Solving the Lane–Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme

Hashemi, Mir Sajjad; Akgül, Ali; İnç, Mustafa; Mustafa, Idrees; Băleanu, Dumitru

doi:10.3390/math5040077

Cited by 19 publications

(8 citation statements)

References 32 publications

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“…A quick overview of the reproducing kernel results can be viewed from previous studies, 28–30 and an overview of its application fields can be collected from previous studies 31–57 . This modern numerical method is based on its structure on pointwise evaluation, successive approximations, and the Green functions approach.…”

Section: Preface and Showmentioning

confidence: 99%

Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator

Arqub

Singh

Maayah

et al. 2021

Math Methods in App Sciences

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Atangana-Baleanu-Caputo differential operators are analytically and numerically treated using extended reproducing kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions, analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three-dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

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Section: Preface and Showmentioning

confidence: 99%

Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator

Arqub

Singh

Maayah

et al. 2021

Math Methods in App Sciences

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“…e author's show many advantages of the proposed scheme like to start the procedure and choose any point lies in the limits of integration, and it requires less effort to investigate the results. Later, various researchers used this strategy to explore the two-point fuzz BVPs model [20,21], fuzzy differential model [22], periodic first-order BVPs of integro-differential model Fredholm type [23], systems of periodic second-order BVPs [24], Lane-Emden equation, and fractional-order model of Lane-Emden [25,26].…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis Hilbert Space Approach for Fuzzy Integro‐Differential Models

Zhang

2022

Journal of Mathematics

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In this paper, we present and demonstrate an innovative numerical method, which makes use of fuzzy numbers and fuzzy parameters that is effective in the solution of fuzzy type Volterra integro-differential equations, which was previously thought to be impossible using conventional methods. The first application of a technique for solving Volterra integro-differential equations of the fuzzy type, which was devised and tested in this paper, is shown here. This is the first time that this approach has been used. This system’s overall quality may be improved as a consequence of the use of the Hilbert space replicating kernel idea, which is a possibility. Separate evaluations are made of the algorithms’ correctness and sloppiness, as well as their foundations in the computationally effective kernel Hilbert space, which has been extensively researched in the past. Numerical examples are provided of the article to demonstrate how the technique outlined before may achieve convergence and accuracy. Here are a few illustrations to help understand that it is possible to deal with physical issues that require complicated geometric calculations with the assistance of the method explained in this article.

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“…A variety of schemes have been implemented to solve the pantograph types of system like the one-dimensional transformation method, Direchlet series approach, the Taylor polynomial scheme [40][41][42][43][44][45]. In order to solve the singular systems, few common schemes have been presented like the Haar Adomian method is presented by Saeed et al [47], the series scheme is implemented by Romas et al [46], the differential transformation scheme is explored by Sabir et al [49] and the reproduced kernel approach applied by Hashemi et al [48]. Moreover, the singular system has been numerically treated by the swarming/heuristic approaches have been given in these citations [50][51][52][53].…”

Section: Introductionmentioning

confidence: 99%

Design of Artificial Neural Networks for the Novel Applications of the Sixth Order Singular Nonlinear Pantograph Engineering Model

Sabir

Ali

Raja

et al. 2021

Preprint

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This aim of this study is to design and numerically investigate a novel nonlinear singular sixth order (SSO) pantograph differential system (PDS), i.e., SSO-PDS along with its five categories. The design of the novel nonlinear SSO-PDS is presented using the concepts of the standard Emden-Fowler along with the pantograph differential system. The singular systems are applied in many engineering and mathematical applications, like inverse models and viscoelasticity systems. The shape factor, singular point and pantograph terms are provided for each category of the novel nonlinear SSO-PDS. The numerical investigations of the nonlinear SSO-PDS are presented through the artificial neural networks (ANNs) using the Levenberg-Marquardt backpropagation (LMB), i.e., ANNs-LMB. The proposed results have been compared with the reference solutions to check the correctness of the designed novel model. The proposed solutions of the novel SSO-PDS are assessed using the training, testing and verification procedures in the form of mean square error (MSE). For the capability, efficacy and accuracy of the novel SSO-PDS through the ANNs, the numerical results are drawn using the MSE, error histograms and regression as well as correlation graphs.

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Solving the Lane–Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme

Cited by 19 publications

References 32 publications

Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator

Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator

Convergence Analysis Hilbert Space Approach for Fuzzy Integro‐Differential Models

Design of Artificial Neural Networks for the Novel Applications of the Sixth Order Singular Nonlinear Pantograph Engineering Model

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