We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventhorder boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), variation of parameters method (VPM), and homotopy analysis method (HAM). Obtained results show that our method is very effective.
Consider the seventhorder boundary value problem [
The paper is organized as follows. Section
In this section, we define some useful reproducing kernel spaces.
We define the space
We define the space
The space
The solution of (
For
The following formula holds:
Consider the following:
The orthonormal system
In the following, we will give the representation of the exact solution of (
If
From the (
Now the approximate solution
Assume
From (
The seventhorder boundary value problems have come out in construction engineering, beam column theory, and chemical reactions. Therefore solutions of the seventhorder boundary value problems are very important in the literature. The reproducing kernel function for seventhorder boundary value problem has not been calculated till now. All computations are performed by Maple 16. The RKHSM does not require discretization of the variables, that is, time and space, and it is not affected by computational roundoff errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKHSM for the seventhorder boundary value problems is controllable and absolute errors are small with present choice of
Numerical results for Example

Exact solution  Approximate solution  Absolute error  Relative error 

0.0  1.0  1.0  0.0  0.0 
0.1  1.1051709180756476248  1.1051709180727538232 


0.2  1.2214027581601698339  1.2214027581330885422 


0.3  1.3498588075760031040  1.3498588075089984856 


0.4  1.4918246976412703178  1.4918246975765762858 


0.5  1.6487212707001281468  1.6487212707474912552 


0.6  1.8221188003905089749  1.8221188006600701498 


0.7  2.0137527074704765216  2.0137527079512967073 


0.8  2.2255409284924676046  2.2255409289689961077 


0.9  2.4596031111569496638  2.4596031113394646558 


1.0  2.7182818284590452354  2.7182818284590452354 


Comparison of absolute error of HPM, VIM, and RKHSM for Example

Absolute error [ 
Absolute error [ 
Absolute error [RKHSM] 

0.0  0.0  0.0  0.0 
0.1 



0.2 



0.3 



0.4 



0.5 



0.6 



0.7 



0.8 



0.9 



1.0 



Numerical results for Example

Exact solution  Approximate solution  Absolute error 

0.0  0.0  0.0  0.0 
0.1 



0.2 



0.3 



0.4 



0.5 



0.6 



0.7 



0.8 



0.9 



1.0 



Comparison of absolute error of VPM, ADM, HAM, and RKHSM.

Absolute error [ 
Absolute error [ 
Absolute error [ 
Absolute error [RKHSM] 

0.0  0.0  0.0  0.0  0.0 
0.1 




0.2 




0.3 




0.4 




0.5 




0.6 




0.7 




0.8 




0.9 




1.0 




Numerical results for Example

Exact solution  Approximate solution  Absolute error 

0.0  1.0  1.0  0.0 
0.1  0.81435367623236361584  0.81435367623236697064 

0.2  0.65498460246238548694  0.65498460246237032242 

0.3  0.51857275447720250625  0.51857275447711724998 

0.4  0.40219202762138358044  0.40219202762145521926 

0.5  0.30326532985631671180  0.30326532985631655134 

0.6  0.21952465443761057305  0.21952465443760682026 

0.7  0.14897559113742285441  0.14897559113743255406 

0.8  0.089865792823444318286  0.089865792823449635588 

0.9  0.040656965974059911188  0.040656965974059021714 

1.0  0.0 


Comparison of absolute error of ADM, HAM, and RKHSM.

Absolute error [ 
Absolute error [ 
Absolute error [RKHSM] 

0.0 

0.0 

0.1 



0.2 



0.3 



0.4 



0.5 



0.6 



0.7 



0.8 



0.9 



1.0 



Comparison of analytical solution and RKHSM solution for Example
Comparison of absolute error of VIM, HPM, and RKHSM for Example
Comparison of analytical solution and RKHSM solution for Example
Comparison of absolute error of ADM, VPM, HAM, and RKHSM for Example
Comparison of analytical solution and RKHSM solution for Example
Comparison of absolute error of ADM, HAM, and RKHSM for Example
In this section, three numerical examples are provided to show the accuracy of the present method.
We first consider the seventhorder nonlinear boundary value problem:
After homogenizing the boundary conditions of (
We now consider the seventhorder linear BVP
After homogenizing the boundary conditions of (
Consider the following seventhorder nonlinear BVP
After homogenizing the boundary conditions of (
where we used the following transformation:
Using RKHSM for this example we obtain Tables
Using our method we chose
In this paper, we introduced an algorithm for solving the seventhorder problem with boundary conditions. For illustration purposes, we chose three examples which were selected to show the computational accuracy. It may be concluded that the RKHSM is very powerful and efficient in finding exact solution for a wide class of boundary value problems. The approximate solution obtained by the present method is uniformly convergent. Clearly, the series solution methodology can be applied to much more complicated nonlinear differential equations and boundary value problems. However, if the problem becomes nonlinear, then the RKHSM does not require discretization or perturbation and it does not make closure approximation. Results of numerical examples show that the present method is an accurate and reliable analytical method for the seventhorder boundary value problem.
Let
This paper is a part of the Ph.D. thesis of Ali Akgül.
The authors declare that they do not have any competing or conflict of interests.