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

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.

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

Abstract and Applied Analysis ◽

10.1155/2013/768963 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 5

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.

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Explicit Solution of Telegraph Equation Based on Reproducing Kernel Method

Journal of Function Spaces and Applications ◽

10.1155/2012/984682 ◽

2012 ◽

Vol 2012 ◽

pp. 1-23 ◽

Cited By ~ 17

Author(s):

Mustafa Inc ◽

Ali Akgül ◽

Adem Kiliçman

Keyword(s):

Exact Solution ◽

Explicit Solution ◽

Reproducing Kernel ◽

Initial Conditions ◽

Kernel Method ◽

Telegraph Equation ◽

Reproducing Kernel Method ◽

Numerical Examples ◽

Reproducing Kernel Theory ◽

Kernel Theory

We propose a reproducing kernel method for solving the telegraph equation with initial conditions based on the reproducing kernel theory. The exact solution is represented in the form of series, and some numerical examples have been studied in order to demonstrate the validity and applicability of the technique. The method shows that the implement seems easy and produces accurate results.

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

Mathematical Modelling and Analysis ◽

10.3846/13926292.2014.909897 ◽

2014 ◽

Vol 19 (2) ◽

pp. 180-198 ◽

Cited By ~ 20

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.

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

Abstract and Applied Analysis ◽

10.1155/2013/239454 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 9

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.

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Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method

Fractal and Fractional ◽

10.3390/fractalfract4020027 ◽

2020 ◽

Vol 4 (2) ◽

pp. 27 ◽

Cited By ~ 1

Author(s):

Onur Saldır ◽

Mehmet Giyas Sakar ◽

Fevzi Erdogan

Keyword(s):

Fractional Order ◽

Reproducing Kernel ◽

Burgers Equation ◽

Kernel Method ◽

Reproducing Kernel Hilbert Space ◽

Neumann Boundary ◽

Kernel Functions ◽

Iterative Approach ◽

Reproducing Kernel Method ◽

Reproducing Kernel Spaces

In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.

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The Solution of a Class of Singularly Perturbed Two-Point Boundary Value Problems by the Iterative Reproducing Kernel Method

Abstract and Applied Analysis ◽

10.1155/2012/984057 ◽

2012 ◽

Vol 2012 ◽

pp. 1-7 ◽

Cited By ~ 5

Author(s):

Zhiyuan Li ◽

YuLan Wang ◽

Fugui Tan ◽

Xiaohui Wan ◽

Tingfang Nie

Keyword(s):

Singular Perturbation ◽

Boundary Layers ◽

Present Method ◽

Reproducing Kernel ◽

Kernel Method ◽

Singularly Perturbed ◽

Point Boundary ◽

Reproducing Kernel Method ◽

Numerical Examples ◽

Perturbation Problems

In (Wang et al., 2011), we give an iterative reproducing kernel method (IRKM). The main contribution of this paper is to use an IRKM (Wang et al., 2011), in singular perturbation problems with boundary layers. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that the method is simple and effective.

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Solving non-local fractical heat equations based on the reproducing kernel method

Thermal Science ◽

10.2298/tsci16s3711l ◽

2016 ◽

Vol 20 (suppl. 3) ◽

pp. 711-716

Author(s):

Xiuying Li ◽

Boying Wu

Keyword(s):

Boundary Conditions ◽

Reproducing Kernel ◽

Kernel Method ◽

Nonlocal Conditions ◽

Heat Equations ◽

Reproducing Kernel Method ◽

Local Boundary ◽

Non Local ◽

Reproducing Kernel Theory ◽

Kernel Theory

In this paper, a numerical method is proposed for 1-D fractional heat equations subject to non-local boundary conditions. The reproducing kernel satisfying nonlocal conditions is constructed and reproducing kernel theory is applied to solve the considered problem. A numerical example is given to show the effectiveness of the method.

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A high order approach for nonlinear Volterra-Hammerstein integral equations

AIMS Mathematics ◽

10.3934/math.2022086 ◽

2021 ◽

Vol 7 (1) ◽

pp. 1460-1469

Author(s):

Jian Zhang ◽

◽

Jinjiao Hou ◽

Jing Niu ◽

Ruifeng Xie ◽

...

Keyword(s):

Integral Equation ◽

Nonlinear Integral Equation ◽

Homotopy Perturbation Method ◽

Reproducing Kernel ◽

Kernel Method ◽

Linear Equations ◽

Reproducing Kernel Method ◽

Numerical Examples ◽

Hammerstein Integral Equations ◽

Homotopy Perturbation

<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>

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A modified reproducing kernel method for a time-fractional telegraph equation

Thermal Science ◽

10.2298/tsci160615037w ◽

2017 ◽

Vol 21 (4) ◽

pp. 1575-1580 ◽

Cited By ~ 3

Author(s):

Yulan Wang ◽

Mingjing Du ◽

Chaolu Temuer

Keyword(s):

Numerical Solution ◽

Present Method ◽

Reproducing Kernel ◽

Kernel Method ◽

Telegraph Equation ◽

Reproducing Kernel Method ◽

Numerical Examples ◽

Fractional Telegraph Equation

The aim of this work is to obtain a numerical solution of a time-fractional telegraph equation by a modified reproducing kernel method. Two numerical examples are given to show that the present method overcomes the drawback of the traditional reproducing kernel method and it is an easy and effective method.

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