We use the reproducing kernel Hilbert space method to solve the fifthorder boundary value problems. The exact solution to the fifthorder boundary value problems is obtained in reproducing kernel space. The approximate solution is given by using an iterative method and the finite section method. The present method reveals to be more effective and convenient compared with the other methods.
The reproducing kernel Hilbert space method has been shown [
Singular fifthorder boundary value problems arise in the fields of gas dynamics, Newtonian fluid mechanics, fluid mechanics, fluid dynamics, elasticity, reactiondiffusion processes, chemical kinetics, and other branches of applied mathematics.
Let us consider the following class of singular fifthorder mixed boundary value problems:
If
Let
Consider
Let
A real Hilbert space
One defines that the inner product space
The inner product in
The space
For studying the solution of (
Let
Consider
If
For any
Consider
Let
The space
Let
If
For each fixed
If
We define an approximate solution
Let
We have
For any
The numerical solution to (
In this section, two numerical examples are studied to demonstrate the accuracy of the present method.
Consider the following fifthorder boundary value problem with nonclassical side condition (the righthand side of this problem has a singularity at
The numerical results of Example








0  0  0  0  0  0  0 
0.08  −0.00150556  −0.00149854 




0.16  −0.00300419  −0.00299716 




0.24  −0.000739712  −0.000738437 




0.32  0.00926282  0.00925363 




0.4  0.0316161  0.0315926 




0.48  0.0717954  0.0717548 




0.56  0.136337  0.136276 




0.64  0.23304  0.232959 




0.72  0.371189  0.371084 




0.8  0.561801  0.561673 




0.88  0.817902  0.817739 




0.96  1.15484  1.15458 




Consider the following fifthorder boundary value problem (the righthand side of this problem has a singularity at
Comparison of the absolute error of Example

Solution  Absolute error  


Reference [ 


Reference [ 


0.0  0.0  0.0  0.0  0.0  0.0  0.0 
0.1249  0.0000752  0.0000754  0.0000754 



0.2431  0.0013039  0.0013043  0.0013037 



0.3806  0.0080242  0.0080249  0.0080244 



0.4195  0.0116531  0.0116538  0.0116533 



0.5  0.0220970  0.0220978  0.0220972 



0.6923  0.0588207  0.0588201  0.0588209 



0.7854  0.0723723  0.0723726  0.0723724 



0.8917  0.0646361  0.0646363  0.0646366 



1.0  0.0  0.0  0.0  0.0  0.0  0.0 
The numerical results of Example








0.0  0  0  0  0  0  0 
0.1  0.0012491  0.00125306 

0.0419792  0.0420291 

0.2  0.0121642  0.0121721 

0.193196  0.193223 

0.3  0.0421473  0.0421562 

0.410381  0.410375 

0.4  0.0930975  0.0931043 

0.591978  0.591944 

0.5  0.15468  0.154682 

0.596621  0.59657 

0.6  0.200775  0.200773 

0.250969  0.250915 

0.7  0.186533  0.186526 

−0.645692  −0.645732 

0.8  0.0457947  0.0457844 

−2.31836  −2.31836 

0.9  −0.311216  −0.311224 

−5.01403  −5.01398 

1.0  −1  −1 

−9  −8.99988 

The numerical results of Example








0.1  0.971215  0.971117 

11.8289  11.8244 

0.2  1.97221  1.9719 

7.04361  7.0429 

0.3  2.19979  2.19947 

−3.23499  −3.2345 

0.4  1.19534  1.19511 

−17.4321  −17.431 

0.5  −1.39212  −1.39222 

−34.8029  −34.8014 

0.6  −5.85595  −5.8559 

−54.8995  −54.8978 

0.7  −12.4526  −12.4524 

−77.4172  −77.4153 

0.8  −21.4126  −21.4122 

−102.132  −102.13 

0.9  −32.9466  −32.9459 

−128.873  −128.871 

1.0  −47.25  −47.2491 

−157.5  −157.498 

In this paper, a new reproducing kernel space satisfying mixed boundary value conditions is constructed skillfully. This makes it easy to solve such kind of problems. Furthermore, the exact solution of the problem can be expressed in series form. The numerical results demonstrate that the new method is quite accurate and efficient for singular problems of fifthorder ordinary differential equations. All computations have been performed using the Mathematica 7.0 software package.
The authors would like to express their thanks to the unknown referees for their careful reading and helpful comments. This paper is supported by the Natural Science Foundation of Inner Mongolia (2013MS0109) and Project Application Technology Research and Development Foundation of Inner Mongolia.