Reproducing kernel method to solve non-local fractional boundary value problem

2021 ◽  
Author(s):  
Raziye Mohammad Hosseiny ◽  
Tofigh Allahviranloo ◽  
Saeid Abbasbandy ◽  
Esmail Babolian
Keyword(s):  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Reproducing Kernel Method ◽  
Non Local

scholarly journals APPROXIMATE SOLUTION TO A MULTI-POINT BOUNDARY VALUE PROBLEM INVOLVING NONLOCAL INTEGRAL CONDITIONS BY REPRODUCING KERNEL METHOD

2013 ◽  
Vol 18 (4) ◽  
pp. 529-536 ◽  
Author(s):  
Kemal Ozen ◽  
Kamil Orucoglu
Keyword(s):  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Point Boundary ◽  
Integral Conditions ◽  
Reproducing Kernel Space ◽  
Order Ordinary Differential Equation ◽  
Reproducing Kernel Method ◽  
Nonlocal Integral Conditions

In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coeffcients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions.

scholarly journals A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Minqiang Xu ◽  
Jing Niu ◽  
Li Guo
Keyword(s):  
Nonlinear System ◽  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Linear Equations ◽  
Second Order ◽  
High Order ◽  
Reproducing Kernel Method

This paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.

scholarly journals CONVERGENCE ORDER OF THE REPRODUCING KERNEL METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS

2016 ◽  
Vol 21 (4) ◽  
pp. 466-477 ◽  
Author(s):  
Zhihong Zhao ◽  
Yingzhen Lin ◽  
Jing Niu
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problems ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Piecewise Linear ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Reproducing Kernel Method ◽  
Rate Analysis ◽  
Reproducing Kernel Spaces

In this paper, convergence rate of the reproducing kernel method for solving boundary value problems is studied. The equivalence of two reproducing kernel spaces and some results of adjoint operator are proved. Based on the classical properties of piecewise linear interpolating function, we provide the convergence rate analysis of at least second order. Moreover, some numerical examples showing the accuracy of the proposed estimations are also given.

scholarly journals Solving non-local fractical heat equations based on the reproducing kernel method

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 711-716
Author(s):  
Xiuying Li ◽  
Boying Wu
Keyword(s):  
Boundary Conditions ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Nonlocal Conditions ◽  
Heat Equations ◽  
Reproducing Kernel Method ◽  
Local Boundary ◽  
Non Local ◽  
Reproducing Kernel Theory ◽  
Kernel Theory

In this paper, a numerical method is proposed for 1-D fractional heat equations subject to non-local boundary conditions. The reproducing kernel satisfying nonlocal conditions is constructed and reproducing kernel theory is applied to solve the considered problem. A numerical example is given to show the effectiveness of the method.

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