Reproducing kernel functions for difference equations

Akgül, Ali; İnç, Mustafa; Karatas, Esra

doi:10.3934/dcdss.2015.8.1055

Cited by 27 publications

(15 citation statements)

References 18 publications

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“…Various Investigators have created some other techniques based on Homotopy are HPM [10] , [14] , [15] , [16] , OHAM [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , and Optimal Homotopy Perturbation Method (OHPM) [27] , [28] , [29] , [30] to achieve the solution of BVPs. The new relevant work can also be seen in [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] , [45] , [46] . Some work about the applied method can be seen in [47] , [48] , [49] , [50] , [51] , [52] , [53] .…”

Section: Introductionmentioning

confidence: 97%

Solutions of nonlinear real world problems by a new analytical technique

Ali

Islam

Gul

et al. 2018

Heliyon

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Here a new analytical scheme is presented to solve nonlinear boundary value problems (BVPs) of higher order occurring in nonlinear phenomena. This method is called second alternative of Optimal Homotopy Asymptotic Method. It converts a complex nonlinear problem into zeroth order and first order problem. A homotopy and auxiliary functions which are consisted of unknown convergence controlling parameters are used in this technique. The unknown parameters are determined by minimizing the residual. Many methods are used to determine these parameters. Here Galerkin's method is used for this purpose. It is applied to solve non-linear BVPs of order four, five, six, and seven. The Consequences are compared with other methods e.g., Differential Transform Method (DTM), Adomain Decomposition Method (ADM), Variational Iteration Method (VIM), and Optimal Homotopy Asymptotic Method (OHAM). It gives efficient and accurate first-order approximate solution. The achieved results are compared with the exact solutions as well as with other methods to authenticate the applied technique. This method is very simple and easy but more operative.

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Section: Introductionmentioning

confidence: 97%

Solutions of nonlinear real world problems by a new analytical technique

Ali

Islam

Gul

et al. 2018

Heliyon

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“…Kashkari et al have applied reproducing kernel method for solving nonlinear fractional Fredholm integro‐differential equation. Fore more details, see the previous studies …”

Section: Introductionmentioning

confidence: 99%

Solutions of fractional gas dynamics equation by a new technique

Akgül

Cordero

Torregrosa

2019

Math Methods in App Sciences

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In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined.Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section. KEYWORDS fractional gas dynamics equation, Hilbert space, operators MSC CLASSIFICATION46E22; 34A08 Recently, many numerical methods have been implemented to solve fractional differential equations. These techniques are Adomian decomposition method, 17 homotopy analysis method, 18 homotopy perturbation method, 19 and variational iteration method. 20 Singh et al 21 investigated the fractional partial differential equations showing up in biological populations. Generalized differential transform method has been implemented to search the solution of time-fractional reaction diffusion equations. 22 Singh et al 23 have investigated many nonlinear fractional differential equations. Bhrawy et al 24 have applied the operational matrix concept for solving the fractional diffusion equations. Machado et al 25 and Yang et al 26 have implemented local fractional series method to investigate fractional differential equations.

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“…Aronszajn [4] reduced the studies of the formers and presented a systematic reproducing kernel theory containing the Bergman kernel function. For more details see [5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

New method for investigating the density-dependent diffusion Nagumo equation

et al. 2018

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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.

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Reproducing kernel functions for difference equations

Cited by 27 publications

References 18 publications

Solutions of nonlinear real world problems by a new analytical technique

Solutions of nonlinear real world problems by a new analytical technique

Solutions of fractional gas dynamics equation by a new technique

New method for investigating the density-dependent diffusion Nagumo equation

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