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IEEE/CAA Journal of Automatica Sinica 2019年第1期6卷 177-187页

作者： najeeb alam khan Tooba Hameed the Department of Mathematics University of Karachi

The objective of this paper is to solve the timefractional Schr¨odinger and coupled Schr¨odinger differential equations(TFSE) with appropriate initial conditions by using the Haar wavelet approximation. For the most part, this endeavor is made to enlarge the pertinence of the Haar wavelet method to solve a coupled system of time-fractional partial differential equations. As a general rule, piecewise constant approximation of a function at different resolutions is presentational characteristic of Haar wavelet method through which it converts the differential equation into the Sylvester equation that can be further simplified easily. Study of the TFSE is theoretical and experimental research and it also helps in the development of automation science,physics, and engineering as well. Illustratively, several test problems are discussed to draw an effective conclusion, supported by the graphical and tabulated results of included examples, to reveal the proficiency and adaptability of the method.

关键词： Fractional calculus Haar wavelets operational matrix wavelets

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

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Acta Mechanica Sinica 2011年第2期27卷 222-227页

作者： Amir Mahmood Saima Parveen najeeb alam khan Department of Mathematics COMSATS Institute of Information Technology Lahore Pakistan Abdus Salam School of Mathematical Sciences Lahore Pakistan Department of Mathematics University of Karachi Karachi-75270 Pakistan

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 ＋ , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.

关键词： Second grade fluid · Velocity field · Shear stress · Longitudinal oscillatory flow · Laplace and Hankel transforms

Axial Couette flow of an Oldroyd-B fluid in an annulus

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Theoretical & Applied Mechanics Letters 2012年第1期2卷 49-54页

作者： Muhammad Jamil najeeb alam khan A bdus Salam School of Mathematical Sciences GC University Department of Mathematics NED University of Engineering&TechnologyKarachi-75270Pakistan Department of Mathematics University of Karachi

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated.

关键词： Oldroyd-B fluid velocity field time-dependent shear stress Hankel transform

Behavioral effects of a four-wing attractor with circuit realization: a cryptographic perspective on immersion

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Communications in Theoretical Physics 2020年第12期72卷 35-70页

作者： najeeb alam khan Muhammad Ali Qureshi Tooba Hameed Saeed Akbar Saif Ullah Department of Mathematics University of KarachiKarachi 75270Pakistan Department of Physics University of KarachiKarachi 75270Pakistan Department of Mathematics Government College UniversityLahore 54000Pakistan

In this paper,we propose an innovative chaotic system,combining fractional derivative and sinehyperbolic nonlinearity with circuit *** study of this system is conducted via an analog circuit simulator,using two anti-parallel semiconductor diodes to provide hyperbolic sine nonlinearity,and to function as operational *** multi-stability of the system is also enhanced with the help of multi-equilibrium points for distinct real orders of *** system explores the generation of a four-wing attractor in different phases,both numerically and *** changing the input parameters of the system,different graphs are generated for current flow in state,phase,and space,to confirm the precision of the fractional order derivatives.A reasonable simulation shows that the deliberate circuit provides effective chaos in terms of speed and accuracy,which is comensurate with the numerical *** nonlinear chaotic behavior is utilized to encrypt sound(.wav),images(.jpg),and animated(.gif)data which are a major requirement for the security of communication *** proposed circuit performs chaos and cryptographic tasks with high-effective analog computation,and constitutes a novel approach to this area of research.

关键词： multi-wing attractors hyperbolic sine circuit realization encryption

A Metaheuristic Approach to Optimize European Call Function with Boundary Conditions

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Journal of Systems Science and Information 2018年第3期9卷 260-268页

作者： najeeb alam khan Oyoon Abdul RAZZAQ Tooba HAMEED Department of Mathematics University of Karachi Department of Humanities & Natural Sciences Bahria University

The purpose of this paper is to investigate the pricing European call option valuation problems under the exercise price,maturity,risk-free interest rate,and the volatility function. An advance methodology,Chebyshev simulated annealing neural network(Ch SANN),is enforced for the Black-Scholes(B-S) model with boundary conditions. Our scheme is stable and easy to implement on B-S equation,for arbitrary volatility and arbitrary interest rate values. Also,the comparative results demonstrate that the attained approximate solutions are converging towards the exact solution. The graphical results show that the increasing flow of the European call option as the exponential increase takes place in assets. The presented algorithm can be further applied to other financial models with certain boundary conditions. The algorithm of the method shows that the approach can also be easily employed on time-fractional B-S equation.

关键词： neural network numerical approximation Chebyshev polynomials

Oscillations of a Punctual Charge in the Electric Field of a Charged Ring: A Comparative Study

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Journal of Electromagnetic Analysis and Applications 2013年第5期5卷 229-235页

作者： najeeb alam khan Asmat Ara Nasir Uddin khan Nadeem alam khan Department of Mathematical Sciences University of Karachi Karachi Pakistan.

We applied multiple parameters method (MPM) to obtain natural frequency of the nonlinear oscillator with rational restoring force. A frequency analysis is carried out and the relationship between the angular frequency and the initial amplitude is obtained in analytical/numerical form. This equation is analyzed in three cases: the relativistic harmonic oscillator, a mass attached of a stretched elastic wire and oscillations of a punctual charge in the electric field of charged ring. The three and four parameters solutions are obtained. The results obtained are compared with the numerical solution, showing good agreement.

关键词： Oscillator Stretched Wire Parameters Electric Field

Homotopy perturbation aided optimization procedure with applications to oscillatory fractional order nonlinear dynamical systems

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International Journal of Modeling, Simulation, and Scientific Computing 2019年第4期10卷 107-128页

作者： najeeb alam khan Tooba Hameed Samreen Ahmed Department of Mathematics University of KarachiKarachi 75270Pakistan

This paper presents an approximate solution of nonlinear fractional differential equations(FDEs)that exhibit an oscillatory behavior by using a metaheuristic *** solutions of the governing equations are approximated by using homotopy perturbation method(HPM)along with the fractional derivative in the Caputo *** designed methodology is based on a weighted series of HPM in conjunction with a nature-inspired *** idea is instantly fascinated by the researchers on the consequent implementation of nature-inspired learning algorithms such as a Cuckoo search algorithm(CSA).The usage of CSA has accelerated the minimized search path of error to the convergent values of the *** validity and accuracy of the proposed technique are ascertained by calculating the approximate solution and the error norms which ensure the convergence of the approximation that can be further *** critical analysis is also provided by the numerical simulation of two different test *** of key points has been determined by the tabulation of numerical values and *** study of the results with known numerical technique is also performed.

关键词： Lotka-Volterra model Brusselator model Caputo fractional derivative Cuckoo search algorithm