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We present an efficient numerical scheme for solving three-point boundary value problems of nonlinear fractional differential equation. The main idea of this method is to establish a favorable reproducing kernel space that satisfies the complex boundary conditions. Based on the properties of the new reproducing kernel space, the approximate solution is obtained by searching least value techniques. Moreover, uniformly convergence and error estimation are provided for our method. Numerical experiments are presented to illustrate the performance of the method and to confirm the theoretical results.

Fractional differential equations have gained considerable importance due to their frequent applications in various fields of science and engineering including physics [

Finding exact solutions in closed forms for most differential equations of fractional order is a difficult task. As a result, a number of methods have been proposed and applied successfully to approximate fractional differential equations, such as Adomian decomposition method [

In contrast to the initial value and two-point boundary value problems, not much attention has been paid to the multipoint fractional boundary value problem (MFBVP). Ahmed and Wang [

In the present work, we are concerned with the numerical solution of the following three-point boundary value problem of fractional differential equation [

Recently, the reproducing kernel space method (RKSM) has been used for obtaining approximate solutions of differential and integral equations [

The aim of this work is to extend the RKSM to derive the numerical solutions of (

The organization of this paper is as follows. In Section

We give some basic definitions and theories which are used further in this paper.

The Riemann-Liouville fractional derivative of order

Let

We now define a new reproducing kernel space

For any

By integrating repeatedly

Plot of distribution for

The method consists of two steps. In the first step, a normal orthogonal basis is established in the reproducing kernel space

Define a bounded linear operator

The proof of existence and uniqueness of solution for (

In fact, according to the properties of reproducing kernel and bounded linear operator, we have

Orthogonalization can be obtained by the following cycle.

Firstly, let

If

According to the orthogonal basis

The approximate solution

For any

pick any initial value

substitute

substitute

if

calculate

if

Firstly, because of denseness, for any

In this section, we give some computational results of three numerical experiments with methods based on preceding sections, to support our theoretical discussion.

Consider the following three-point boundary value problem of nonlinear fractional differential equation [

For this example, we solved the three-point boundary value problem, by applying the technique described in preceding section as following.

By using the representation formula for calculation of reproducing kernel in Section

According to the numerical algorithm in Section

The obtained numerical results are displayed in Table

Due to the multipoint boundary value conditions, to our knowledge, there is no the same example as (

Numerical results for

Node | Exact solution | Approximate solution | Absolute error |
---|---|---|---|

1/8 | 0.00871094 | 0.0087107 | |

1/4 | 0.0727344 | 0.0727329 | |

3/8 | 0.25418 | 0.254176 | |

1/2 | 0.61750 | 0.617491 | |

5/8 | 1.22070 | 1.22069 | |

3/4 | 2.10305 | 2.10302 | |

7/8 | 3.26922 | 3.26918 | |

1 | 4.67000 | 4.66995 |

Comparison of the approximate solution

In this example, we consider the following nonlinear fourth-order fractional integrodifferential equation of the form:

The approximate solution in our method and in [

Node | Present method | AD method in [ |
---|---|---|

0.1 | 1.12257 | 1.1202485492579 |

0.2 | 1.27641 | 1.2624009459051 |

0.3 | 1.42190 | 1.4293559759369 |

0.4 | 1.64819 | 1.6248578688995 |

0.5 | 1.90926 | 1.8534779619798 |

0.6 | 2.18762 | 2.1206550639506 |

0.7 | 2.50183 | 2.4327620675062 |

0.8 | 2.83318 | 2.7971904644190 |

0.9 | 3.29015 | 3.2224499000323 |

Consider the nonlinear fifth-order fractional differential equation with three-point boundary value conditions as following:

Numerical results for

Node | Exact solution | Approximate solution | Absolute error |
---|---|---|---|

1/8 | 0.00533087 | 0.00533148 | |

1/4 | 0.0194715 | 0.0194721 | |

3/8 | 0.135717 | 0.135719 | |

1/2 | 0.529906 | 0.529911 | |

5/8 | 1.48634 | 1.48635 | |

3/4 | 3.34208 | 3.3421 | |

7/8 | 6.59401 | 6.59403 | |

1 | 10.8280 | 10.8280 |

Comparison of the approximate solution

In this paper, the RKSM is applied to derive approximate analytical solution of nonlinear fractional-order differential equations with three-point boundary value conditions. We have constructed a novel reproducing kernel space and give the way to express the reproducing kernel function, while traditional RKSM still can not be mentioned. The explicit series solution is obtained using the orthogonal basis established in the new reproducing kernel space. The numerical results given in the previous section demonstrate the better accuracy of our algorithms. Moreover, the numerical algorithms introduced in this paper can be well suited for handling general linear and nonlinear fractional-order differential equations with multipoint boundary conditions. We note that the corresponding analytical and numerical solutions are obtained using Mathematica 7.0.

This paper was supported by Youth Foundation of Heilongjiang Province under grant QC2010036 and also supported by Fundamental Research Funds for the Central Universities under Grant no.HIT.NSRIF.2009050 and Academic Foundation for Youth of Harbin Normal University 11KXQ-04 and 10KXQ-05.