_{1}

A numerical method based on septic B-spline function is presented for the solution of linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization technique to reduce the nonlinear problems to linear problems and use B-spline collocation method, which leads to a seven nonzero bands linear system. Illustrative example is included to demonstrate the validity and applicability of the proposed techniques.

Consider the following fifth-order boundary value problem.

With boundary conditions

where

B-spline functions based on piece polynomials are useful wavelet basis functions, the resulting matrices are sparse, but always, banded. And that possess attractive properties: piecewise smooth, compact support, symmetry, rapidly decaying, differentiability, linear combination, B-splines were introduced by Schoenberg in 1946 [

In this paper, the septic B-spline function is used as a basis function and the B-spline collocation method is studied to solve the linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization technique to reduce the nonlinear problems to linear problems. The present method is tested for its efficiency by considering two examples.

An arbitrary Nth order spline function with compact support of N. It is a concatenation of N sections of (N-1)th order polynomials, continuous at the junctions or “knots”, and gives continuous (N-1)th derivatives at the junctions.

Let

The set of splines

We seek the approximation

which satisfies the following interpolation conditions:

where

Using the septic B-spline function Equation (3) and the approximate solution Equation (4), the nodal values

0 | 1 | 120 | 1191 | 2416 | 1191 | 120 | 1 | 0 | |

0 | 0 | 0 | |||||||

0 | 0 | ||||||||

0 | 0 | 0 | |||||||

0 | 0 | 0 | 0 | ||||||

0 | 0 | 0 |

From Equations (4)-(7), we have

Using operator notations

Expanding them in powers of

Hence we get

From Equation (1) and Equation (12), we can get

Using the boundary conditions and by neglecting the error of Equation (13), we can obtain following linear equations

Or

where

where

T denoting transpose.

In which B is a square matrix of order N + 7 with seven nonzero bands. Since B is nonsingular, after solving the linear system Equation (15) for

solution

Consider the nonlinear fifth order boundary value problem

with boundary conditions

We use the quesilinearization technique to reduce the above nonlinear problem to a sequence of linear problems. Expanding the right hand side of Equation (16), we have

Equation (18) can be rewritten as

where

Equation (19) once the initial values (k = 0,

Subject to the boundary conditions

Instead of solving nonlinear problem (16) with boundary conditions (17), we solve a sequence of linear problems (19) with boundary conditions (20), we consider

The relative error of numerical solution is given by

The pointwise errors are given by

The maximum pointwise errors are given by

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

Example 1. Consider the following equation [

With boundary conditions

The exact solution is given by

The numerical results are shown in

Example 2. Consider the following nonlinear equation [

h | CPU time (seconds) | ||
---|---|---|---|

1/8 | 1.718286042191597e−004 | 3.294552154146086e−004 | 3.218 |

1/10 | 6.447839923018339e−005 | 1.351847113984665e−004 | 8.172 |

1/16 | 1.028885985859818e−005 | 2.068672045465237e−005 | 2.953 |

1/20 | 4.192815893866442e−006 | 8.479871033239017e−006 | 5.391 |

1/32 | 6.368030709968942e−007 | 1.296221338426021e−006 | 6.922 |

1/40 | 2.475231232201836e−007 | 5.045908672756208e−007 | 8.688 |

1/50 | 1.671824484961171e−007 | 3.420027458407504e−007 | 8.172 |

1/64 | 3.108106372273767e−008 | 6.343766826003454e−008 | 6.921 |

h | |||||
---|---|---|---|---|---|

Our method | Caglar et al. [ | Shahid.et al. [ | Khan et al. [ | ||

1/10 | 6.447839923018339E−5 | 0.1570 | 2.259E−4 | 4.025E−3 | |

1/20 | 4.192815893866442E−6 | 0.0747 | 1.33E−5 | 3.911E−3 | |

1/40 | 2.475231232201836E−7 | 0.0208 | 5.2812E−7 | 1.145E−2 | |

With boundary conditions

The exact solution is given by

Comparison of numerical results and pointwise errors are given in

X | Numerical | Exact | Our errors | Errors of [ | Errors of [ | Errors of [ | Errors of [ |
---|---|---|---|---|---|---|---|

0.1 | 1.10517089134327 | 1.10517091807565 | 2.673237986527965e−008 | 7.0e−4 | 1.3e−7 | 2.3e−7 | 0 |

0.2 | 1.22140266137546 | 1.22140275816017 | 9.678471002416700e−008 | 7.2e−4 | 4.2e−7 | 1.6e−6 | 1.0e−5 |

0.3 | 1.34985864259601 | 1.34985880757600 | 1.649799901137783e−007 | 4.1e−4 | 7.2e−7 | 4.6e−6 | 1.0e−5 |

0.4 | 1.49182448262282 | 1.49182469764127 | 2.150184499338792e−007 | 4.6e−4 | 9.4e−7 | 8.9e−6 | 1.0e−4 |

0.5 | 1.64872127070013 | 1.64872103467892 | 2.360212099095094e−007 | 4.7e−4 | 1.0e−6 | 1.3e−5 | 3.2e−4 |

0.6 | 1.82211858273888 | 1.82211880039051 | 2.176516300522735e−007 | 4.8e−4 | 9.3e−7 | 1.6e−5 | 3.6e−4 |

0.7 | 2.01375254082381 | 2.01375270747048 | 1.666466702410219e−007 | 3.9e−4 | 7.1e−7 | 1.6e−6 | 1.4e−4 |

0.8 | 2.22554083163489 | 2.22554092849247 | 9.685758017852209e−008 | 3.1e−4 | 4.1e−7 | 1.2e−5 | 3.1e−4 |

0.9 | 2.45960308048908 | 2.45960311115695 | 3.066787002126148e−008 | 1.6e−4 | 1.3e−7 | 5.1e−6 | 5.8e−4 |

In the paper, the fifth-order boundary value problems are solved by means of septic B-splines collocation method. We use the quesilinearization technique to reduce the nonlinear problems to linear problems and reduce a boundary value problem to the solution of algebraic equations with seven nonzero bands. The numerical results show that the present method is relatively simple to collocate the solution at the mesh points and easily carried out by a computer and approximates the exact solution very well.

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 (2015A030313827).

Bin Lin, (2016) Septic B-Spline Solution of Fifth-Order Boundary Value Problems. Journal of Applied Mathematics and Physics,04,1446-1454. doi: 10.4236/jamp.2016.48149