Numerical algorithm to solve a coupled system of fractional order using a novel reproducing kernel method

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.

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

Thermal Science ◽

10.2298/tsci200302021d ◽

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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