Mathematics And Statistics - Research Publications
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Item The (3 + 1)-dimensional Benjamin-Ono equation: Painlev� analysis, rogue waves, breather waves and soliton solutions(World Scientific, 2022-06-08T00:00:00) Kumar, Sachin; Malik, SandeepIn this paper, we analyzed the (3 + 1)-dimensional Benjamin-Ono (BO) equation. We first demonstrated that the governing model is not integrable in the Painlev� sense. The rogue wave and the breather wave solutions are then achieved with the use of bilinear form. Furthermore, using a combination of Lie symmetry analysis with the new Kudryshov method, and the Riccati equation technique, the abundant soliton and singular periodic solutions were derived. The criteria for existence of such solutions are also provided. Consequently, the derived solutions are presented graphically through 3D, 2D and contour plots, which describe useful physical phenomena due to existence of the free parameters. Corresponding to the one-reduction, power series solution of BO equation is also obtained. � 2022 World Scientific Publishing Company.Item Application of new Kudryashov method to various nonlinear partial differential equations(Springer, 2022-11-13T00:00:00) Malik, Sandeep; Hashemi, Mir Sajjad; Kumar, Sachin; Rezazadeh, Hadi; Mahmoud, W.; Osman, M.S.The purpose of this work is to seek various innovative exact solutions using the new Kudryashov approach to the nonlinear partial differential equations (NLPDEs). This technique obtains novel exact solutions of soliton types. Moreover, several 3D and 2D plots of the higher dimensional Klein-Gordon, Kadomtsev-Petviashvili, and Boussinesq equations are demonstrated by considering the relevant values of the aforementioned parameters to exhibit the nonlinear wave structures more adequately. The new Kudryashov technique is an effective, and simple technique that provides new generalized solitonic wave profiles. It is anticipated that these novel solutions will enable a thorough understanding of the development and dynamic nature of such models. � 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.Item A Bivariate Teissier Distribution: Properties, Bayes Estimation and Application(Springer, 2023-08-29T00:00:00) Sharma, Vikas Kumar; Singh, Sudhanshu Vikram; Pathak, Ashok KumarThis article presents a bivariate extension of the Teissier distribution, whose univariate marginal distributions belong to the exponentiated Teissier family. Analytic expressions for the different statistical quantities such as conditional distribution, joint moments, and quantile function are explicitly derived. For the proposed distribution, the concepts of reliability and dependence measures are also explored in details. Both the maximum likelihood technique and the Bayesian approach are utilised in the process of parameter estimation for the proposed distribution with unknown parameters. Several numerical experiments are reported to study the performance of the classical and Bayes estimators for varying sample size. Finally, a bivariate data is fitted using the proposed distribution to show its applicability over the bivariate exponential, Rayleigh, and linear exponential distributions in real-life situations. � 2023, Indian Statistical Institute.Item A case study of 2019-nCOV cases in Argentina with the real data based on daily cases from March 03, 2020 to March 29, 2021 using classical and fractional derivatives(Springer Science and Business Media Deutschland GmbH, 2021-07-20T00:00:00) Kumar, Pushpendra; Erturk, Vedat Suat; Murillo-Arcila, Marina; Banerjee, Ramashis; Manickam, A.In this study, our aim is to explore the dynamics of COVID-19 or 2019-nCOV in Argentina considering the parameter values based on the real data of this virus from March�03, 2020 to March�29, 2021 which is a data range of more than one complete year. We propose a Atangana�Baleanu type fractional-order model and simulate it by using predictor�corrector (P-C) method. First we introduce the biological nature of this virus in theoretical way and then formulate a mathematical model to define its dynamics. We use a well-known effective optimization scheme based on the renowned trust-region-reflective (TRR) method to perform the model calibration. We have plotted the real cases of COVID-19 and compared our integer-order model with the simulated data along with the calculation of basic reproductive number. Concerning fractional-order simulations, first we prove the existence and uniqueness of solution and then write the solution along with the stability of the given P-C method. A�number of graphs at various fractional-order values are simulated to predict the future dynamics of the virus in Argentina which is the main contribution of this paper. � 2021, The Author(s).Item Comment on “Lie symmetries and group classification of a class of time fractional evolution systems” [J. Math. Phys. 56, 123504 (2015)](Springer, 2017) Singla, Komal; Gupta, R. K.In this article, the prolongation formulae proposed in Huang and Shen [J. Math. Phys. 56, 123504 (2015)] for the analysis of time fractional systems of are proved incomplete and the required correct prolongation operators are suggested. With the help of some examples, the efficiency of the operators introduced in this study is illustrated.Item Comment on “Time fractional third-order variant Boussinesq system: Symmetry analysis, explicit solutions, conservation laws and numerical approximations” by Fairouz Tchier et al.(Springer Verlag, 2019) Kour B.; Kumar S.[No abstract available]Item Constant curvature conditions for generalized kropina spaces(RAMANUJAN SOCIETY OF MATHEMATICS AND MATHEMATICAL SCIENCES, 2021-03-01T00:00:00) Shanker, Gauree; Sharma, Ruchi KaushikThe classification of Finsler spaces of constant curvature is an interesting and important topic of research in differential geometry. In this paper we obtain necessary and sucient conditions for generalized Kropina space to be of constant flag curvature. � 2021, RAMANUJAN SOCIETY OF MATHEMATICS AND MATHEMATICAL SCIENCES. All rights reserved.Item Dispersion analysis and improved F-expansion method for space–time fractional differential equations(Springer, 2019) Kaur, B; Gupta, R.K.In this article, an improved F-expansion method with the Riccati equation is suggested for space–time fractional differential equations for exact solutions. The fractional complex transformation is used to convert the space–time fractional differential equations into ordinary differential equations. The application of the method is described by solving space–time fractional potential Yu–Toda–Sasa–Fukuyama equation, and the solutions of the equation are successfully established in terms of the hyperbolic, trigonometric and rational types of functions. The graphical analysis describes the effect of fractional orders α, β, γ, δ of time and space derivatives, respectively, on the wave profile of solutions. The dispersion relation is obtained using the linear analysis, and it shows that waves follow anomalous or normal dispersion depending upon space or time fractional-order values. © 2019, Springer Nature B.V.Item Doubly periodic wave structure of the modified Schr�dinger equation with fractional temporal evolution(Elsevier B.V., 2022-01-01T00:00:00) Kumari, Pinki; Gupta, R.K.; Kumar, Sachin; Nisar, K.S.Abundant Jacobi elliptic type solutions with distinct physical structures of complex nonlinear conformable time-fractional modified Schr�dinger equation are obtained by using the generalized Jacobi elliptic function (GJEF) method. The Jacobi function expansions may lead to new doubly periodic wave solutions, soliton solutions, and triangular periodic solutions. Nowadays the conformable operator is being used for a better description of the dynamical systems. Motivated by the potential applications of the governed equation in nonlinear optics, biological sciences, and fluid dynamics, these solutions may be significant in the study of wave propagation in the desired field. Symbolic computations are made with the aid of Maple. � 2022 The AuthorsItem Exact Solutions of Some Complex Non-Linear Equations," Lecture Notes in Engineering and Computer Science(Newswood Limited, 2018) Kumar, SachinExact solutions of the coupled Higgs and Maccari system are obtained. Travelling wave solutions of coupled Higgs equation and Maccari system in the form of Jacobi’s elliptical functions are presented.Item Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations(Springer, 2020) Bedi, P; Kumar, A; Abdeljawad, T; Khan, A.In this paper, we investigate the existence of mild solutions for neutral Hilfer fractional evolution equations with noninstantaneous impulsive conditions in a Banach space. We obtain the existence results by applying the theory of resolvent operator functions, Hausdorff measure of noncompactness, and Sadovskii's fixed point theorem. We also present an example to show the validity of obtained results. 2020, The Author(s).Item Existence of solutions by fixed point theorem of general delay fractional differential equation with p-Laplacian operator(American Institute of Mathematical Sciences, 2023-02-27T00:00:00) Kaushik, Kirti; Kumar, Anoop; Khan, Aziz; Abdeljawad, ThabetIn this manuscript, the main objective is to analyze the existence, uniqueness,(EU) and stability of positive solution for a general class of non-linear fractional differential equation (FDE) with fractional differential and fractional integral boundary conditions utilizing ?p-Laplacian operator. To continue, we will apply Green�s function to determine the suggested FDE�s equivalent integral form. The Guo-Krasnosel�skii fixed point theorem and the properties of the p-Laplacian operator are utilized to derive the existence results. Hyers-Ulam (HU) stability is additionally evaluated. Further, an application is presented to validate the effectiveness of the result. � 2023 the Author(s), licensee AIMS Press.Item First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications; [Premi�re in�galit� de Chen pour des produits tordus de sous-vari�t�s d�un espace forme riemannien et applications](ISTE Group, 2023-06-15T00:00:00) Mustafa, Abdulqader; �zel, Cenap; Pigazzini, Alexander; Kaur, Ramandeep; Shanker, GaureeIn this paper, the first Chen inequality is proved for general warped product submanifolds in Riemannian space forms, this inequality involves intrinsic invariants (?-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Chen�s Problem 1 relating to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold. As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. For further research directions, we address a couple of open problems; namely Problem 3 and Problem 4. � 2023 ISTE OpenScience � Published by ISTE Ltd. London, UK.Item Four-dimensional conformally flat Berwald and Landsberg spaces(Informatics Publishing Limited and The Indian Mathematical Society, 2018) Shanker, G.The problem of conformal transformation and conformal atness of Finsler spaces has been studied in [6], [16], [17], [20], [21]. Recently, Prasad et. al [19] have studied three dimensional conformally at Landsberg and Berwald spaces and have obtained some important results. The purpose of the present paper is to extend the idea of conformal change to four dimensional Finsler spaces and find the suitable conditions under which a four dimensional conformally at Landsberg space becomes a Berwald space. ? 2018 Indian Mathematical Society.Item Fractional dynamics of 2019-nCOV in Spain at different transmission rate with an idea of optimal control problem formulation(Elsevier B.V., 2021-07-29T00:00:00) Kumar, Pushpendra; Erturk, Vedat Suat; Nisar, Kottakkaran Sooppy; Jamshed, Wasim; Mohamed, Mohamed S.In this article, we studied the fractional dynamics of the most dangerous deathly disease which outbreaks have been recorded all over the world, called 2019-nCOV or COVID-19. We used the numerical values of the given parameters based on the real data of the 2019-nCOV cases in Spain for the time duration of 25 February to 9 October 2020. We performed our observations with the help of the Atangana-Baleanu (AB) non-integer order derivative. We analysed the optimal control problem in a fractional sense for giving the information on all necessary health care issues. We applied the Predictor-Corrector method to do the important graphical simulations. Also, we provided the analysis related to the existence of a unique solution and the stability of the proposed scheme. The aim and the main contribution of this research is to analyse the structure of novel coronavirus in Spain at different transmission rate and to indicate the danger of this deathly disease for future with the introduction of some optimal controls and health care measures. � 2021Item Fractional time-delay mathematical modeling of Oncolytic Virotherapy(Elsevier Ltd, 2021-06-19T00:00:00) Kumar, Pushpendra; Erturk, Vedat Suat; Yusuf, Abdullahi; Kumar, SunilAn emerging treatment tool which uses replication-competent viruses to dissipate cancers without causing deficit to normal tissues, named as oncolytic virotherapy, is discussed in the article. We analysed a fractional delay dynamical model on the oncolytic virotherapy compositing viral lytic cycle and virus-specific cytotoxic T lymphocyte (CTL) response. We used a well known Caputo fractional derivative to analyse the structure of the given dynamical model. Using the literature of fixed-point theory, the given time-delay model is specified to have existence of a unique solution. We established different types of graphical simulations for the various values of R0 and R1. We observed a different behaviour of the given fractional model as compare to the integer order model. The given algorithm is smooth in use and reliable to apply on different delay dynamical models. � 2021 Elsevier LtdItem General form of axially symmetric stationary metric: exact solutions and conservation laws in vacuum fields(Institute of Physics, 2023-06-24T00:00:00) Jyoti, Divya; Kumar, SachinThe invariant non-static solutions of Einstein�s vacuum field equations, corresponding to the most general form of axially symmetric stationary line element that represents a non conformally flat semi-Riemannian spacetime in cylindrical coordinates, are investigated. Lie symmetry method is used for symmetry reduction as well as for obtaining exact solutions in terms of arbitrary functions. The conservation laws are obtained for vacuum equations in axially symmetric gravitational fields. The solutions of Lewis metric and Chandrasekhar metric, are derived from the obtained solutions. By considering the possibilities of arbitrary functions, the graphical behaviour of the solutions is also shown. � 2023 IOP Publishing Ltd.Item Highly dispersive W�shaped and other optical solitons with quadratic�cubic nonlinearity: Symmetry analysis and new Kudryashov's method(Elsevier Ltd, 2023-06-24T00:00:00) Yadav, Ravindra; Malik, Sandeep; Kumar, Sachin; Sharma, Rajesh; Biswas, Anjan; Y?ld?r?m, Yakup; Gonz�lez-Gaxiola, O.; Moraru, Luminita; Alghamdi, Abdulah A.Lie symmetry analysis is utilized in this paper to explore the properties of highly dispersive optical solitons that exhibit quadratic�cubic self-phase modulation. The use of Lie symmetry analysis enables the reduction of the governing partial differential equation to an ordinary differential equation, which is then integrated using an enhanced Kudryashov's approach to obtain solitons with the model. The analysis presented in this paper does not explicitly discuss the formation and dynamics of soliton radiation. � 2023Item Invariance analysis, exact solution and conservation laws of (2 + 1) dim fractional kadomtsev-petviashvili (kp) system(MDPI AG, 2021-03-16T00:00:00) Kumar, Sachin; Kour, Baljinder; Yao, Shao-Wen; Inc, Mustafa; Osman, Mohamed S.In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether�s operator, the conservation laws of the KP system are obtained. � 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item Invariant Analysis for Space�Time Fractional Three-Field Kaup�Boussinesq Equations(Springer Science and Business Media Deutschland GmbH, 2020-09-30T00:00:00) Kaur, Jaskiran; Kumar Gupta, Rajesh; Kumar, SachinSymmetries of the nonlinear fractional�differential equations are an interesting�and important topic. In this paper, space�time fractional three-field Kaup�Boussinesq equations with Riemann�Liouville fractional derivative are studied for invariant analysis. Symmetries are obtained by using classical Lie�s symmetry approach. Using obtained symmetries, the governing equations reduce to system of fractional ordinary differential equations which contains left and right-sided Erde� lyi- Kober (EK) fractional operators. � 2021, Springer Nature Singapore Pte Ltd.