Reproducing Kernel Method for Fractional Derivative with Non-local and Non-singular Kernel

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Ali Akgül
Keyword(s):  
Fractional Derivative ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Singular Kernel ◽  
Reproducing Kernel Method ◽  
Non Local

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.

scholarly journals A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Xinjian Zhang ◽  
Xiongwei Liu
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear Differential Equation ◽  
Error Estimation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Inner Product ◽  
Reproducing Kernel Method ◽  
Linear Differential ◽  
Polynomial Reproducing

A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

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