Papers by Azhar Ali Zafar
Springer proceedings in mathematics & statistics, 2024
In this paper, we present generalizations of some fixed point theorems using the notion of w-dist... more In this paper, we present generalizations of some fixed point theorems using the notion of w-distance on a metric space. The results herein contain the work of many authors including Reich, Morales, Rakotch, Chu and Diaz.
AIMS mathematics, 2023
As an exponentially growing sensitivity to modest perturbations, chaos is pervasive in nature. Ch... more As an exponentially growing sensitivity to modest perturbations, chaos is pervasive in nature. Chaos is expected to provide a variety of functional purposes in both technological and biological systems. This work applies the time-fractional Caputo and Caputo-Fabrizio fractional derivatives to the Chua type nonlinear chaotic systems. A numerical analysis of the mathematical models is used to compare the chaotic behavior of systems with differential operators of integer order versus systems with fractional differential operators. Even though the chaotic behavior of the classical Chua's circuit has been extensively investigated, our generalization can highlight new aspects of system behavior and the effects of memory on the evolution of the chaotic generalized circuit.
Chaos Solitons & Fractals, Nov 1, 2020
Abstract In this article, the dynamics of blood with suspended magnetic particles in coronal and ... more Abstract In this article, the dynamics of blood with suspended magnetic particles in coronal and femoral arteries are investigated. The flow of blood is examined in the presence of external magnetic field, periodic body acceleration and a pressure gradient of an oscillating type. Expressions for the velocity of blood and velocity of magnetic particles will be yielded by employing integral transforms. The analytical results will be expressed in terms of steady-state and transient parts. Moreover, to get insight of the control of the material parameters such as amplitude, the lead angle, frequency of body acceleration, magnetic field and particles’ concentration parameter, numerical simulations and graphical illustrations will be used and useful consequences will be summarized.
Computers & Mathematics with Applications, 2011
The objective of this paper is to study the unsteady flow of an Oldroyd-B fluid with fractional d... more The objective of this paper is to study the unsteady flow of an Oldroyd-B fluid with fractional derivative model, between two infinite coaxial circular cylinders, using Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t = 0 + , applies a time dependent longitudinal shear stress to the fluid. Velocity field and the adequate shear stress are presented in series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional Maxwell, ordinary Maxwell, fractional second grade, ordinary second grade and Newtonian fluids performing the same motion are obtained as limiting cases of general solutions. In particular, the existing solutions for ordinary Oldroyd-B and second grade fluids are compared with the present solutions. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between models is underlined by graphical illustrations.
Case Studies in Thermal Engineering, Oct 1, 2021
Abstract On a porous layered inclined plate, magneto-free-convection flow of a rate type fluid is... more Abstract On a porous layered inclined plate, magneto-free-convection flow of a rate type fluid is investigated. The problem is modeled as natural convection for oscillating and general movements of an inclined plate moving in a magnetized medium with a slanted external magnetic field that is either stationary or moving in tandem with the plate. The plate is subjected to a symmetric temperature field with double-sided thermal operation. Constant concentration, first order chemical reaction, and thermal conductivity as a general feature of time are used in heat transfer measurement. On the velocity of the fluid, the impact of varying plate angles with the vertical as well as the slanted angle of the magnetic field with the plate are studied. The motion of the plate and temperature distribution in specific situations was discussed. We can retrieve many results by changing the values of the functions and parameters in our general solutions, including the related results for the viscous fluid from the literature as a limiting case. As a result, the matter of identical models is no longer an issue. Furthermore, graphical software is used to do parametric analysis on fluid dynamics.
Alexandria Engineering Journal
Fractals
The theoretical study focuses on the examination of the convective flow of Oldroyd-B fluid with r... more The theoretical study focuses on the examination of the convective flow of Oldroyd-B fluid with ramped wall velocity and temperature. The fluid is confined on an extended, unbounded vertical plate saturated within the permeable medium. To depict the fluid flow, the coupled partial differential equations are settled by using the Caputo (C) and Caputo Fabrizio (CF) differential time derivatives. The mathematical analysis of the fractionalized models of fluid flow is performed by Laplace transform (LT). The complexity of temperature and velocity profile is explored by numerical inversion algorithms of Stehfest and Tzou. The fractionalized solutions of the temperature and velocity profile have been traced out under fractional and other different parameters considered. The physical impacts of associated parameters are elucidated with the assistance of the graph using the software MATHCAD 15. We noticed the significant influence of the fractional parameter (memory effects) and other param...
Alexandria Engineering Journal, 2015
Unsteady Taylor-Couette flows of an Oldroyd-B fluid, which fills a straight circular cylinder of ... more Unsteady Taylor-Couette flows of an Oldroyd-B fluid, which fills a straight circular cylinder of radius R, are studied. Flows are generated by the oscillating azimuthal tension which is given on the cylinder surface. As a novelty, authors used in this paper the governing equation related to the tension field. The closed forms of the shear stress and velocity fields corresponding to the flow problems are obtained by means of the integral transforms method. Expressions for the azimuthal tension and fluid velocity were written as sums between the ''permanent component" (the steady-state component) and the transient component. By customizing values of parameters from the mathematical model were obtained the corresponding solutions of other types of fluids, namely, Maxwell fluids. By using numerical simulations and diagrams of the azimuthal stress, the fluid behavior has been analyzed. The necessary time to achieve the ''steady-state" was, also, determined.
Mathematical Modelling of Natural Phenomena, 2018
In this work, we study the flow of both blood and magnetic particles using Caputo-Fabrizio fracti... more In this work, we study the flow of both blood and magnetic particles using Caputo-Fabrizio fractional derivative model approach. The fluid flow through a circular cylinder is influenced by an external magnetic field which is perpendicular to the circular tube and an oscillating pressure gradient. Integral transforms are used to find solutions for the blood and magnetic particle velocities. Comparison of profiles of velocities for different values of α, the impact of physical variables on the dynamics of fluid and magnetic parameters are highlighted graphically.
Lecture notes in networks and systems, Oct 14, 2022
Advances in the Theory of Nonlinear Analysis and its Application, 2021
Two dimensional ow of mixed convection nanouid on horizontal plate with the eect of nonlinear Ros... more Two dimensional ow of mixed convection nanouid on horizontal plate with the eect of nonlinear Rosseland thermal radiation has been investigated. Mathematical model of the problem is based on partial dierential equations and optimal homotopy analysis method (OHAM) is applied to sort out solutions. Moreover, comprehensive study of inuence of emerging parameters is carried out via graphical interpretation and tables.
Computers & Mathematics with Applications, 2011
The unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudd... more The unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied using Fourier sine and Laplace transforms. The obtained solutions for the velocity field and shear stress, written in terms of generalized G functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian contributions. The non-Newtonian contributions, as expected, tend to zero for λ → 0. Furthermore, the solutions for ordinary Maxwell fluid, performing the same motion, are obtained as limiting cases of general solutions and verified by comparison with previously known results. Finally, the influence of the material and the fractional parameters on the fluid motion, as well as a comparison among fractional Maxwell, ordinary Maxwell and Newtonian fluids is also analyzed by graphical illustrations.
Communications in Nonlinear Science and Numerical Simulation, 2012
The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is stu... more The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is studied using Laplace and finite Hankel transforms. The motion of the fluid is due to the inner cylinder that, at time t = 0 + begins to rotate around its axis, and to slide along the same axis due to hyperbolic sine or cosine shear stresses. The components of the velocity field and the resulting shear stresses are presented in series form in terms of Bessel functions J 0 (), Y 0 (), J 1 (), Y 1 (), J 2 () and Y 2 (). The solutions that have been obtained satisfy all imposed initial and boundary conditions and are presented as a sum of large-time and transient solutions. Furthermore, the solutions for Newtonian fluids performing the same motion are also obtained as special cases of general solutions. Finally, the solutions that have been obtained are compared and the influence of pertinent parameters on the fluid motion is discussed. A comparison between second grade and Newtonian fluids is analyzed by graphical illustrations.
Applied Mathematics and Computation, 2011
The velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B flui... more The velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders are established by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder that applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in
Mathematical Problems in Engineering
Heat and mass transfer combined effects on MHD natural convection for a viscoelastic fluid flow a... more Heat and mass transfer combined effects on MHD natural convection for a viscoelastic fluid flow are investigated. The dynamics of the fluid are controlled by the motion of the plate with arbitrary velocity along with varying temperature and mass diffusion. The non-dimensional forms of the governing equations of the model are developed along with generalized boundary conditions and the resulting forms are solved by the classical integral (Laplace) transform technique/method and closed-form solutions are developed. Obtained generalized results are very important due to their vast applications in the field of engineering and applied sciences; few of them are highlighted here as limiting cases. Moreover, parametric analysis of system parameters P r , S , K c , G T , G c , M , S c , λ is done via graphical simulations.
Advances in Difference Equations
This research note’s objective is to elaborate on the study of the unsteady MHD natural convectiv... more This research note’s objective is to elaborate on the study of the unsteady MHD natural convective flow of the Jeffery fluid with the fractional derivative model. The fluid flow phenomenon happens between two vertical parallel plates immersed in a porous medium. The one plate is moving with the time-dependent velocity $U_{0} f(t)$ U 0 f ( t ) , while the other is fixed. The mathematical model is presented with the system of the partial differential equation along with physical conditions. Appropriate dimensionless variables are employed in the system of equations, and then this dimensionless model is transformed into the Caputo fractional-order model and solved analytically by the Laplace transform. The exact expressions for velocity and temperature, which satisfy the imposed initial and boundary conditions, are obtained. Memory effects in the fluid are observed which the classical model fails to elaborate. Interesting results are revealed from the investigation of emerging paramete...
4th International Conference on Computational Mathematics and Engineering Sciences (CMES-2019), 2020
The purpose of this article is to study analytically the hydromagnetic natural convection flow of... more The purpose of this article is to study analytically the hydromagnetic natural convection flow of an electrically conducting, incompressible viscous fluid over a moving infinite inclined plate. Moreover, the dynamic of fluid is studied under the influence of exponential heating and constant concentration. Porous effects are taken into consideration and in order to investigate the influence of the transverse magnetic field, two cases when the transverse magnetic field is held fixed to the fluid or to the plate are considered. The Laplace transform technique is used to obtain exact solutions for such motions. The dimensionless Latin symbols velocity, and also the corresponding skin friction, is presented as sum of mechanical, thermal and concentration components. Finally, for illustration, as well as for a check of results, some special cases with applications in engineering are considered and influence of the system parameters is graphically brought to light.
In this paper, we establish some new Hermite-Hadamard integral inequalities for log-φ-convex and ... more In this paper, we establish some new Hermite-Hadamard integral inequalities for log-φ-convex and φ-convex functions in the framework of multiplicative calculus. Furthermore, some results related to differentiable log-φ-invex functions are also obtained.
Physica A: Statistical Mechanics and its Applications, 2020
Abstract This study investigates the unsteady magnetohydrodynamics (MHD) flow of a viscous fluid.... more Abstract This study investigates the unsteady magnetohydrodynamics (MHD) flow of a viscous fluid. The fluid is passing over a vertical plate through porous medium. Additionally conjugate effects of heat and mass transfer with ramped temperatures, slip effect and influence of thermal radiation in the energy equation are taken into account. The dimensionless fractional-order governing equations, in the Caputo–Fabrizio sense, are solved with the help of Laplace transformation. Moreover, semi analytical technique is used to investigate the velocity field. Some results which present in literature are recovered as limiting cases. Influences of different parameters on the velocity profiles for the case of f ( t ) = t and f ( t ) = sin ω t are highlighted. The novelty of the manuscript is the use of the most recent definition of the non integer order derivative operator i.e. Caputo–Fabrizio derivative operator, the use of generalized boundary conditions in terms of general function f ( t ) , from our general results, several particular cases for instance when f ( t ) is a linear or sinusoidal function could be recovered.
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Papers by Azhar Ali Zafar