Browsing by Author "Kumar, Sachin"

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Citation Count: Kumar, Sachin; Baleanu, Dumitru (2020). "A New Numerical Method for Time Fractional Non-linear Sharma-Tasso-Oliver Equation and Klein-Gordon Equation With Exponential Kernel Law", Frontiers in Physics, Vol. 8.
A New Numerical Method for Time Fractional Non-linear Sharma-Tasso-Oliver Equation and Klein-Gordon Equation With Exponential Kernel Law
(2020) Kumar, Sachin; Baleanu, Dumitru; 56389
In this work, we derived a novel numerical scheme to find out the numerical solution of fractional PDEs having Caputo-Fabrizio (C-F) fractional derivatives. We first find out the formula of approximation for the C-F derivative of the function f(t) = t(k). We approximate the C-F derivative in time direction with the help of Legendre spectral method and approximation formula of t(k). The unknown function and their derivatives in spatial direction are approximated with the help of the method which is based on a quasi wavelet. We implement this newly derived method to solve the non-linear Sharma-Tasso-Oliver equation and non-linear Klein-Gordon equation in which time-fractional derivative is of C-F type. The accuracy and validity of this new method are depicted by giving the numerical solution of some numerical examples. The numerical results for the particular cases of Klein-Gordon equation are compared with the existing exact solutions and from the obtained error we can conclude that our proposed numerical method achieves accurate results. The effect of time-fractional exponent alpha on the solution profile is characterized by figures. The comparison of solution profile u(x, t) for different type time-fractional derivative (C-F vs. Caputo) is depicted by figures.
Citation Count: Pandey, Prashant...et al. (2020). "An efficient technique for solving the space-time fractional reaction-diffusion equation in porous media", Chinese Journal of Physics, Vol. 68, pp. 483-492.
An efficient technique for solving the space-time fractional reaction-diffusion equation in porous media
(2020) Pandey, Prashant; Kumar, Sachin; Gomez-Aguilar, J. F.; Baleanu, Dumitru; 56389
In this paper, we obtained the approximate numerical solution of space-time fractional-order reaction-diffusion equation using an efficient technique homotopy perturbation technique using Laplace transform method with fractional-order derivatives in Caputo sense. The solution obtained is very useful and significant to analyze the many physical phenomenons. The present technique demonstrates the coupling of the homotopy perturbation technique and Laplace transform using He's polynomials for finding the numerical solution of various non-linear fractional complex models. The salient features of the present work are the graphical presentations of the approximate solution of the considered porous media equation for different particular cases and reflecting the presence of reaction terms presented in the equation on the physical behavior of the solute profile for various particular cases.
Citation Count: Kumar, Sachin...et al. (2020). "Derivation of operational matrix of Rabotnov fractional-exponential kernel and its application to fractional Lienard equation", Alexandria Engineering Journal, Vol. 59, No. 5, pp. 2991-2997.
Derivation of operational matrix of Rabotnov fractional-exponential kernel and its application to fractional Lienard equation
(2020) Kumar, Sachin; Gomez-Aguilar, J. F.; Lavin-Delgado, J. E.; Baleanu, Dumitru; 56389
Our motive in this contribution is to find out the operational matrix of fractional derivative having non singular kernel namely Rabotnov fractional-exponential (RFE) kernel which is recently introduced and seeking numerical solution of non-linear Lienard equation which have Rabotnov fractional-exponential kernel fractional derivative. First we derive an approximation formula of the fractional order derivative of polynomial function z(k) in term of RFE kernel. Using this formula and some properties of shifted Legendre polynomials, we find out the operational matrix of fractional order differentiation. In the author of knowledge this operational matrix of RFE kernel fractional derivative is derived first time. We solve a new class of fractional partial differential equation (FPDEs) by implementation of this newly derived operational matrix. We show that our newly derived operational matrix is valid by taking an fractional derivative of a polynomial. Also, we study a new model of Lienard equation with RFE kernel fractional derivative and we can easily predict the feasibility of our numerical method to this new model. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
Citation Count: Kumar, Sachin...et al. (2020). "DOUBLE-QUASI-WAVELET NUMERICAL METHOD FOR THE VARIABLE-ORDER TIME FRACTIONAL AND RIESZ SPACE FRACTIONAL REACTION-DIFFUSION EQUATION INVOLVING DERIVATIVES IN CAPUTO-FABRIZIO SENSE", Fractals-Complex Geometry Patterns and Scaling in Nature and Society, Vol. 28, No. 8.
DOUBLE-QUASI-WAVELET NUMERICAL METHOD FOR THE VARIABLE-ORDER TIME FRACTIONAL AND RIESZ SPACE FRACTIONAL REACTION-DIFFUSION EQUATION INVOLVING DERIVATIVES IN CAPUTO-FABRIZIO SENSE
(2020) Kumar, Sachin; Pandey, Prashant; Gomez-Aguilar, J. F.; Baleanu, Dumitru; 56389
Our motive in this scientific contribution is to deal with nonlinear reaction-diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo-Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional type. To approximate the variable-order time fractional derivative, we used a difference scheme based upon the Taylor series formula. While approximating the variable order spatial derivatives, we apply the quasi-wavelet-based numerical method. Here, double-quasi-wavelet numerical method is used to investigate this type of model. The discretization of boundary conditions with the help of quasi-wavelet is discussed. We have depicted the efficiency and accuracy of this method by solving the some particular cases of our model. The error tables and graphs clearly show that our method has desired accuracy.
Citation Count: Kumar, Sachin;...et.al. (2022). "Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations", Symmetry, Vol.14, No.3.
Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations
(2022) Kumar, Sachin; Kumar Dhiman, Shubham; Baleanu, Dumitru; Osman, M.S.; 56389
This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with timedependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and nonvanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models.
Citation Count: Singh, Jaskirat Pal...et al. (2023). "MONKEYPOX VIRAL TRANSMISSION DYNAMICS AND FRACTIONAL-ORDER MODELING WITH VACCINATION INTERVENTION", Fractals, Vol. 31, No. 10.
MONKEYPOX VIRAL TRANSMISSION DYNAMICS AND FRACTIONAL-ORDER MODELING WITH VACCINATION INTERVENTION
(2023) Singh, Jaskirat Pal; Kumar, Sachin; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; 56389
A current outbreak of the monkeypox viral infection, which started in Nigeria, has spread to other areas of the globe. This affects over 28 nations, including the United Kingdom and the United States. The monkeypox virus causes monkeypox (MPX), which is comparable to smallpox and cowpox (MPXV). The monkeypox virus is a member of the Poxviridae family and belongs to the Orthopoxvirus genus. In this work, a novel fractional model for Monkeypox based on the Caputo derivative is explored. For the model, two equilibria have been established: disease-free and endemic equilibrium. Using the next-generation matrix and Castillo's technique, if R0 < 1 the global asymptotic stability of disease-free equilibrium is shown. The linearization demonstrated that the endemic equilibrium point is locally asymptotically stable if R0 > 1. Using the parameter values, the model's fundamental reproduction rates for both humans and non-humans are calculated. The existence and uniqueness of the solution are proved using fixed point theory. The model's numerical simulations demonstrate that the recommended actions will cause the infected people in the human and non-human populations to disappear.
Citation Count: Kumar, Sachin; Baleanu, Dumitru (2020). "Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel",Mathematical Methods in the Applied Sciences, Vol. 43, No. 15, pp. 8348-8362.
Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel
(2020) Kumar, Sachin; Baleanu, Dumitru; 56389
The main motive of this article is to study the recently developed Atangana-Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag-Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two-dimensional cable equation in which time-fractional derivative is of Mittag-Leffler kernel type. First, we derive an approximation formula of the fractional-order ABC derivative of a function t(k) using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two-dimensional model of the time-fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two-dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag-Leffler fractional derivative.
Citation Count: Ghanbari, Behzad...et al. (2021). "The Lie symmetry analysis and exact Jacobi elliptic solutions for the Kawahara–KdV type equations", Results in Physics, Vol. 23.
The Lie symmetry analysis and exact Jacobi elliptic solutions for the Kawahara–KdV type equations
(2021) Ghanbari, Behzad; Kumar, Sachin; Niwas, Monika; Baleanu, Dumitru; 56389
In this article, we aim to employ two analytical methods including, the Lie symmetry method and the Jacobi elliptical solutions finder method to acquire exact solitary wave solutions in various forms of (1+1)-dimensional Kawahara–KdV type equation and modified Kawahara–KdV type equation. These models are famous models that arise in the modeling of many complex physical phenomena. At the outset, we have generated geometric vector fields and infinitesimal generators of Kawahara–KdV type equations. The (1+1)-dimensional Kawahara–KdV type equations reduced into ordinary differential equations (ODEs) using Lie symmetry reductions. Furthermore, numerous exact solitary wave solutions are obtained utilizing the Jacobi elliptical solutions finder method with the help of symbolic computation with Maple. The obtained results are new in the formulation, and more useful to explain complex physical phenomena. The results reveal that these mathematical approaches are straightforward, effective, and powerful methods that can be adopted for solving other nonlinear evolution equations.
Citation Count: Singh, Jaskirat Pal...et.al. (2023). "Transmission dynamics of a novel fractional model for the Marburg virus and recommended actions", European Physical Journal: Special Topics, Vol.232, No.14-15, pp.2645-2655.
Transmission dynamics of a novel fractional model for the Marburg virus and recommended actions
(2023) Singh, Jaskirat Pal; Abdeljawad, Thabet; Baleanu, Dumitru; Kumar, Sachin; 56389
Marburg virus disease is a particularly virulent illness that causes hemorrhagic fever and has a fatality rate of up to 88%. It belongs to the same family of pathogens as the Ebola virus. The disease was first identified in 1967 as a result of two significant epidemics that happened concurrently in Marburg, hence the name Marburg, Frankfurt, both in Germany, and Belgrade, Serbia. This work proposes a unique fractional model for the Marburg virus based on the Atangana–Baleanu derivative in the Caputo sense. For the model, two equilibrium states have been founded: endemic equilibrium and disease-free equilibrium. If R< 1 , Castillo’s method and the next-generation matrix are used to demonstrate the disease-free equilibrium’s asymptotic global stability. When R> 1 , the endemic equilibrium point is locally asymptotically stable, according to the linearization. The model’s basic reproduction rates for both humans and bats are calculated using the parameter values. Fixed point theory is used to demonstrate the solution’s existence and uniqueness. Number of infected bats should be controlled and interaction with just recovered individuals should be avoided as these are the main contributors in the infection rate. These recommended actions will make the infected persons in the humans disappear, as demonstrated by the model’s numerical simulations.