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An approach for approximate solution of fractional-order smoking model with relapse class

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International Journal of Biomathematics 2018年第6期11卷 31-57页

作者： Anwar Zeb Vedat Suat Erturk Umar Khan gul zaman Shaher Momani Department of Mathematics COMSATS University Islamabad Abbottabad Campus Pakistan Department of Mathematics Faculty of Arts and Sciences Ondokuz Mayis University 55139 Samsun Turkey Department of Mathematics University of Malakand Chakdara Dir (Lower) KPK PakistantDepartment of Mathematics Faculty of Arts and Sciences Ondokuz Mayis University 55139 Samsun Turkey Department of Mathematics Faculty of Science The University of Jordan Amman 1195 Jordan Nonlinear Analysis and Applied Mathematics (NAAM)Research Group Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia

In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Griinwald-Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann-Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain trans-formations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model A comparative study between Griinwald-Letnikov method and Runge-Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.

关键词： Mathematical model reproductive number next generation matrix method stability analysis Griinwald-Letnikov method numerical simulation.

The Homotopy Analysis Method for Approximating of Giving Up Smoking Model in Fractional Order

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Applied Mathematics 2012年第8期3卷 914-919页

作者： Anwar Zeb M. Ikhlaq Chohan gul zaman Buraimi University College Department of Business Administration and Accounting Al-Buraimi Oman Department of Mathematics University of Malakand Chakdara Dir (Lower) Khyber Pakhtunkhawa Pakistan

In this paper, we consider the giving up smoking model. First, we present the giving up smoking model in fractional order. Then the homotopy analysis method (HAM) is employed to compute an approximate and analytical solution of the model in fractional order. The obtained results are compaired with those obtained by forth order Runge-Kutta method and nonstandard numerical method in the integer case. Finally, we present some numerical results.

关键词： Fractional Differential Equations Epidemic Model Homotopy Analysis Method

Exact Solution for a Class of Stiff Systems by Differential Transform Method

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Applied Mathematics 2013年第3期4卷 440-444页

作者： Muhammad Idrees Fazle Mabood Asar Ali gul zaman City University of Science and Information Technology Peshawar Pakistan Edwardes College Peshawar Pakistan University of Malakand Chakdara Dir (Lower) Pakistan

In this paper, Differential Transform Method (DTM) is proposed for the closed form solution of linear and non-linear stiff systems. First, we apply DTM to find the series solution which can be easily converted into exact solution. The method is described and illustrated with different examples and figures are plotted accordingly. The obtained result confirm that DTM is very easy, effective and convenient.

关键词： Differential Transform Method Stiff System Inverse Differential Transform

The Oscillatory Motion of Oldroyd-B Fluid by Incorporating Some of the Mechanical Factors

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Journal of Applied Mathematics and Physics 2017年第12期5卷 2402-2410页

作者： Amir Khan gul zaman Yongjin Li Saeed Ahmad Altaf Hussain Department of Mathematics and Statistics University of Swat Khyber Pakhtunkhwa Pakistan Department of Mathematics University of Malakand Chakdara Dir (Lower) Khyber Pakhtunkhwa Pakistan Department of Mathematics Sun Yat-sen University Guangzhou China

The aim of this paper is to present a numerical study of oscillatory motion of Oldroyd-B fluid in a uniform magnetic field through a small circular pipe. First, we derive the orientation stress tensor by considering the Brownian force. Then, the orientation stress tensor is incorporated by taking Hookean dumbbells on Brownian configuration fields in the Oldroyd-B model. The Oldroyd-B model is then reformulated coupled with the momentum equation and the total stress tensor. Finally, we analyze the orientation stress tensor in the pipe by the numerical simulations of the model and showed that the effect of orientation stress tensor is considerable although the Brownian force is sufficiently small.

关键词： Oldroyd-B Fluid Brownian Force Finite Difference Oscillatory Flows

Square-Root Dynamics of a SIR-Model in Fractional Order

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Applied Mathematics 2012年第12期3卷 1882-1887页

作者： Young Il Seo Anwar Zeb gul zaman Il Hyo Jung Department of Mathematics Pusan National University Busan South Korea Department of Mathematics University of Malakand Chakdara Pakistan National Fisheries Research and Development Institute Busan South Korea

In this paper, we consider an SIR-model for which the interaction term is the square root of the susceptible and infected individuals in the form of fractional order differential equations. First the non-negative solution of the model in fractional order is presented. Then the local stability analysis of the model in fractional order is discussed. Finally, the general solutions are presented and a discrete-time finite difference scheme is constructed using the nonstandard finite difference (NSFD) method. A comparative study of the classical Runge-Kutta method and ODE45 is presented in the case of integer order derivatives. The solutions obtained are presented graphically.

关键词： Mathematical Model Square Root Dynamics Fractional Derivative Non-Standard Finite Difference Scheme Numerical Analysis

Some Exact Solutions of Generalized Jeffrey Fluid Using N-Transform

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American Journal of Computational Mathematics 2017年第4期7卷 402-412页

作者： Amir Khan gul zaman Saeed Ahmad M. Ikhlaq Chohan Department of Mathematics and Statistics University of Swat Swat Pakistan Department of Mathematics University of Malakand Lower Dir Pakistan Department of Business Administration and Accounting Buraimi University College Al-Buraimi Oman

In this paper, we presents some new exact solutions corresponding to three unsteady flow problems of a generalized Jeffrey fluid produced by a flat plate between two side walls perpendicular to the plate. The fractional calculus approach is used in the governing equations. The exact solutions are established by means of the Fourier sine transform and N-transform. The series solutions of velocity field and associated shear stress in terms of Fox H-function, satisfying all imposed initial and boundary conditions, have been obtained. The similar solutions for ordinary Jeffrey fluid, performing the same motion, appear as limiting case of the solutions are also obtained. Also, the obtained results are analyzed graphically through various pertinent parameters.

关键词： Generalized Jeffrey Fluid Fractional Derivatives Fox H-Functions N-Transform

Optimal Campaign in Leptospirosis Epidemic by Multiple Control Variables

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Applied Mathematics 2012年第11期3卷 1655-1663页

作者： Muhammad Altaf Khan gul zaman Saeed Islam Muhammad Ikhlaq Chohan Department of Business Administration and Accounting Buraimi University College Al-Buraimi Oman Department of Mathematics Abdul Wali Khan University Mardan Pakistan Department of Mathematics University of Malakand Chakdara Lower Dir District of Khyber Pakhtunkhwa Pakistan

In this paper, we consider a leptospirosis epidemic model to implement optimal campaign by using multiple control variables. First, we show the existence of the control problem. Then we derive the conditions under which it is optimal to eradicate the leptospirosis infection and examine the impact of a possible educatioal/vaccinaction campaign using Pontryagin’s Maximum Principle. We completely characterize the optimal control problem and compute the numerical solution of the optimality system using an iterative method. The results obtained from the numerical simulations of the model show that a possible educational/vaccinaction combined with effective treatment regime would reduce the spread of the leptospirosis infection appreciably.

关键词： Mathematical Model Optimal Control Pontryagin’s Maximum Principle Numerical Results

潜伏期和染病期均具有传染性的媒介传染病模型的全局稳定性分析(英文)

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应用泛函分析学报 2012年第4期14卷 321-329页

作者： Abid Ali Lashari Muhammad Ozair gul zaman 李学志 Centre for Advanced Mathematics and Physics National University of Sciences and Technology H-12IslamabadPakistan Department of Mathematics University of Malakand Chakdara Dir(Lower) Khyber PukhtoonkhwaPakistan 信阳师范学院数学学院信阳464000

讨论潜伏期和染病期均具有传染性的媒介传染病模型.得到模型基本再生数的表达式,证明了当基本再生数小于1时,无病平衡点是全局渐近稳定的,此时疾病消亡;当基本再生数大于1时,无病平衡点是不稳定的,系统存在全局渐近稳定的地方病平衡点,... 详细信息

讨论潜伏期和染病期均具有传染性的媒介传染病模型.得到模型基本再生数的表达式,证明了当基本再生数小于1时,无病平衡点是全局渐近稳定的,此时疾病消亡;当基本再生数大于1时,无病平衡点是不稳定的,系统存在全局渐近稳定的地方病平衡点,此时,疾病将在人群中持续存在,数值模拟验证了理论结果.

关键词：媒介传染病模型平衡点全局稳定性

Investigation of Optical and Luminescence Properties of Soda Lime Borate Glasses Doped with Sm^(3+) Ion

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Journal of the Chinese Ceramic Society 2016年第1期3卷 26-32页

作者： FALAK zaman gul Rooh NATTAPON Srisittipokakun JACKRAPONG Kaewkhao Department of Physics Faculity of Science Abdul Wali Khan University Center of Excellence in Glass Technology and Materials Science Nakhon Pathom Rajabhat University Physics Program Faculty of Science and TechnologyNakhon Pathom Rajabhat University

This paper was to investigate the optical and luminescence properties of Sm^(3+) doped SLB glasses by a melt quenching technique. The optical and luminescence properties of the prepared glass samples were investigated via absorption and photoluminescence spectra, respectively. The related physical and optical parameters were also calculated. From optical absorption measurements, the transition ~6H_(5/2) → ~6P_(3/2) at 403 nm has a higher spectral intensity and is a hypersensitive transition. From photoluminescence spectra, four prominent emission spectra appear. The most intense band is located at ~4G_(5/2) →~6H_(9/2)(599 nm), which is the characteristic emission range of Sm^(3+) ions with the reddish orange color. The experimental decay time of the ~4G_(5/2) level of Sm^(3+) SLB glasses was determined. The decay time decreases from 1.367 to 0.333 ms with increasing the content of Sm^(3+). The color coordinates(x, y) of the prepared glasses passes through the reddish-orange region in the CIE 1931 diagram and are suitable for orange LEDs, optoelectronics, and solidstate lighting. The further investigation on the optimization of the dopant content in the SLB glasses was suggested.

关键词： soda lime borate glass absorption luminescence spectra