scholarly journals Explicit Solution of Telegraph Equation Based on Reproducing Kernel Method

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
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.

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

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.

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

2017 ◽  
Vol 21 (4) ◽  
pp. 1575-1580 ◽  
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.

scholarly journals Reproducing Kernel Method for Solving Nonlinear Fractional Fredholm Integrodifferential Equation

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Bothayna S. H. Kashkari ◽  
Muhammed I. Syam
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Integrodifferential Equation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Integrodifferential Equations ◽  
Reproducing Kernel Method ◽  
Numerical Studies ◽  
Uniformly Convergence ◽  
Present Algorithm

This article is devoted to both theoretical and numerical studies of nonlinear fractional Fredholm integrodifferential equations. In this paper, we implement the reproducing kernel method (RKM) to approximate the solution of nonlinear fractional Fredholm integrodifferential equations. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the solution of the nonlinear fractional Fredholm integrodifferential equation. Uniformly convergence of the approximate solution produced by the RKM to the exact solution is proven.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
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.

scholarly journals A high order approach for nonlinear Volterra-Hammerstein integral equations

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>

scholarly journals New method for investigating the density-dependent diffusion Nagumo equation

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 143-152 ◽  
Author(s):  
Ali Akgul ◽  
Mir Hashemi ◽  
Mustafa Inc ◽  
Dumitru Baleanu ◽  
Hasib Khan
Keyword(s):  
Exact Solution ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Kernel Functions ◽  
New Method ◽  
Series Solutions ◽  
Powerful Method ◽  
Reproducing Kernel Method ◽  
Density Dependent ◽  
Nagumo Equation

We apply reproducing kernel method to the density-dependent diffusion Nagumo equation. Powerful method has been applied by reproducing kernel functions. The approximations to the exact solution are obtained. In particular, series solutions are obtained. These solutions demonstrate the certainty of the method. The results acquired in this work conceive many attracted behaviors that assure further work on the Nagumo equation.

scholarly journals GENERALIZED JACOBI REPRODUCING KERNEL METHOD IN HILBERT SPACES FOR SOLVING THE BLACK-SCHOLES OPTION PRICING PROBLEM ARISING IN FINANCIAL MODELLING

2018 ◽  
Vol 23 (4) ◽  
pp. 538-553
Author(s):  
Mohammadreza Foroutan ◽  
Ali Ebadian ◽  
Hadi Rahmani Fazli
Keyword(s):  
Option Pricing ◽  
Reproducing Kernel ◽  
Initial Conditions ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Kernel Functions ◽  
Dirichlet Boundary Conditions ◽  
Financial Modelling ◽  
Black Scholes ◽  
Reproducing Kernel Theory

Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.

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