Browsing by Author "Kumar, Rakesh"

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Citation - WoS: 7
Citation - Scopus: 3
Artificial Neural Networks for the Wavelet Analysis of Lane-Emden Equations: Exploration of Astrophysical Enigma
(Taylor & Francis inc, 2024) Aeri, Shivani; Baleanu, Dumitru; Kumar, Rakesh
The equations of Lane-Emden (LE) can be visualized in various phenomena of astrophysics, fluid mechanics, polymer science and material science, thus the main concern of the present study is to put a novel effort to resolve these equations by utilizing the artificial neural networking approach incorporation with Vieta-Lucas wavelets called as VLW-ANN method. This unique combination of neural networking and Vieta-Lucas wavelets has been prepared to reduce the computational challenges as well as to overcome the obstacles while dealing with singularity. Many examples of the LE variety are solved by this approach. The effectiveness, accuracy and simplicity of the VLW-ANN scheme are demonstrated by a comparative study between the VLW-ANN results and existing results. Additionally, the results are shown in tables and figures, which give a more favorable impression of the scheme's dependability. VLW-ANN scheme will provide interesting results for other non-linear models.
Citation - WoS: 11
Citation - Scopus: 21
Normalized Lucas Wavelets: an Application To Lane-Emden and Pantograph Differential Equations
(Springer Heidelberg, 2020) Koundal, Reena; Srivastava, K.; Baleanu, D.; Kumar, Rakesh
In this paper, a novel normalized Lucas wavelet scheme based on tau approach is proposed for the two classes of second-order differential equations, namely Lane-Emden and pantograph equations. The introduced scheme depends on shifted Lucas polynomials (SLPs) and their operational matrix of derivative (which are developed here). The weight function for the orthogonality of Lucas polynomials, and Rodrigues formula are proposed for the first time, which form the basis for the construction of SLPs. Normalized Lucas wavelets are constructed by utilizing SLPs and their novel properties. Literally, the present scheme transforms the given method to a set of nonlinear algebraic equations with undetermined coefficients which are here tackled by tau method. Meanwhile, new treatment of convergence and error analysis is provided for the established approach. Finally, the accuracy and applicability of present scheme is ensured by considering several examples.