scholarly journals Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Li-Hong Yang ◽  
Hong-Ying Li ◽  
Jing-Ran Wang
Keyword(s):  
Integral Equations ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Numerical Technique ◽  
Volterra Integral Equations ◽  
Reproducing Kernel Space ◽  
Approximation Solution ◽  
Reproducing Kernel Method ◽  
Modified Method ◽  
Linear Volterra Integral Equations

A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations system of the second kind is given. The traditional reproducing kernel method requests that operator a satisfied linear operator equationAu=f, is bounded and its image space is the reproducing kernel spaceW21[a,b]. It limits its application. Now, we modify the reproducing kernel method such that it can be more widely applicable. Then-term approximation solution obtained by the modified method is of high accuracy. The numerical example compared with other methods shows that the modified method is more efficient.

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 Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Xiaoli Zhang ◽  
Haolu Zhang ◽  
Lina Jia ◽  
Yulan Wang ◽  
Wei Zhang
Keyword(s):  
Numerical Solution ◽  
Fractional Order ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Diffusion Equations ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Reproducing Kernel Method ◽  
Reproducing Kernel Spaces ◽  
Methods Numerical

In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.

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 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. 

scholarly journals Iterative Reproducing Kernel Method for Solving Second-Order Integrodifferential Equations of Fredholm Type

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Iryna Komashynska ◽  
Mohammed AL-Smadi
Keyword(s):  
Reproducing Kernel ◽  
Kernel Method ◽  
Simple Algorithm ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Reproducing Kernel Space ◽  
Fredholm Type ◽  
Reproducing Kernel Method

We present an efficient iterative method for solving a class of nonlinear second-order Fredholm integrodifferential equations associated with different boundary conditions. A simple algorithm is given to obtain the approximate solutions for this type of equations based on the reproducing kernel space method. The solution obtained by the method takes form of a convergent series with easily computable components. Furthermore, the error of the approximate solution is monotone decreasing with the increasing of nodal points. The reliability and efficiency of the proposed algorithm are demonstrated by some numerical experiments.

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