A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space

2010 ◽  
Vol 34 (1) ◽  
pp. 44-47 ◽  
Author(s):  
Yingzhen Lin ◽  
Minggen Cui
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Reproducing Kernel ◽  
Boundary Value ◽  
Point Boundary ◽  
Kernel Space ◽  
Reproducing Kernel Space

scholarly journals A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems

2020 ◽  
Vol 4 (4) ◽  
pp. 53
Author(s):  
Mehmet Giyas Sakar ◽  
Onur Saldır
Keyword(s):  
Exact Solutions ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Legendre Polynomials ◽  
Fractional Derivatives ◽  
Reproducing Kernel ◽  
Boundary Value ◽  
Point Boundary ◽  
Shifted Legendre Polynomials ◽  
Kernel Approach

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to α and β are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.

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 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.

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