scholarly journals Numerical Solutions for the Three-Point Boundary Value Problem of Nonlinear Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
C. P. Zhang ◽  
J. Niu ◽  
Y. Z. Lin
Keyword(s):  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Main Idea ◽  
Point Boundary ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Nonlinear Fractional Differential Equation ◽  
Uniformly Convergence ◽  
Fractional Differential

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.

scholarly journals APPROXIMATE SOLUTION OF NONLINEAR MULTI-POINT BOUNDARY VALUE PROBLEM ON THE HALF-LINE

2012 ◽  
Vol 17 (2) ◽  
pp. 190-202 ◽  
Author(s):  
Jing Niu ◽  
Ying Zhen Lin ◽  
Chi Ping Zhang
Keyword(s):  
Approximate Solution ◽  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Boundary Value ◽  
Half Line ◽  
Point Boundary ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Uniformly Convergence ◽  
Reproducing Kernel Function

In this work, we construct a novel weighted reproducing kernel space and give the expression of reproducing kernel function skillfully. Based on the orthogonal basis established in the reproducing kernel space, an efficient algorithm is provided to solve the nonlinear multi-point boundary value problem on the half-line. Uniformly convergence of the approximate solution and convergence estimation of our algorithm are studied. Numerical results show our method has high accuracy and efficiency.

scholarly journals Approximate Solutions to Three-Point Boundary Value Problems with Two-Space Integral Condition for Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Jing Niu ◽  
Yingzhen Lin ◽  
Minggen Cui
Keyword(s):  
Parabolic Equations ◽  
Reproducing Kernel ◽  
Boundary Value ◽  
Integral Condition ◽  
Approximate Solutions ◽  
Orthogonal Basis ◽  
Point Boundary ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
The Common

We construct a novel reproducing kernel space and give the expression of reproducing kernel skillfully. Based on the orthogonal basis of the reproducing kernel space, an efficient algorithm is provided firstly to solve a three-point boundary value problem of parabolic equations with two-space integral condition. The exact solution of this problem can be expressed by the series form. The numerical method is supported by strong theories. The numerical experiment shows that the algorithm is simple and easy to implement by the common computer and software.

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Jiang ◽  
Zhong Chen ◽  
Ning Hu ◽  
Yali Chen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Orthonormal Bases ◽  
Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

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