_{1}

We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical results are found in good agreement with exact solutions.

Consider following singularly perturbed boundary value problem

with boundary conditions

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 [

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

The fifth order B-spline function

The region [a,b] is partitioned into uniformly sized finite elements of length h by the knots

In the proposed algorithm, The fifth order B-spline function

where

So the global approximation

where

Using the fifth order B-spline function and the approximate solution Equation (5), the nodal values

where the symbols

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

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

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

Example 1. Consider the convention-dominated equation:

with boundary conditions:

The exact solution is given by

where

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

It observed that

1) when h decreases (i.e. collocation number increases) for fixed

2) when

3) when

Example 2. Solve the following non-homogeneous equation:

with boundary conditions

The analytical solution is given by

where

And

Approximation solutions for different values of

X | e = 0.1, h = 1/32 | e = 0.1, h = 1/128 | e = 0.01 h = 1/32 | e = 0.01, h = 1/128 | e = 0.0015, h = 1/1024 | ||||
---|---|---|---|---|---|---|---|---|---|

error | numerical | Exact | error | error | Numerical | Exact | error | error | |

1/16 | 0.0036 | 0.0565 | 0.0556 | 0.0009 | 0.0095 | 0.0613 | 0.0600 | 0.0013 | 0.0001560 |

2/16 | 0.0037 | 0.1090 | 0.1082 | 0.0008 | 0.0092 | 0.1176 | 0.1164 | 0.0012 | 0.0001465 |

4/16 | 0.0033 | 0.2053 | 0.2045 | 0.0007 | 0.0082 | 0.2204 | 0.2193 | 0.0011 | 0.0001294 |

6/16 | 0.0030 | 0.2907 | 0.2901 | 0.0006 | 0.0074 | 0.3111 | 0.3102 | 0.0010 | 0.0001142 |

12/16 | 0.0021 | 0.0004 | 0.4582 | 0.4578 | 0.0052 | 0.5248 | 0.5241 | 0.0007 | 0.0000785 |

14/16 | 0.0014 | 0.3984 | 0.3981 | 0.0003 | 0.0025 | 0.5801 | 0.5795 | 0.0006 | 0.0000693 |

1 | 0.0004 | 0.0462 | 0.0462 | 0 | 0.0199 | 0.3402 | 0.3401 | 0.0001 | 0.0107 |

1) when

The numerical results show clearly the effect of

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).

Lin, B. (2016) B-Spline Collocation Method for Solving Singularly Perturbed Boundary Value Problems. Journal of Applied Mathematics and Physics, 4, 1699-1704. http://dx.doi.org/10.4236/jamp.2016.49178