Numerical study of self-adjoint singularly perturbed two-point boundary value problems using collocation method with error estimation

Khlid K. Ali , A.R. Hdhoud , M.A. Shln , ?

a Faculty of Science, Al-Azhar University, Cairo, Egypt

b Faculty of Science, Menoufia University, Shebeen El-Koom, Egypt

c Higher Technological Institute, 10th of Ramadan City, Egypt

Abstract A numerical treatment for self-adjoint singularly perturbed second-order two-point boundary value problems using trigonometric quintic B-splines is presented, which depend on different engineering applications. The method is found to have a truncation error of O ( h 6 ) and converges to the exact solution at O ( h 4 ). The numerical examples show that our method is very effective and the maximum absolute error is acceptable.

Keywords: Self-adjoint singularly-perturbation problems; Two-point boundary value problems; Trigonometric quintic B-spline collocation method.

1.Introduction

We consider self-adjoint singularly-perturbed two-point boundary value problem, given by

subject to boundary conditions

where α0, αn∈R, 0 ＜ ε＜ 1 is a small positive perturbation parameter andq(t),g(t) are sufficiently smooth functions. These problems appear a lot in different fields of science and engineering such as heat transfer problem with large Peclet numbers, Navier–Stokes flows with large Reynolds numbers [1] ,chemical reactor theory, reaction-diffusion process, quantum mechanics, optimal control etc. [2–4] . The numerical solution of singularly perturbed two-point boundary value problems has been discussed with exponential splines [5–8] , fitted mesh B-spline [9] , Haar wavelet approach [10] , uniformly convergent non-standard finite different methods [11] , initial value technique [12] , variable mesh finite difference method [13] ,variable mesh difference scheme [14] , a spline based computational simulations [15] , parametric cubic spline function[16] , quintic spline [17,18] , sextic spline [19] , septic spline[20] , a parameter-uniform Schwarz method [21] , a patching approach [22] , trigonometric quintic B-spline is presented[23,24] and B-spline functions are used to solve many types of partial differential equations [25,26] . However, the aim of this paper is to treat the self-adjoint singularly perturbed twopoint boundary value problems numerically using the trigonometric quintic B-spline method and also to derive the errors estimate of the method. This paper is organized as follows:In Section 2 , we introduce the analysis of the method. In Section 3 , we find the error estimation of the trigonometric quintic B-spline collocation method. In Section 4 , some numerical examples are discussed. Finally, the conclusion of this study is given in Section 5 .

2.Trigonometric quintic B-spline collocation method

Firstly, we assume that the problem domain [a,b]is equally divided intonsubintervals [tj,tj+1] ,j=0, 1 , 2, . . . ,n?1 by the knotstj=a+jhwherea=t0＜t1＜ ···＜tn=band the step sizeOn the above partition together with additional knotst?5 ,t?4 ,t?3 ,t?2 ,t?1 ,tn+1 ,tn+2 ,tn+3 ,tn+4 ,tn+5 outside the problem domain, we present the trigonometric quintic B-splineTQBj(t) collocation method for solving self-adjoint singularly perturbed two point value problem (1) numerically as follows:

Table 1 Values of TQB j ( t ) and its principle two derivatives at the knot points.

where,

An approximate solutions(t) to the unknownz(t), of the problem (1) and (2) , is written in terms of the expansion of theTQBj(t) as

where constants τj’s are be determined from the collocation pointstj,j= 0, 1 , ...,nand the boundary and initial conditions. The values of the trigonometric quintic B-spline functionsTQBj(t) and its principle four derivativesattheknots andtheirval-ues are summarized in Table 1 . where

Using Eqs. (3) and (4) , the values ofsjand their derivatives up to fourth order at the knots are

Substituting from Eq. (5) in Eqs. (1) and (2) we find,

After simplifying above equation, we get

and the boundary conditions (2) are given as

So, two equations are still required. By differentiate Eq.(1) with respect tot, we get

Puttingt=ain the above equation and using Eq. (5) , we get

Then from Eqs. (7), (8), (10) and (11) , we have the following matrix form:

whereAis an (n+ 5 ) ×(n+ 5 ) non-singular square matrix.T is an (n+ 5 ) dimensional vector with components τjand the right hand sideBis an (n+ 5 ) dimensional vector as shown:

3.Error estimation of trigonometric quintic B-spline method

In this section, we find the truncation error for the trigonometric quintic B-spline method in the interval [a,b]. Assume thatz(t) has continuous derivatives for allt∈ [a,b]. Using the Eq. (5) , we have the following relationships as follows:

Theorem 3.1.Letz(t)betheexactsolutionands(t)be theapproximatesolutionofself-adjointsingularlyperturbed second-ordertwo-pointboundaryvalueproblemhavethe form(1)and(2),thenthetrigonometricquinticB-spline methodhasatruncationerrorofO(h6)andtheconvergence ofthismethodisO(h4)forsufficientlysmallh.

4.Numerical examples

In this section, we present some examples of self-adjoint singularly perturbed second-order two-point boundary value problems. We divide the interval [0, 1] intonsubintervals and the results are generated with Mathematica to solve the emerging algebraic equations. At each different values ofnand perturbation parameter εwe evaluated the difference between approximate solution and exact solution, and then take the absolute errors of this difference.

Example 1.We consider singularly perturbed two point boundary value problem [29]

whereg(t) = ? cos2( πt) ? 2επ2cos ( 2πt) and the exact solution is given by

Thus, the Tables 2 , 3 and Fig. 1 show that our numerical estimates using trigonometric quintic B-spline method are acceptable comparing with other methods [5–8,14,16] and [18] .

Example 2.We present another singularly perturbed two point boundary value problem [20]

Table 2 Maximum absolute errors of Example 1 .

Table 3 Maximum absolute errors of Example 1 .

Fig. 1. Numerical solution of Example 1 by the present method for n = 256 and ε = 1 / 16 . .

Example 3.We investigate nonlinear singularly perturbed two point boundary value problem [22]

Table 4 Maximum absolute errors of Example 2 .

Fig. 2. Numerical solution of Example 2 by the present method for n = 64 and ε = 1 / 1024. .

Table 5 Maximum absolute errors of Example 3 .

Fig. 3. Numerical solution of Example 3 by the present method for n = 16 and ε = 1 / 10. .

Table 6 Maximum absolute errors of Example 4 for ε = 2 ?12 .

Fig. 4. Numerical solution of Example 4 by the present method for n = 32 and ε = 2 ?12 .

Example 4.We examine the reaction diffusion problem[21]

5.Conclusion

In this paper, we presented a numerical treatment of selfadjoint singularly perturbed second-order two-point boundary value problems using trigonometric quintic B-splines. From the computational results, the proposed method demonstrates efficient solutions of the considered problems at different values ofnand perturbation parameter ε. Four numerical examples are presented, which compare with the analytic solutions by finding the maximum absolute errors and we can see that the results of numerical experiments are convenient to be implemented and the proposed method effective numerical technique for solving two-point boundary value problems as in the Tables 2 ,–6 and the Figs. 1 –4 .

Acknowledgment

The authors would like to express their sincere thanks to the reviewers for their careful reading, additions valuable scientific comments and suggestions.