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Search Results: 1 - 10 of 1969 matches for " Mujeeb ur Rehman "
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Haar Wavelet Operational Matrix Method for Fractional Oscillation Equations
Umer Saeed,Mujeeb ur Rehman
International Journal of Mathematics and Mathematical Sciences , 2014, DOI: 10.1155/2014/174819
Abstract: We utilized the Haar wavelet operational matrix method for fractional order nonlinear oscillation equations and find the solutions of fractional order force-free and forced Duffing-Van der Pol oscillator and higher order fractional Duffing equation on large intervals. The results are compared with the results obtained by the other technique and with exact solution. 1. Introduction Haar wavelet is the lowest member of Daubechies family of wavelets and is convenient for computer implementations due to availability of explicit expression for the Haar scaling and wavelet functions [1]. Operational approach is pioneered by Chen and Hsiao [2] for uniform grids. The basic idea of Haar wavelet technique is to convert differential equations into a system of algebraic equations of finite variables. The Haar wavelet technique for solving linear homogeneous/inhomogeneous, constant, and variable coefficients has been discussed in [3]. The fractional order forced Duffing-Van der Pol oscillator is given by the following second order differential equation [4]: where is the Caputo derivative; represents the periodic driving function of time with period , where is the angular frequency of the driving force; is the forcing strength; and is the damping parameter of the system. Duffing-Van der Pol oscillator equation can be expressed in three physical situations: (1)single-well , ;(2)double-well , ;(3)double-hump , . The quasilinearization approach was introduced by Bellman and Kalaba [5, 6] as a generalization of the Newton-Raphson method [7] to solve the individual or systems of nonlinear ordinary and partial differential equations. The quasilinearization approach is suitable to general nonlinear ordinary or partial differential equations of any order. The Haar wavelets with quasilinearization technique [8–10] are applied for the approximate solution of integer order nonlinear differential equations. In [11], we extend the Haar wavelet - quasilinearization technique for fractional nonlinear differential equations. The aim of the present work is to investigate the solution of the higher order fractional Duffing equation, fractional order force-free and forced Duffing-Van der pol (DVP) oscillator using Haar wavelet-quasilinearization technique. We have discussed the three special situations of DVP oscillator equation such as single-well, double-well, and double- hump. 2. Preliminaries In this section, we review basic definitions of fractional differentiation and fractional integration [12].(1)Riemann-Liouville fractional integral operator of order is as follows: the
Hermite Wavelet Method for Fractional Delay Differential Equations
Umer Saeed,Mujeeb ur Rehman
Journal of Difference Equations , 2014, DOI: 10.1155/2014/359093
Abstract: We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method. 1. Introduction The future state of a physical system depends not only on the present state but also on its past history. Functional differential equations provide a mathematical model for such physical systems in which the rate of change of the system may depend on the influence of its hereditary effects. Delay differential equations have numerous applications in mathematical modeling [1], for example, physiological and pharmaceutical kinetics, chemical kinetics, the navigational control of ships and aircrafts, population dynamics, and infectious diseases. Delay differential equation is a generalization of the ordinary differential equation, which is suitable for physical system that also depends on the past data. During the last decade, several papers have been devoted to the study of the numerical solution of delay differential equations. Therefore different numerical methods [2–7] have been developed and applied for providing approximate solutions. Method of steps is easy to understand and implement. In the method of steps [8], we convert the delay differential equation to a nondelay differential equation. The method of steps is utilized in [9] for solving integer order delay differential equations. Hermite wavelet method [10] is implemented for finding the numerical solution of the boundary value problems and compares the obtained solutions with exact solution. In [11], authors utilized the physicists Hermite wavelet method for solving linear singular differential equations. According to our information, Hermite wavelet method has not been implemented for delay differential equations. In the present work, we established a technique by combining both the method of steps and the Hermite wavelets method for solving the fractional delay differential equation. We also implemented the Hermite wavelet method for solving fractional delay differential equation, as described in Example 6, which was not implemented before. Shifted Chebyshev nodes are used as the collocation points. Comparison of solutions by these two methods, proposed method and Hermite wavelet method, with each other and with exact
Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations
Mujeeb Ur Rehman,Rahmat Ali Khan
Abstract and Applied Analysis , 2010, DOI: 10.1155/2010/501230
Abstract: We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type , , . , , , , where, , , , the boundary parameters and is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results. 1. Introduction Fractional calculus goes back to the beginning of the theory of differential calculus and is developing since the 17th century through the pioneering work of Leibniz, Euler, Abel, Liouville, Riemann, Letnikov, Weyl, and many others. Fractional calculus is the generalization of ordinary integration and differentiation to an arbitrary order. For almost 300 years, it was seen as interesting but abstract mathematical concept. Nevertheless the applications of fractional calculus just emerged in the last few decades in various areas of physics, chemistry, engineering, biosciences, electrochemistry, and diffusion processes. For details, we refer the readers to [1–5]. The existence and uniqueness of solutions for fractional differential equations is well studied in [6–10] and references therein. It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of initial value problems for fractional differential equations. In contrast, the theory of boundary value problems for nonlinear fractional differential equations has received attention quiet recently, and many aspects of the theory need to be further investigated. There are some recent development dealing with the existence and multiplicity of positive solutions to nonlinear boundary value problems for fractional differential equations, see, for example, [11–18] and the reference therein. However, few results can be found in the literature concerning the existence of positive solutions to nonlinear three-point boundary value problems for fractional differential equations. For example, Li and coauthors [19] obtained sufficient conditions for the existence and multiplicity results to the following three point fractional boundary value problem where is standard Riemann-Liouville fractional order derivative. Bai [20] studied the existence and uniqueness of positive solutions to the following three-point boundary value problem for fractional differential equations where , , is standard
A typical case of hydrallantois accompanied by fetal monstrosity in a local ewe of Kashmir
Hiranya Kumar Bhattacharyya,Shahid Hussain Dar,Mujeeb-ur-Rehman Fazili,Abdul Hafiz
Veterinary Research Forum , 2012,
Abstract: A full termed local ewe with the history of continuous straining with labored breathing for last 24 hours was presented. The animal was disinclined to move with tense and round abdomen which developed rapidly during last two weeks. Caesarean section revealed hydrallantois accompanied by multiple fetal congenital abnormalities. The ewe was under observation for four weeks. Metritis developed 12 days post-operation and was treated successfully. The ewe was found active on 25 days post-surgery with gain of extra 3 kg bodyweight.
Management of obstructive urolithiasis in dairy calves with intact bladder and urethra by Fazili’s minimally invasive tube cystotomy technique
Mujeeb ur Rehman Fazili,Hiranya Kumar Bhattacharyya,Bashir Ahmad Buchoo,Hamid Ullah Malik
Veterinary Science Development , 2012, DOI: 10.4081/vsd.2012.e11
Abstract: The present study was planned to evaluate minimally invasive tube cystotomy technique in calves suffering from obstructive urolithiasis having intact urinary bladder and urethra. Fifteen male non-castrated calves with age ranging from 1-10 months (mean 4.05 months), presented for treatment within one to three days (mean 2.2 days) of complete urinary tract obstruction due to urethral calculi with intact bladder and urethra, were included in this study. Under light sedation and local infiltration anaesthesia, all the animals were subjected through left paralumbar fossa, to a minimally invasive surgical tube cystotomy in which catheter was placed in the bladder lumen through a metallic cannula and fixed to the skin with a stay suture (Fazili’s technique). All the animals were discharged the same day. Time taken for the procedure varied from 8 to 17 minutes (mean 11.0 minutes). Normal urination resumed in twelve (80.0%) calves. Mean time taken for normal urination was 10.50 days. In two of the remaining calves, urine flow stopped through the catheter prematurely and they were then subjected to standard surgical tube cystotomy. One more calf did not urinate normally for 30 postoperative days and was lost to the follow up thereafter. Recurrence of the obstruction was not detected in ten and nine animals observed up to six and 12 months respectively. In conclusion, the outcome of this minimally invasive technique is similar to the standard tube cystectomy. Additionally, the procedure is cost effective, quick, simple and field applicable. It also minimizes exposure of abdominal cavity of metabolically compromised animals. However, the technique needs to be tried in larger number of such calves wherein better quality catheter of larger diameter be used before recommending its extensive use.
Jovian Problem: Performance of Some High-Order Numerical Integrators  [PDF]
Shafiq Ur Rehman
American Journal of Computational Mathematics (AJCM) , 2013, DOI: 10.4236/ajcm.2013.33028
Abstract:

N-body simulations of the Sun, the planets, and small celestial bodies are frequently used to model the evolution of the Solar System. Large numbers of numerical integrators for performing such simulations have been developed and used; see, for example, [1,2]. The primary objective of this paper is to analyse and compare the efficiency and the error growth for different numerical integrators. Throughout the paper, the error growth is examined in terms of the global errors in the positions and velocities, and the relative errors in the energy and angular momentum of the system. We performed numerical experiments for the different integrators applied to the Jovian problem over a long interval of duration, as long as one million years, with the local error tolerance ranging from 10-16 to 10-18.

Accuracy and Computational Cost of Interpolation Schemes While Performing N-Body Simulations  [PDF]
Shafiq Ur Rehman
American Journal of Computational Mathematics (AJCM) , 2014, DOI: 10.4236/ajcm.2014.45037
Abstract: The continuous approximations play a vital role in N-body simulations. We constructed three different types, namely, one-step (cubic and quintic Hermite), two-step, and three-step Hermite interpolation schemes. The continuous approximations obtained by Hermite interpolation schemes and interpolants for ODEX2 and ERKN integrators are discussed in this paper. The primary focus of this paper is to measure the accuracy and computational cost of different types of interpolation schemes for a variety of gravitational problems. The gravitational problems consist of Kepler’s two-body problem and the more realistic problem involving the Sun and four gas-giants—Jupiter, Saturn, Uranus, and Neptune. The numerical experiments are performed for the different integrators together with one-step, two-step, and three-step Hermite interpolation schemes, as well as the interpolants.
SIMULTANEOUS ESTIMATION OF SIMVASTATIN AND EZETIMIBE IN PHARMACEUTICAL TABLET DOSAGE FORMS BY RP-HPLC
Mujeeb ur Rahman
International Journal of Pharmaceutical Research and Development , 2010,
Abstract: This work is concerned with application of simple, accurate, precise and highly selective reverse phase high performance liquid chromatographic (RP-HPLC) method for simultaneous estimation of simvastatin and ezetimibe in combined dosage form. Chromatographic separation was achieved isocratically phenomenax C18 column (250 ? 4.6 mm i.d.) with a mobile phase composed of 75:20:5 of acetonitrile:methanol:orthophosphoric acid (0.1%) % v/v/v at flow rate of 1 ml/min. Detection is carried out using a UV-vis detector at 238 nm. The retention time of simvastatin and ezetimibe was found to be 3.701 min and 5.975 min. respectively. The method was found to be linear in the range of 50-175 μg/ml with mean recovery of 99.78% for simvastatin and 98.81% for ezetimibe. The correlation coefficients for all components are close to 1. The developed method was validated according to ICH guidelines and values of accuracy, precision and other statistical analysis were found to be in good accordance with the prescribed values. Thus the proposed method was successfully applied for simultaneous determination of simvastatin and ezetimibe in routine analysis.
Safety Management in a Manufacturing Company: Six Sigma Approach  [PDF]
Lateef Ur Rehman Ateekh-ur-Rehman
Engineering (ENG) , 2012, DOI: 10.4236/eng.2012.47053
Abstract: The manufacturing company under consideration recorded the high accident rates for last few years. These accidents cause the organization the heavy man-day loss, the production loss and heavy costs of insurance. The objective of health and safety department at the manufacturing company was to set and improve accidents prevention system. The paper presents how does the six-sigma technique will help to evaluate the safety and environmental hazards in performance of organizations? It is observed that the study helped the management to measure, analyze and improve overall safety plan to protect the life and health of the employees.
Inhibitory Effects of Essential Oil of Psamogeton canescens on Asexsual Reproduction of Toxigenic Fungi (Strains of Aspergillus)
Mujeeb ur Rahman,Shereen Gul
Pakistan Journal of Biological Sciences , 2000,
Abstract: The effects of essential oil of the plant species Psamogeton canescens ( a member of family Umbelliferae) and of an antifungal medicine Taeniafex on spore germination , mycelium elongation and sporulation were studied on five toxigenic fungi. All the three stages of fungal asexual reproduction were effected but the mycelium elongation was the most sensitive followed by sporulation and the spore germination. Aspergillus oryzae and Aspergillus flavus were found to be the most sensitive, followed by Aspergillus niger, Aspergillus awamori and the Aspergillus foetidus. The antifungal activity at all the three stages of asexual reproduction was shown to be co-related to the oil composition.
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