Browsing by Author "Kumar, Amit"

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Citation Count: Kumar, Sunil...et al. (2020). "A modified analytical approach with existence and uniqueness for fractional Cauchy reaction-diffusion equations", Advances in Difference Equations, Vol. 2020, No. 1.
A modified analytical approach with existence and uniqueness for fractional Cauchy reaction-diffusion equations
(2020) Kumar, Sunil; Kumar, Amit; Abbas, Syed; Al Qurashi, Maysaa; Baleanu, Dumitru; 56389
This article mainly explores and applies a modified form of the analytical method, namely the homotopy analysis transform method (HATM) for solving time-fractional Cauchy reaction-diffusion equations (TFCRDEs). Then mainly we address the error norms L2 and L infinity for a convergence study of the proposed method. We also find existence, uniqueness and convergence in the analysis for TFCRDEs. The projected method is illustrated by solving some numerical examples. The obtained numerical solutions by the HATM method show that it is simple to employ. An excellent conformity obtained between the solution got by the HATM method and the various well-known results available in the current literature. Also the existence and uniqueness of the solution have been demonstrated.
Citation Count: Kumar, Amit; Baleanu, Dumitru (2021). "An analysis for Klein-Gordon equation using fractional derivative having Mittag-Leffler-type kernel", Mathematical Methods in the Applied Sciences, Vol. 44, No. 7, pp. 5458-5474.
An analysis for Klein-Gordon equation using fractional derivative having Mittag-Leffler-type kernel
(2021) Kumar, Amit; Baleanu, Dumitru; 56389
Within this paper, we present an analysis of the fractional model of the Klein-Gordon (K-G) equation. K-G equation is considered as one of the significant equations in mathematical physics that describe the interaction of soliton in a collision less plasma. In a novel aspect of this work, we have used the latest form of fractional derivative (FCs), which contains the Mittag-Leffler type of kernel. The homotopy analysis transform method (HATM) is being taken to solve the fractional model of the K-G equation. A convergence study of HATM has been studied. The existence and uniqueness of the solution for the fractional K-G equation are presented. For verifying the obtained numerical outcomes regarding accuracy and competency, we have given different graphical presentations. Figures are reflecting that a novel form of the technique is a good organization in respect of proficiency and accurateness to solve the mentioned fractional problem.
Citation Count: Zhang, Yu; Kumar, Amit; Kumar, Sunil; et al., "Residual power series method for time-fractional Schrodinger equations", Journal of Nonlinear Sciences and Applications, Vol. 9, No. 11, pp. 5821-5829, (2016).
Residual power series method for time-fractional Schrodinger equations
(Int Scientific Research Publications, 2016) Zhang, Yu; Kumar, Amit; Kumar, Sunil; Baleanu, Dumitru; 56389
In this paper, the residual power series method (RPSM) is effectively applied to find the exact solutions of fractional-order time dependent Schrodinger equations. The competency of the method is examined by applying it to the several numerical examples. Mainly, we find that our solutions obtained by the proposed method are completely compatible with the solutions available in the literature. The obtained results interpret that the proposed method is very effective and simple for handling different types of fractional differential equations (FDEs). (C) 2016 All rights reserved.
Citation Count: Vats, R.K...et al. (2014). Triple fixed point theorems via alpha-series in partially ordered metric spaces. Triple fixed point theorems via alpha-series in partially ordered metric spaces. http://dx.doi.org/10.1186/1029-242X-2014-176
Triple fixed point theorems via alpha-series in partially ordered metric spaces
(Springer International Publishing, 2014) Vats, Ramesh Kumar; Taş, Kenan; Sihag, Vizender; Kumar, Amit; 4971
This manuscript has two aims: first we extend the definitions of compatibility and weakly reciprocally continuity, for a trivariate mapping F and a self-mapping g akin to a compatible mapping as introduced by Choudhary and Kundu (Nonlinear Anal. 73:2524-2531, 2010) for a bivariate mapping F and a self-mapping g. Further, using these definitions we establish tripled coincidence and fixed point results by applying the new concept of an alpha-series for sequence of mappings, introduced by Sihag et al. (Quaest. Math. 37:1-6, 2014), in the setting of partially ordered metric spaces.
Citation Count: Kumar, S., Kumar, A., Baleanu, D. (2016). Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger's equations arise in propagation of shallow water waves. Nonlinear Dynamics, 85(2), 699-715. http://dx.doi.org/10.1007/s11071-016-2716-2
Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger's equations arise in propagation of shallow water waves
(Springer, 2016) Kumar, Sunil; Kumar, Amit; Baleanu, Dumitru
In this paper, an analytical method based on the generalized Taylors series formula together with residual error function, namely residual power series method (RPSM), is proposed for finding the numerical solution of the coupled system of time-fractional nonlinear Boussinesq-Burger's equations. The Boussinesq-Burger's equations arise in studying the fluid flow in a dynamic system and describe the propagation of the shallow water waves. Subsequently, the approximate solutions of time-fractional nonlinear coupled Boussinesq-Burger's equations obtained by RPSM are compared with the exact solutions as well as the solutions obtained by modified homotopy analysis transform method. Then, we provide a rigorous convergence analysis and error estimate of RPSM. Numerical simulations of the results are depicted through different graphical representations and tables showing that present scheme is reliable and powerful in finding the numerical solutions of coupled system of fractional nonlinear differential equations like Boussinesq-Burger's equations