in Maple 13 on a PC with an Intel Core 2 Quad 2.66 GHz CPU. The variable Digits in the experiments is to control the number of digits when calculating with software floating point numbers in Maple.

In this paper, an efficient modification of HAM is proposed for solving boundary layer problems. Using the derived orthonormal functions, the right-hand sides of highorder deformation equations are approximated to reduce the rapid growth of terms in high-order approximate solution. Two typical examples show that the new approach can greatly reduce the terms in the approximate solution; meanwhile the accuracy can be largely retained. The new approach needs less time to get high-order approximation than the standard HAM. However, one unsolved problem

0          50          100         150        200 CPU (s)

Figure 2. Residual error versus CPU time in Example 2. Solid line: MHAM; Dash-dotted line: HAM.

of this approach is that there is so far no estimation theory on how many orthonormal functions should be used to approximate when accuracy is prior given. We will try to generalize this truncation technique to solve PDEs in the next step.

The first author would like to thank Dr. Z. Liu for valuable discussion. This work is supported by State Key Laboratory of Ocean Engineering (Approve No. GKZD- 010053).