ABSTRACT

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.

1. Introduction

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.

2. Analysis of the Method

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

, (1)

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

(2)

such that

(3)

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

, (4)

where

, (5)

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

(6)

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

, (7)

where

, (8)

and

(9)

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

(10)

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

, (11)

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

(12)

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

. (13)

The inner product is defined as

. (14)

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

(15)

to proceed the computation in HAM.

3. Numerical Experiment

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.

3.1. Example 1

Let us consider the Blasius Equation (9)

, (16)

subject to the boundary conditions

, (17)

where the prime denotes the derivative with respect to. Following Liao [9], we seek the solution in the form

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 Table 3, we can see that the number of terms in high-order approximation given by MHAM is 39, while that given by HAM grows exponentially. Moreover, it shows that MHAM needs less than ninth the CPU time used by HAM to get 50th-order approximate solution. The curve of residual error and CPU time is plotted in Figure 1. From it we can see that the truncation technique greatly improves the efficiency of HAM.

3.2. Example 2

Consider a set of two coupled nonlinear differential equations (see Kuiken [10] for details)

(18)

(19)

with the boundary conditions

, , where is the Prandtl number.

Under the transformation

, Equations (18) and (19) become

, (20)

, (21)

with the boundary conditions

Table 1. Comparison of given by MHAM and HAM in Example 1.

Table 2. Comparison of given by MHAM and HAM in Example 1.

Table 3. Comparison of CPU time (seconds) and number of terms appearing in mth order approximation given by MHAM and HAM in Example 1.

0        500      1000      1500      2000     2500 CPU (s)

Figure 1. Residual error versus CPU time in Example 1. Solid line: MHAM; Dash-dotted line: HAM.

, ,.

We seek the solutionandin the form

, where, are coefficients.

Following Liao [2], the auxiliary linear operators are chosen as

,

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

, (22)

and

, (23)

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 Table 4, and the residual error for Equation (20) is compared in Table 5. We can see that although we use to proceed the computation instead of, the accuracy of the approximate solution is well retained. From Table 6, we can see that the number of terms in the high-order approximation given by MHAM was kept within 16, while that

Table 4. Comparison of given by MHAM and HAM in Example 2.

Table 5. Comparison of given by MHAM and HAM in Example 2.

Table 6. Comparison of CPU time (seconds) and number of terms appearing in given by MHAM and HAM in Example 2.

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 Figure 2. From it we can see that the truncation technique is more powerful to get higher-order approximation than the standard HAM.

4. Conclusion and Discussions

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.

5. Acknowledgements

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).

REFERENCES