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

2020 ◽  
Vol 4 (2) ◽  
pp. 27 ◽  
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.

scholarly journals Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Xiaoli Zhang ◽  
Haolu Zhang ◽  
Lina Jia ◽  
Yulan Wang ◽  
Wei Zhang
Keyword(s):  
Numerical Solution ◽  
Fractional Order ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Diffusion Equations ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Reproducing Kernel Method ◽  
Reproducing Kernel Spaces ◽  
Methods Numerical

In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.

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 CONVERGENCE ORDER OF THE REPRODUCING KERNEL METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS

2016 ◽  
Vol 21 (4) ◽  
pp. 466-477 ◽  
Author(s):  
Zhihong Zhao ◽  
Yingzhen Lin ◽  
Jing Niu
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problems ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Piecewise Linear ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Reproducing Kernel Method ◽  
Rate Analysis ◽  
Reproducing Kernel Spaces

In this paper, convergence rate of the reproducing kernel method for solving boundary value problems is studied. The equivalence of two reproducing kernel spaces and some results of adjoint operator are proved. Based on the classical properties of piecewise linear interpolating function, we provide the convergence rate analysis of at least second order. Moreover, some numerical examples showing the accuracy of the proposed estimations are also given.

scholarly journals Numerical Simulation Characteristics of Logging Response in Water Injection Well by Reproducing Kernel Method

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Ming-Jing Du ◽  
Yu-Lan Wang ◽  
Temuer Chaolu
Keyword(s):  
Temperature Field ◽  
Water Temperature ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Hilbert Space ◽  
Water Injection ◽  
Kernel Functions ◽  
Injection Well ◽  
Two Phase Flows ◽  
Two Phase

Reproducing kernel Hilbert space method (RKHSM) is an effective method. This paper, for the first time, uses the traditional RKHSM for solving the temperature field in two phase flows of multilayer water injection well. According to 2D oil-water temperature field mathematical model of two phase flows in cylindrical coordinates, selecting the properly initial and boundary conditions, by the process of Gram-Schmidt orthogonalization, the analytical solution was given by reproducing kernel functions in a series expansion form, and the approximate solution was expressed byn-term summation. The satisfied numerical results were carried out by Mathematica 7.0, showing that the larger the difference between injected water temperature and initial borehole temperature or water injection conditions, the more obvious the indication of water accepting zones. The numerical examples evidence the feasibility and effectiveness of the proposed method of the two phase flows in engineering.

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 The piecewise reproducing kernel method for the time variable fractional order advection-reaction-diffusion equations

2021 ◽  
pp. 21-21
Author(s):  
Dan-Dan Dai ◽  
Ting-Ting Ban ◽  
Yu-Lan Wang ◽  
Wei Zhang
Keyword(s):  
Fractional Order ◽  
Legendre Polynomials ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reaction Diffusion ◽  
Time Variable ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Variable Order ◽  
Reproducing Kernel Method

This paper structures some new reproductive kernel spaces based on Legendre polynomials to solve time variable order fractional advection-reaction-diffusion equations. Some examples are given to show the effectiveness and reliability of the method.

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