Browsing by Author "Kumar, R."

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Citation - WoS: 101
Citation - Scopus: 116
An Efficient Computational Approach for a Fractional-Order Biological Population Model With Carrying Capacity
(Pergamon-elsevier Science Ltd, 2020) Dubey, V. P.; Kumar, R.; Singh, J.; Kumar, D.; Baleanu, D.; Srivastava, H. M.
In this article, we examine a fractional-order biological population model with carrying capacity. The blended homotopy techniques pertaining to the Sumudu transform are utilized to explore the solutions of a nonlinear fractional-order population model with carrying capacity. The fractional derivative of the Caputo type is utilized in the proposed investigation. The numerical computations indicate the sufficiency of the iterations for the improved estimations of the solutions of this fractional-order biological population model which exemplifies the potency and soundness of the utilized schemes. The analysis explored through the utilization of the projected methods reveals that both of the schemes are in a great agreement with each other. The variations of the prey and predator populations with respect to time and fractional order of the Caputo derivative are presented and graphically illustrated. (c) 2020 Elsevier Ltd. All rights reserved.
Citation - Scopus: 16
Lucas Wavelet Scheme for Fractional Bagley–torvik Equations: Gauss–jacobi Approach
(Springer, 2022) Koundal, R.; Kumar, R.; Srivastava, K.; Baleanu, D.
A novel technique called as Lucas wavelet scheme (LWS) is prepared for the treatment of Bagley–Torvik equations (BTEs). Lucas wavelets for the approximation of unknown functions appearing in BTEs are introduced. Fractional derivatives are evaluated by employing Gauss–Jacobi quadrature formula. Further, well-known least square method (LSM) is adopted to compute the residual function, and the system of algebraic equation is obtained. Convergence criterion is derived and error bounds are obtained for the established technique. The scheme is investigated by choosing some reliable test problems through tables and figures, which ensures the convenience, validity and applicability of LWS. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.
Citation - Scopus: 8
A Novel Collocated-Shifted Lucas Polynomial Approach for Fractional Integro-Differential Equations
(Springer, 2021) Kumar, R.; Srivastava, K.; Baleanu, D.; Koundal, R.
In current analysis, a novel computational approach depending on shifted Lucas polynomials (SLPs) and collocation points is established for fractional integro-differential equations (FIDEs) of Volterra/Fredholm type. A definition for integer order derivative and the lemma for fractional derivative of SLPs are developed. To convert the given equations into algebraic set of equations, zeros of the Lucas polynomial are used as collocation points. Novel theorems for convergence and error analysis are developed to design rigorous mathematical basis for the scheme. Accuracy is proclaimed through comparison with other known methods. © 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited.
Citation - Scopus: 10
Solving System of Fractional Differential Equations Via Vieta-Lucas Operational Matrix Method
(Springer, 2024) Aeri, S.; Bala, A.; Kumar, R.; Baleanu, D.; Chaudhary, R.
Vieta-Lucas polynomials (VLPs) belong to the class of weighted orthogonal polynomials, which can be used to effectively handle various natural and engineered problems. The classical fractional derivative due to Caputo is used to write the emerging operational matrices. These matrices are developed and evaluated by using the properties of VLPs. The residuated functions are mapped to zero by the tools of the Tau algorithm. Convergence and error analysis are thoroughly explored. Test examples for a fractional system of differential equations are borrowed from literature. The theoretical and simulated exercise on these examples authenticate the relevance of this scheme. Here, novel inclusion of Vieta-Lucas polynomials has been ensured in combination with the Tau approach. The operational matrix approach which provides extensive information about the fractional derivatives of different terms of Vieta-Lucas polynomial expansion, is ensured to operate to reduce the problem into an algebraic setup. The novelty is further enhanced by comparing the present scheme with the fourth-order Runge–Kutta method. © 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited.