1 Introduction

Singular perturbation problems (SPPs) frequently occur in fluid mechanics and other branches of applied mathematics. Generally,classical numerical methods fail to achieve high accuracy because of the singularly perturbed nature. In recent years,many special methods have been undertaken for precise numerical solutions^{[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]}. However,a universal way to solve SPPs is still needed at present.

The multiplication perturbation method^[12] proposed by Tan and Zhong can be considered as a first-order multiplication perturbation method for time-varying systems. Fu et al.^[13] further proposed a high order multiplication perturbation method (HOMPM),which enabled the precise integration method (PIM) for time-varying ordinary differential equations (ODEs) to reach the same accuracy as that of the PIM proposed by Zhong and Williams^[14] for constant coefficient problems.

In this paper,a coupling technique of the HOMPM and the reduction method is developed to provide an efficient method for the variable coefficient singularly perturbed two-point boundary value problems (TPBVPs) with one boundary layer. The present method can have both the improved accuracy with increased orders of the HOMPMand the high efficiency of the reduction method. Thus,it has a bright application future.

2 Method description

Consider the following variable coefficient SPP:

where ε is a small positive parameter (0<ε <<1),and α and β are known constants. We transform Eq. (1) into a system of first-order ODEs by introducing p(x) = y′(x). Thus, where Conducting dimensional expanding^[15] with Eq. (3),we have where

Thus,in the following,we only need to solve the homogeneous equation (4).

First,divide the interval ^{[0, 1]} with τ = 1/m. Let x_i = iτ (i = 0,1,· · · ,m) and c_i= x_i + τ/2 .Then,discompose H*(x) as

where

Let

Substituting Eq. (6) into Eq. (3),we obtain where

Then,discompose H₁*(x) as

where

Furthermore,let

in which N is the final times of perturbation.

Obviously,

in which

Similarly,H_k*(x) is discomposed as

where

Ignoring the high order item ,we obtain

The solution of Eq. (12) is represented as where C is to be determined.

Define

We have Let x = x_i in Eq. (15). We get where

Let x = x_i+1 in Eq. (15). Then,

whereis the transfer matrix.

Equation (17) cannot be solved directly by the HOMPM. Thus,we use an efficient reduction method to solve it. Let m = 2^M + 1,and substitute Eq. (2) into Eq. (17). Then,

where We add the (2^k−1 + 1)th equation multiplied by −C^(k−1) _L to the first equation,and add the (m−2^k−1)th equation multiplied by −(A^(k−1) _RB +A^(k−1) _RS ) to the last one. At the same time,add each (2^kj+1−2^k−1)th equation multiplied by −( A^(k−1) _(2^kj)B + A^(k−1) (2^kj)S) and the (2^kj+1+2^k−1)th equation multiplied by −C^(k−1) _2^kj to the (2^kj + 1)th equation (j = 1,2,· · · ,2^M−k − 1,k = 1,2,· · · ,M),which gives where in which After M times of iterations,the system becomes in which

The solution of Eq. (20) is After u₁^(M) is given,all the required solutions of Eq. (1) can be worked out through the reversed reduction process.

3 Numerical examples

Example 1 Consider the SPP

It has one boundary layer at x = 1,and the exact solution is y(x) =. To solve this problem,we let N = 0 and M = 0 for ε = 10⁻² and ε = 10⁻³ (when N = 0,the method of this paper degrades into the method in Ref. ^[16]). After y′(1) is given,redivide the mesh and use v*(x_i) = P⁻¹ _N(i+1)v*(x_i+1) to get the solutions from right to left with τ = 0.01. The numerical solutions given by the present method are displayed in Table 1 together with Chawla’s and Andargie’s solutions.

From Table 1 ,we can see that the present solutions are exactly the same as the exact solu- tions. Obviously,the present method is much more precise than the other numerical methods. Therefore,the present method has an outstanding advantage in solving constant coefficient SPPs.

To show the efficiency of the present method,we compare its computation time with that of Ref. ^[4] (see Table 2 ).

Example 2 Consider this variable coefficient SPP

It has one boundary layer at x = 0,and the analytical solution is ^[17] First,let N = 2 for ε = 10⁻² with M = 3 and ε = 10⁻³ with M = 6. Remesh the grid with τ = 0.01 after y′(0) is obtained to compute the required solutions in the desired points. To show how precise the present method is,we compare the numerical solutions of the present method with those of Refs. ^[4] and ^[1] (see Table 3).

It can be easily noticed from Table 3 that in the boundary layer,the solutions of the present method are obviously much better than those of Refs. ^[4] and ^[1].

The computation time of the present method is also compared with that of Ref. ^[1] (see Table 4 ).

It can be noticed from Table 4 that the computation time of the present method is a bit longer than that of Ref. ^[1],which is partly due to the redivision of grids for computation after y′(0) is worked out.

4 Conclusions

We present a coupling technique of the HOMPM and the reduction method for variable coefficient singularly perturbed TPBVPs with one boundary layer. This method combines the advantages of the HOMPM and the reduction method. Thus,it is highly accurate and efficient. The present method extends the application of the PIM to the solution for variable coefficient SPPs and also provides a new way to solve other variable coefficient TPBVPs.