scholarly journals Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Sedigheh Farzaneh Javan ◽  
Saeid Abbasbandy ◽  
M. Ali Fariborzi Araghi
Keyword(s):  
Hilbert Space ◽  
Integral Equations ◽  
Satisfactory Result ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Nonlinear Integral Equations ◽  
New Approach ◽  
Effectiveness And Efficiency ◽  
The Given ◽  
Space Method

A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.

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 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.

Sign in / Sign up

Export Citation Format

Share Document