scholarly journals A New Application of the Reproducing Kernel Hilbert Space Method to Solve MHD Jeffery-Hamel Flows Problem in Nonparallel Walls

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kılıçman
Keyword(s):  
Hilbert Space ◽  
Fluid Flow ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Nonlinear Problems ◽  
New Method ◽  
Reproducing Kernel Method ◽  
Rigid Plane ◽  
Space Method

The present paper emphasizes Jeffery-Hamel flow: fluid flow between two rigid plane walls, where the angle between them is 2α. A new method called the reproducing kernel Hilbert space method (RKHSM) is briefly introduced. The validity of the reproducing kernel method is set by comparing our results with HAM, DTM, and HPM and numerical results for different values ofH,α, and Re. The results show up that the proposed reproducing kernel method can achieve good results in predicting the solutions of such problems. Comparison between obtained results showed thatRKHSMis more acceptable and accurate than other methods. This method is very useful and applicable for solving nonlinear problems.

scholarly journals Numerical Solutions of the Second-Order One-Dimensional Telegraph Equation Based on Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kılıçman
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Telegraph Equation ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
Reproducing Kernel Theory ◽  
Space Method

We investigate the effectiveness of reproducing kernel method (RKM) in solving partial differential equations. We propose a reproducing kernel method for solving the telegraph equation with initial and boundary conditions based on reproducing kernel theory. Its exact solution is represented in the form of a series in reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of this method. The results obtained from this method are compared with the exact solutions and other methods. Results of numerical examples show that this method is simple, effective, and easy to use.

scholarly journals A Novel Method for Solving KdV Equation Based on Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Mustafa Inc ◽  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Kdv Equation ◽  
Reproducing Kernel Hilbert Space ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
The Kdv Equation ◽  
Novel Method ◽  
Reproducing Kernel Theory

We propose a reproducing kernel method for solving the KdV equation with initial condition based on the reproducing kernel theory. The exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have also been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is effective.

scholarly journals A Reproducing Kernel Hilbert Space Method for Solving Systems of Fractional Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Samia Bushnaq ◽  
Banan Maayah ◽  
Shaher Momani ◽  
Ahmed Alsaedi
Keyword(s):  
Hilbert Space ◽  
Fractional Calculus ◽  
Present Method ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Nonlinear Problems ◽  
Convergent Series ◽  
Integrodifferential Equations ◽  
Space Method

We present a new version of the reproducing kernel Hilbert space method (RKHSM) for the solution of systems of fractional integrodifferential equations. In this approach, the solution is obtained as a convergent series with easily computable components. Several illustrative examples are given to demonstrate the effectiveness of the present method. The method described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

scholarly journals A REPRODUCING KERNEL METHOD FOR SOLVING A CLASS OF NONLINEAR SYSTEMS OF PDES

2014 ◽  
Vol 19 (2) ◽  
pp. 180-198 ◽  
Author(s):  
Maryam Mohammadi ◽  
Reza Mokhtari
Keyword(s):  
Hilbert Space ◽  
Nonlinear Systems ◽  
System Performance ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Coupled System ◽  
Reproducing Kernel Method ◽  
Nonlinear Hyperbolic System ◽  
Systems Of Pdes

This paper is concerned with a technique for solving a class of nonlinear systems of partial differential equations (PDEs) in the reproducing kernel Hilbert space. The analytical solution is represented in the form of series. An iterative method is given to obtain the approximate solution. The convergence analysis is established theoretically. The proposed method is successfully used for solving a coupled system of viscous Burgers’ equations and a nonlinear hyperbolic system. Performance of the method is tested in terms of various error norms. In the case of non-availability of exact solution, it is compared with the existing methods.

scholarly journals Approximation scheme for handling coupled systems of differential equations within reproducing kernel method

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 599-615
Author(s):  
Ali Ateiwi ◽  
édamat Al ◽  
Asad Freihat ◽  
Iryna Komashynska
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Coupled Systems ◽  
Reproducing Kernel Space ◽  
Reproducing Kernel Method ◽  
And Performance ◽  
Space Method

This paper proposes an efficient numerical method to obtain analytical-numerical solutions for a class of system of boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. The analytical solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential 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.

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