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 Solving a Class of Singular Fifth-Order Boundary Value Problems Using Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yulan Wang ◽  
Shuai Lu ◽  
Fugui Tan ◽  
Mingjing Du ◽  
Hao Yu
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Boundary Value ◽  
Reproducing Kernel Space ◽  
Finite Section ◽  
Finite Section Method ◽  
Fifth Order ◽  
Space Method

We use the reproducing kernel Hilbert space method to solve the fifth-order boundary value problems. The exact solution to the fifth-order boundary value problems is obtained in reproducing kernel space. The approximate solution is given by using an iterative method and the finite section method. The present method reveals to be more effective and convenient compared with the other methods.

scholarly journals Reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation

2018 ◽  
Vol 8 (2) ◽  
pp. 145-151 ◽  
Author(s):  
Esra Karatas Akgül
Keyword(s):  
Hilbert Space ◽  
Inverse Problem ◽  
Kinetic Equation ◽  
Inverse Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Coefficient Inverse Problem ◽  
Space Method

On the basis of a reproducing kernel Hilbert space, reproducing kernel functions for solving the coefficient inverse problem for the kinetic equation are given in this paper. Reproducing kernel functions found in the reproducing kernel Hilbert space imply that they can be considered for solving such inverse problems. We obtain approximate solutions by reproducing kernel functions. We show our results by a table. We prove the eciency of the reproducing kernel Hilbert space method for solutions of a coefficient inverse problem for the kinetic equation.

scholarly journals Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Ghaleb Gumah ◽  
Khaled Moaddy ◽  
Mohammed Al-Smadi ◽  
Ishak Hashim
Keyword(s):  
Hilbert Space ◽  
Integral Equations ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Convergent Series ◽  
Kernel Functions ◽  
Orthogonal Basis ◽  
Approximation Solution ◽  
Solution Methodology ◽  
Space Method

We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert spaceW21a,bin order to formulate the analytical solutions in a rapidly convergent series form in terms of theirα-cut representation. The approximation solution is expressed byn-term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

scholarly journals New Numerical Method for Solving Tenth Order Boundary Value Problems

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 245 ◽  
Author(s):  
Ali Akgül ◽  
Esra Karatas Akgül ◽  
Dumitru Baleanu ◽  
Mustafa Inc
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Numerical Results ◽  
Space Method

In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables.

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