scholarly journals A new least square based reproducing kernel space method for solving regular and weakly singular 1D Volterra-Fredholm integral equations with smooth and nonsmooth solutions

2020 ◽  
Author(s):  
Minqiang Xu ◽  
Jing Niu ◽  
Emran Tohidi ◽  
Jinjiao Hou
Keyword(s):  
Integral Equations ◽  
Reproducing Kernel ◽  
Least Square ◽  
Fredholm Integral Equations ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Weakly Singular ◽  
Space Method

scholarly journals The Combined Reproducing Kernel Method and Taylor Series for Solving Weakly Singular Fredholm Integral Equations

2016 ◽  
Vol 5 (3) ◽  
pp. 109
Author(s):  
Azizallah Alvandi ◽  
Mahmoud Paripour
Keyword(s):  
Integral Equations ◽  
Taylor Series ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Fredholm Integral Equations ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Reproducing Kernel Method ◽  
Weakly Singular ◽  
Reproducing Kernel Function

<p>In this paper, a numerical method is proposed for solving weakly singular Fredholm integral equations in Hilbert reproducing kernel space (RKHS). The Taylor series is used to remove singularity and reproducing kernel function are used as a basis. The effectiveness and stability of the numerical scheme is illustrated through two numerical examples.</p>

scholarly journals A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Er Gao ◽  
Songhe Song ◽  
Xinjian Zhang
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Space Method ◽  
Term Approximation

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional orderq∈(1,2]based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so then-term approximation. At the same time, then-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which shows that the new algorithm is efficient and accurate.

scholarly journals Reproducing Kernel Banach Spaces with the ℓ1 Norm II: Error Analysis for Regularized Least Square Regression

2011 ◽  
Vol 23 (10) ◽  
pp. 2713-2729 ◽  
Author(s):  
Guohui Song ◽  
Haizhang Zhang
Keyword(s):  
Banach Spaces ◽  
Reproducing Kernel ◽  
Sampling Error ◽  
Approximation Error ◽  
Learning Rate ◽  
Least Square ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Regularization Error ◽  
Reproducing Kernel Banach Spaces

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error, and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the ℓ1 norm can be applied to improve the learning rate estimate of ℓ1-regularization in machine learning.

scholarly journals Reproducing Kernel Space Method for the Solution of Linear Fredholm Integro-Differential Equations and Analysis of Stability

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xueqin Lv ◽  
Yue Gao
Keyword(s):  
Differential Equation ◽  
Reproducing Kernel ◽  
Simple Algorithm ◽  
Approximate Solutions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
True Solution ◽  
Integro Differential Equation ◽  
Analysis Of Stability ◽  
Space Method

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable.

scholarly journals On Accuracy and Stability Analysis of the Reproducing Kernel Space Method for the Forced Duffing Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Bahram Asadi ◽  
Taher Lotfi
Keyword(s):  
Reproducing Kernel ◽  
Boundary Integral ◽  
Duffing Equation ◽  
Stability And Convergence ◽  
Integral Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
The Stability ◽  
Space Method ◽  
Accuracy And Stability

It is attempted to provide the stability and convergence analysis of the reproducing kernel space method for solving the Duffing equation with with boundary integral conditions. We will prove that the reproducing space method is stable. Moreover, after introducing the method, it is shown that it has convergence order two.

scholarly journals The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

2018 ◽  
Vol 15 ◽  
pp. 8070-8080 ◽  
Author(s):  
Hameeda Oda Al-Humedi
Keyword(s):  
Hilbert Space ◽  
Exact Solutions ◽  
Integral Equations ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Reproducing Kernel Space ◽  
Linear Ordinary Differential Equations ◽  
Weakly Singular

The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find.  The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity.  From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs).   The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization. 

Sign in / Sign up

Export Citation Format

Share Document