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 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 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 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 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 New Reproducing Kernel Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Ali Akgül
Keyword(s):  
Time Scales ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Kernel Functions ◽  
Dynamic Equations ◽  
Reproducing Kernel Method

Some new reproducing kernel functions on time scales are presented. Reproducing kernel functions have not been found on time scales till now. These functions are very important on time scales and they will be very useful for researchers. We need these functions to solve dynamic equations on time scales with the reproducing kernel 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 novel method for the space and time fractional Bloch-Torrey equations

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 253-258 ◽  
Author(s):  
Esra Akgul
Keyword(s):  
Exact Solution ◽  
Reproducing Kernel ◽  
Kernel Functions ◽  
Efficient Technique ◽  
Series Solutions ◽  
Numerical Example ◽  
Space And Time ◽  
Novel Method

Reproducing kernel technique was implemented to solve the fractional Bloch-Torrey equations. This efficient technique was used via some useful reproducing kernel functions, to obtain approximations to the exact solution in form of series solutions. A numerical example has been presented to prove efficiency of developed technique.

scholarly journals A novel method for nonlinear singular oscillators

2020 ◽  
pp. 146134842098053
Author(s):  
Ali Akgül ◽  
Hijaz Ahmad ◽  
Yu-Ming Chu ◽  
Phatiphat Thounthong
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Reproducing Kernel Method ◽  
Novel Method ◽  
Relationship Of

The present work deals with a study of a nonlinear singular oscillator. To approximate the frequency–amplitude relationship of the singular oscillator, reproducing kernel method is employed. The approximate solution is compared with the exact solution as well as the results obtained by the He’s frequency–amplitude formulation, to show the effectiveness of the proposed technique for solving the problem.

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