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 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.

Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method

2019 ◽  
Vol 30 (11) ◽  
pp. 4711-4733 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Numerical Simulation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiphase Flows ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Fractional Partial Differential Equations ◽  
Reproducing Kernel Method ◽  
Content Type

Purpose The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this paper is to present results on the numerical simulation for time-fractional partial differential equations arising in transonic multiphase flows, which are described by the Tricomi and the Keldysh equations of Robin functions types. Design/methodology/approach Those resulting mathematical models are solved by using the reproducing kernel method, which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations and error bounds under some hypotheses, which provide the theoretical basis of the proposed method are also discussed. Findings The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the method are discussed through academic validations. Originality/value In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional PDEs such as those found in the transonic multiphase flows. The authors applied the reproducing kernel method systematically for the numerical solutions of time-fractional Tricomi and Keldysh equations subject to Robin functions types.

Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method

2015 ◽  
Vol 20 (8) ◽  
pp. 3283-3302 ◽  
Author(s):  
Omar Abu Arqub ◽  
Mohammed AL-Smadi ◽  
Shaher Momani ◽  
Tasawar Hayat
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Reproducing Kernel Hilbert Space ◽  
Space Method

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 A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Er Gao ◽  
Songhe Song ◽  
Xinjian Zhang
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Space Method ◽  
Term Approximation

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional orderq∈(1,2]based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so then-term approximation. At the same time, then-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which shows that the new algorithm is efficient and accurate.

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.

scholarly journals A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
F. Z. Geng ◽  
X. M. Li
Keyword(s):  
Differential Equations ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Initial Position ◽  
New Method ◽  
Riccati Differential Equation ◽  
Reproducing Kernel Method ◽  
Riccati Differential Equations ◽  
Linear Problems ◽  
Quasilinearization Technique

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

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 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.

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