B-Spline Collocation Method for Solving Singularly Perturbed Boundary Value Problems

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1. Introduction

Consider following singularly perturbed boundary value problem

(1)

with boundary conditions

(2)

B-spline functions are useful wavelet basis functions; the stiffness matrix is sparse when it is used as trial functions. B-splines were introduced by Schoenberg in 1946 [5] . Up to now, B-spline approximation method for numerical solutions has been researched by various researchers [6] - [8] .

2. Description of the B-Spline Collocation Method

The expression of fifth order B-spline function is as follows:

(3)

The fifth order B-spline function is used to calculate in this work and possesses the following characters: piecewise smooth, compact support, Symmetry, rapidly decaying, differentiability, linear combination.

The region [a,b] is partitioned into uniformly sized finite elements of length h by the knots such that with, ,. Let be fifth order B-spline function with knots at the points,. The set of splines forms a basis for functions defined over [a,b].

In the proposed algorithm, The fifth order B-spline function is used as a single mother wavelet, i.e. and dilation and translation of mother wavelet functions can construct any function of.

(4)

where

So the global approximation to the function can be written in terms of the B-spline as follows

, (5)

where, are unknown real coefficients.

Using the fifth order B-spline function and the approximate solution Equation (5), the nodal values, and at the node are given in terms of element parameters by

(6)

(7)

(8)

where the symbols and denote first and second differentiation with respect to x, respectively.

Substituting Equations (6)-(8) into Equation (1) and Equation (2), we can obtain following linear equations

(9)

where

,

Note

where

It is easily seen that the matrix B is strictly diagonally dominant and hence nonsingular. Since B is nonsingular, we can solve the system for . Hence the method of collocation using the fifth order B-spline function as a basis function applied to the singularly perturbed boundary value problem has a unique solution given by Equation (5).

3. Numerical Results

In the section, we illustrate the numerical techniques discussed in the previous section by the following problems.

Example 1. Consider the convention-dominated equation:

(10)

with boundary conditions:,

,

.

The exact solution is given by

(11)

where

Comparison of the numerical results and point-wise errors is given in Table 1.

It observed that

1) when h decreases (i.e. collocation number increases) for fixed the point-wise errors decrease;

2) when decreases for fixed h the point-wise errors increase;

3) when, the errors are very large.

Example 2. Solve the following non-homogeneous equation:

(12)

with boundary conditions

, ,.

The analytical solution is given by

(13)

where

,.

And and are the real solutions of the characteristic equation.

Approximation solutions for different values of and for fixed p are given in Figure 1. It observed that

Table 1. Example 1. Comparison of results and point-wise errors.

Figure 1. Approximation solutions of example 2 for different values of epsilon g and for fixed p.

1) when and, the approximation solutions are in good agreement with exact solution; 2) when and, and the errors are very large; 3) when decreases for fixed p the width of boundary layer becomes small and wave shape change more and more stiff at and.

4. Conclusion

The numerical results show clearly the effect of on the boundary layer and the B-spline collocation method solving singular boundary value problems is relatively simple to collocate the solution at the mesh points. It is applicable technique and approximates the exact solution very well.

Acknowledgements

The authors would like to thank the editor and the reviewers for their valuable comments and suggestions to improve the results of this paper. This work was supported by the Natural Science Foundation of Guangdong (No. 2015A030313827).

References

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