
Devendra KumarUniversity of Rajasthan · Department of Mathematics
Devendra Kumar
M.Sc., M. Phil, Ph.D.
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265
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Introduction
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June 2018 - present
June 2017 - June 2018
September 2014 - May 2017
Publications
Publications (265)
The aim of the present paper is to obtain a general distribution of the linear combinations of independent non-negative Stochastic variables. This result generalizes a class of distribution of the linear combination of independent non-negative Stochastic variables and further the result of Mathai and Saxena ([8], p.163) as a particular case of the...
In the present paper, we obtain the distribution of
quotient of two independent I-function random variables. We
show that I-function and spherically- invariant random
processes (SIRP) can be used to provide a unified theory on
wireless communication fading statistics. Further, we show
that various performance measures of fading communication
system...
In this paper, a user friendly algorithm based on new homotopy perturbation transform method (HPTM) is proposed to solve nonlinear fractional Fornberg-Whitham equation in wave breaking. The new homotopy perturbation transform method is combined form of Laplace transform, homotopy perturbation method and He's polynomials. The nonlinear terms can be...
An efficient approach based on homotopy perturbation method by using sumudu transform is proposed to solve nonlinear fractional Harry Dym equation. This method is called homotopy perturbation sumudu transform (HPSTM). Furthermore, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreem...
A user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation. The fractional derivative is considered in the Caputo sense. Further, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement...
In this paper, we investigate the fractal nature of the local fractional Landau–Ginzburg–Higgs Equation (LFLGHE) describing nonlinear waves with weak scattering in a fractal medium. The main goal of the paper is to introduce and apply the Local Fractional Elzaki Variational Iteration Method (LFEVIM) for solution of LFLGHE. Convergence analysis of L...
In this paper, the solution and behaviour of local fractional Fokker–Planck equation (LFFPE) is investigated in fractal media. For this purpose, the local fractional homotopy perturbation Elzaki transform method (LFHPETM) is proposed and utilized to explore the solution of LFFPE. The proposed scheme is a merger of well known local fractional homoto...
The approximate solutions for the two dimensional nonlinear PDEs with Liouville-Caputo fractional derivative are determined and presented in this paper. Comparative numerical simulations obtained from alternative models are introduced in order to demonstrate the effectiveness and precision of the proposed techniques. Various source terms are taken...
A new type of methodology termed the local fractional natural homotopy perturbation method (LFNHPM) with the local fractional derivative operator (LFDO) was implemented in this study. The hybrid methodology combines the natural transform method (NTM) with the homotopy perturbation method (HPM).To validate and illustrate the efficacy of the current...
Elliptic-type integrals (ETIs) are very useful in solving many problems related to radiation and nuclear physics. Previously, Many authors have worked on the unification and generalization of ETIs. This work aims to derive some new theorems on generating functions. Further, we derive some more new and known results on Euler-type integrals with the...
This study aims to assess the generalized matrix transform (M-transform) of various incomplete types of special functions named generalized incomplete hypergeometric functions, incomplete H-functions, incomplete H-functions, incomplete I-functions, all of which possess a matrix argument. The matrix argument in this case is a real symmetric positive...
Newly discovered, incomplete forms of special functions are increasing the interest of both pure and applied mathematicians. The main purpose of this work is to derive four theorems on partial derivatives with incomplete Aleph functions of two variables and generalize them up to r-variables. In addition to these theorems, we also established some n...
The natural homotopy perturbation technique (NHPT) is an excellent analytical tool employed in this study to solve the nonlinear differential equations (NDEs). The Antagana-Baleanu sense is used to characterize the fractional derivatives (ABFD). We also discuss the convergence of the NHPT for NDEs. To show the applicability of the recommended techn...
In this paper, a fractional food chain system consisting of a Holling type Ⅱ functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model...
This work presents a numerical approach for handling a fractional Lienard equation (FLE) arising in an oscillating circuit. The scheme is based on the Vieta Lucas operational matrix of the fractional Liouville-Caputo derivative and the collocation method. This methodology involves a systematic approach wherein the operational matrix aids in express...
This paper examined the features of an infection therapy for fractional-order quarry-hunter systems in order to control sickness. It focused especially on how illnesses and several populations combine to affect how well harvesting policies work. We created a new dynamic model full of such ideas by examining systems with fractional-order non-integer...
In this paper, the local fractional natural decomposition method (LFNDM) is used for solving a local fractional Poisson equation. The local fractional Poisson equation plays a significant role in the study of a potential field due to a fixed electric charge or mass density distribution. Numerical examples with computer simulations are presented in...
In the present article, we introduced and explore an integral operator which consist Aleph function in the kernel with fractional calculus. In second section, we construct the characteristics of R-L fractional integral operator I β a+ and derivative operator D β a+ containing the Aleph-function and in third section, we develop the Sumudu transform...
In this paper, Natural transforms of Prabhakar integral, Hilfer–Prabhakar (HP) fractional derivative and regularized Caputo form of HP fractional derivative (HPFD) are computed. Furthermore, we investigate the solution of a fractional free-electron laser equation and fractional Cauchy problems involving the HPFD via the Natural transform. The solut...
Modern developments in nanotechnology have provided a fantastic foundation for creating a better ultra-high-performing coolant known as nanofluids for many applications in manufacturing and engineering. Numerous scholars have been drawn in by the hybrid nanomaterials’ capacity to improve heat transmission more to examine the working fluid. This stu...
In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP fractional derivative via Kharrat–Toma transform. The...
In this manuscript, an approximate analytical solution of the Helmholtz and coupled Helmholtz equations of fractional order is obtained using local fractional Sumudu decomposition method (LFSDM). The Helmholtz equations play an important role in the study of various physical problems such as seismology, tsunamis, optics, acoustics, medical imaging,...
The belongings of radiation and velocity slip on MHD stream and melting
warmth transmission of a micropolar liquid over an exponentially stretched
sheet which is fixed in a porous medium with heat source/sink are
accessible. Homothety transforms the major PDE into a set of non-linear
ODE. Then, by varying the boundary value problem to the initial v...
In this article, we evaluate the approximate solutions of Nonlinear Differential Equations (NoLDEs) with the association of S-function, incomplete H-functions (IHFs) and incomplete I-functions (IIFs) with two variables by using the Hermite, Legendre and Jacobi polynomials. Here, we introduce incomplete I-functions with two variables. The NoLDEs are...
In this paper, we present a reliable numerical algorithm to determine approximate solutions of the two-point boundary value problems having Robin boundary conditions that naturally occur in the investigation of distinct tumor growth issues, the dispersal of heat sources in the person head and steady state oxygen diffusion in spherical cell possessi...
In this paper, we implement the local fractional natural homotopy perturbation method (LFNHPM) to solve certain local fractional partial differential equations (LFPDEs) with fractal initial conditions occurring in physical sciences in a fractal domain. LFPDEs successfully exhibit the important properties of physical models occurring in a fractal me...
In this paper, we present an efficient computational approach named as Sumudu residual power series method (SRPSM) to solve fractional Bloch equations appearing in an NMR flow. This method is a copulation of the residual power series method (RPSM) and the Sumudu transform to construct approximate solution in shapes of sharp convergent series by ado...
The fractional model of diffusion equations is very important in the study of oil pollution in the water. The key objective of this article is to analyze a fractional modification of diffusion equations occurring in oil pollution associated with the Katugampola derivative in the Caputo sense. An effective and reliable computational method q-homotop...
Karst aquifers have a very complex flow system because of their high spatial heterogeneity of void distribution. In this manuscript, flow simulation has been used to investigate the flow mechanism in a fissured karst aquifer with double porosity, revealing how to connect exchange and storage coefficients to the volumetric density of the highly perm...
In this article, we extend the generalized invexity and duality results for multiobjective variational problems with fractional derivative pertaining to an exponential kernel by using the concept of weak minima. Multiobjective variational problems find their applications in economic planning, flight control design, industrial process control, contr...
In this paper, we present the implementation of a local fractional homotopy perturbation method pertaining to the local fractional natural transform (LFNT) operator for local fractional Klein-Gordon equations (LFKGEs) under distinct fractal initial conditions. The Klein-Gordon equation is a relativistic wave equation which estimates the nature and...
In this paper, we implement computational methods, namely the local fractional natural homotopy analysis method (LFNHAM) and local fractional natural decomposition method (LFNDM), to examine the solution for the local fractional Lighthill–Whitham–Richards (LFLWR) model occurring in a fractal vehicular traffic flow. The LWR approach preferably model...
The aim of the current study is to capture the complex behavior of the Ivancevic option pricing (IOP) model using the q-homotopy analysis transform method (q-HATM) with novel fractional operator. The generalization of the Black-Scholes model with the nonlinear Schrödinger equation plays a pivotal role in financial mathematics in studying the option...
The key aim of the present work is to develop extended fractional calculus results associated with product of the generalized extended Mittag–Leffler function, S-function, general class of polynomials and H¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepack...
In this paper, we present the application of local fractional schemes in connection with the local fractional Sumudu transform (LFST) for a local fractional Tricomi equation (LFTE). The numerical simulations for obtained results are presented for the local fractional Tricomi equation with different initial conditions on the Cantor set. The computat...
In this paper, we propose a copulation of the local fractional variational iteration technique and local fractional natural transform (LFNT) for solving the dissipative and damped wave equations with local fractional derivatives. The convergence analysis for the solution obtained through this newly suggested method is also discussed. The numerical...
In this paper, we investigate an equation of nonlinear fractional diffusion with the derivative of Riemann–Liouville. Firstly, we determine the global existence and uniqueness of the mild solution. Next, under some assumptions on the input data, we discuss continuity with regard to the fractional derivative order for the time. Our key idea is to co...
Here, In this work, we derive some novel integrals coupled with incomplete I-functions and the general class of polynomials and that results are demonstrate in various forms of incomplete I-functions. Further, we reconstruct new integrals are in term of Fox’s H function, I-function and incomplete H-functions by putting the parameters value in the i...
In this paper, we present a computational algorithm, namely, local fractional natural homotopy analysis method (LFNHAM) to explore the solutions of local fractional coupled Helmholtz and local fractional coupled Burgers' equations (LFCHEs and LFCBEs). This work also investigates the uniqueness and convergence of the solution of a general local frac...
In this paper, we present a newly proposed local fractional method pertaining to the local fractional Sumudu transform (LFST) for computational study of local fractional Schrödinger’s equations (LFSEs). The error analysis for the present method is also discussed here. The uniqueness and convergence analyses for the solution obtained by using the pr...
In this manuscript, we are concerned with finding approximate solutions to fractional PDEs by using the Daftardar-Jafari method (DJM). The presented method is considered in the Caputo-Fabrizio fractional operator (CFFO). Illustrative examples for handling the FPDEs are given. The obtained results are given to show the sample and efficient features...
In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This mode...
In the present paper, a nonlocal problem for semilinear fractional diffusion equations with Riemann-Liouville derivative is investigated. By applying Banach fixed point theorem combined with some techniques on Mittag-Leffler functions, we establish some results on the existence, uniqueness, and regularity of the mild solutions of our problem in som...
In this paper, an effective analytical scheme based on Sumudu transform known as homotopy perturbation Sumudu transform method (HPSTM) is employed to find numerical solutions of time fractional Schrödinger equations with harmonic oscillator.These nonlinear time fractional Schrödinger equations describe the various phenomena in physics such as motio...
The aim of this paper is to study the calcium profile governed by the advection diffusion equation. The mathematical and computational modeling has provided insights to understand the calcium signalling which depends upon cytosolic calcium concentration. Here the model includes the important physiological parameters like diffusion coefficient, flow...
This book contains several contemporary topics in the areas of mathematical modelling and computation for complex systems. The readers find several new mathematical methods, mathematical models and computational techniques having significant relevance in studying various complex systems. The chapters aim to enrich the understanding of topics presen...
In this paper, we implement the semi‐analytical schemes, namely, local fractional homotopy perturbation Sumudu transform method (LFHPSTM) and local fractional homotopy analysis Sumudu transform method (LFHASTM), for finding the approximate analytical solutions of local fractional Laplace equations under different initial conditions on Cantor sets....
In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In add...
Fractional derivatives are considered as influential weapon in terms of analysis of infectious disease. The research study in fractional calculus with formulation of new definitions and mathematical tools have a great impact in sector of community health by controlling some fatal diseases. In this article, a generalized version of Caputo derivative...
Fractional order calculus and special functions play a great role in scientific, financial and technological fields. In view of considerable impact and applications of fractional derivatives and integrals in real life, we aim to suggest some main formulas for the product of generalized M-series, \({\overline{\text{H}}}\)-function and Aleph function...
In this article, we study certain results connected with a generalized Mittag-Leffler function. A generalized Mittag-Leffler function operator of Laplace and Sumudu conversions are investigated and some applications of the recognized results are also deduced as corollaries in this article. The outcomes of the present study are valuable in solving f...
This article deals with fractional model of Bratu's equation. We have solved fractional model of Bratu's equation using Chebyshev polynomials (CPs). The fractional model of Bratu's equation plays an important role in electrospinning and vibration‐electrospinning process. We have discussed the error analysis of proposed scheme. We have also provided...
In the present study, we apply an analytical scheme to acquire wave solutions of a partial differential equation involving a local fractional derivative. The main idea of this scheme is to generalize the procedure of the well-known generalized exponential rational function technique. To test the method, we have considered the local fractional longi...
In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefle...
The present article attempts to examine fractional order Covid-19 model by employing an efficient and powerful analytical scheme termed as q-homotopy analysis Sumudu transform method (q-HASTM). The q-HASTM is the hybrid scheme based on q-HAM and Sumudu transform technique. Liouville-Caputo approach of the fractional operator has been employed. The...
Our main aim in this paper is to propose and investigate new models of fractional blood ethanol and fractional two-cell cubic autocatalysis reaction models.
In particular, we evaluate the numerical solutions of these models by means
of the power law and the Mittag–Leffler kernel. The numerical solutions are
based mma7188 the fundamental theorem of...
The aim of this paper is to evaluate the potential improvement of classification results in the frame of discrete proportional fractional operator. The nonlocal kernel of the generalized proportional fractional sum depending onĥ-discrete exponential functions defined on time scaleĥZ. This paper deals novel discrete versions of the Po´lya-Szego¨and...
The main objective of the paper is to study the non‐local problem for a pseudo‐parabolic equation with fractional time and space. The derivative of time is understood in the sense of the time derivative of the Caputo fraction of the order α, 0 < α < 1. The first result is an investigation of the existence and uniformity of the solution; the formula...
Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H -functions and the incomplete {\overline{H}} -functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H -functions (IH...
Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and con...
This paper studies a fractional Bloch equation pertaining to Hilfer fractional operator. Bloch equation is broadly applied in physics, chemistry, nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI) and many more. The sumudu transform technique is applied to obtain the analytic solutions for nuclear magnetization . The general solutio...
In this paper, we analyze the concept of observability in the case of conformable time-invariant linear control systems. Also, we study the Gramian observability matrix of the conformable linear system, its rank criteria, null space, and some other conditions. We also discuss some properties of con-formable Laplace transform. 1. Introduction. A new...
The aim of this paper is to evaluate an interesting integral involving generalized hypergeometric function and the multivariable Aleph-function. The integral is evaluated with the help of an integral involving generalized hypergeometric function obtained recently by Kim et al. [8]. The integral is further used to evaluate an interesting summation f...
The main objective of the present investigation is to find the solution for the fractional model of Klein-Gordon-Schrödinger system with the aid of q-homotopy analysis transform method (q-HATM). The projected solution procedure is an amalgamation of q-HAM with Laplace transform. More preciously, to elucidate the effectiveness of the projected schem...
Introduction
Fractional operators find their applications in several scientific and engineering processes. We consider a fractional guava fruit model involving a non-local additionally non-singular fractional derivative for the interaction into guava pests and natural enemies. The fractional guava fruit model is considered as a Lotka-Volterra natur...
The present work aims to solve the fractional reaction-diffusion equation (RDE) using an effective and powerful hybrid analytical scheme, namely q-HASTM. The suggested technique is the combination of Sumudu transform (ST) and HAM technique. The definition of Caputo's fractional derivative has used. The numerical procedure reveals that only few iter...
Most countries around the world are battling to limit the spread of severe acute respiratory syndrome-coronavirus 2 (SARS-CoV-2). As the world strives to get an effective medication to control the disease, appropriate control measures for now remains one of the effective measures to reduce the spread of the disease. In this study, a fractional opti...
In this article, we suggest a numerical approach based on q‐homotopy analysis Elzaki transform method (q‐HAETM) to solve fractional multidimensional diffusion equations which represents density dynamics in a material undergoing diffusion. We take the noninteger derivative in the Caputo–Fabrizio kind. The proposed method, q‐HAETM is an advanced adap...
In this article, we analyze local fractional Poisson equation (LFPE) by employing q‐homotopy analysis transform method (q‐HATM). The PE describes the potential field due to a given charge with the potential field known, one can then calculate gravitational or electrostatic field in fractal domain. It is an elliptic partial differential equations (P...
In this work, we investigate thin film flow of a third grade fluid down a inclined plane. The solution of a nonlinear boundary value problem (BVP) is derived by using an effective well organized computational scheme namely homotopy perturbation Elzaki transform method. Furthermore, this model is also resolved by Elzaki decomposition technique. The...
The primarily object of this article is to derive the solutions of modified fractional kinetic equations (MFKEs) containing the incomplete Aleph functions by using the application of Elzaki and inverse Elzaki transforms and hereto we also established some novel results such as the Elzaki transform of well-known the Riemann–Liouville operator and th...
In this paper, we consider an inverse source problem for the time-space-fractional diffusion equation. Here, in the sense of Hadamard, we prove that the problem is severely ill-posed. By applying the quasi-reversibility regularization method, we propose by this method to solve the problem (1.1). After that, we give an error estimate between the sou...
In this paper we introduce the (k, s)-Hilfer-Prabhakar fractional derivative and discuss its properties. We find the generalized Laplace transform of this newly proposed operator. As an application, we develop the generalized fractional model of the free-electron laser equation, the generalized time-fractional heat equation, and the generalized fra...
In this paper, a fractional order nonlinear mathematical model describing the dynamics of atmospheric concentration of CO2 is investigated and studied through the application of a semi-analytical homotopy scheme combined with Sumudu transform and homotopy polynomials. This study examines the consequences of the variations of forest biomass and huma...
In this work, we study a numerical approach for studying a nonlinear model of fractional optimal control problems (FOCPs). We have taken the fractional derivative in a dynamical system of FOCPs, which is in Liouville–Caputo sense. The presented scheme is a grouping of an operational matrix of integrations for Jacobi polynomials and the Ritz method....