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A new modification of the Homotopy Analysis Method (HAM) is presented for highly nonlinear ODEs on a semi-infinite domain. The main advantage of the modified HAM is that the number of terms in the series solution can be greatly reduced; meanwhile the accuracy of the solution can be well retained. In this way, much less CPU is needed. Two typical examples are used to illustrate the efficiency of the proposed approach.

In 1992, Liao [

To improve the efficiency of HAM, many scholars have proposed different techniques. Yabushita [

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 [

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.

Let us consider the Blasius Equation (9)

subject to the boundary conditions

where the prime denotes the derivative with respect to. Following Liao [

where is the spatial-scale parameter, and is a coefficient. The auxiliary linear operator is chosen as

.

The initial guess is

.

The zeroth-order deformation equation is constructed as in (4), and the th-order deformation equation as in (7) with homogeneous boundary conditions

where

.

For this example, the first kind orthonormal functions are used to approximate every time when is got. Then is used to compute instead of. In the experiment, we set, , , and. From Tables 1 and 2, we can see that though we use approximate (i.e.) to do the computation, the residual error and given by MHAM are almost the same as that given by standard HAM. From

Consider a set of two coupled nonlinear differential equations (see Kuiken [

with the boundary conditions

, , where is the Prandtl number.

Under the transformation

, Equations (18) and (19) become

with the boundary conditions

0 500 1000 1500 2000 2500 CPU (s)

, ,.

We seek the solutionandin the form

, where, are coefficients.

Following Liao [

,

and the initial guess of and as

,.

Then the high-order deformation equations become

,

subject to the homogeneous boundary conditions

and

,

.

We find that and can be expressed in the form of finite combination of functions

and

respectively. Applying the Schmidt-Gram process to the first functions in (22) and (23), respectively, we obtain two set of N orthonormal functions, denoted by and. We use to approximate every time it is got, and to approximate. Then and are used to proceed the computation.

In the experiment, we set, , , , , and. For different order approximation given by the two approaches, the quantity is showed in

given by HAM grows with the order. Moreover, it shows that MHAM needs less CPU time than the standard HAM to get high-order approximate solution. The curve of residual error and CPU time is plotted in

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)

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).