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Qualitative Analysis of Some Higher Order Rational Difference Equations

Thesis Info

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Author

Sadiq, Stephen

Program

PhD

Institute

National College of Business Administration and Economics

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11224/1/Stephen%20Sadiq_Maths_2019_NCBAE_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727051968

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The study of Qualitative Analysis of rational difference equation is very important. However, in Physics, Economics and Biological sciences difference equations are extensively used. Qualitative behavior includes global attractivity, bounded character and periodicity. The motivation is to study the Dynamics of some higher order rational difference equations. In this thesis we have studied the behavior of solutions of some higher order rational difference equations. To confirm the proved results we have used mathematical program Matlab to give graphical examples by assigning different numerical values to initial values. The first chapter presents brief introduction to the problem of study, its objectives and methodology used to achieve the goal. In the second chapter we have presented a comprehensive form of literature review related to the problem understudy. The current literature is discussed in detail and results are summarized to draw conclusions and further directions. The third chapter is preliminaries, showing notions and basic definitions which are associated with this study. In chapter four, we have presented our main results with their proofs for rational difference equation of order twenty 2 9 19 9 19 , 0,1,2,...nnn nn z z z n zz - +- -- b =a + = g +d With initial conditions 19 18 17 16 15 14 13 , , , , , , , z z z z z z z - - - - - - - 12 11,, zz -- 10,z - 9 8 7 6 5 4 3 2 1 0 , , , , , , , , , z z z z z z z z z z R+ - - - - - - - - - Î and the coefficients , , , a b g d are constants. We obtained some special cases of this equation. The numerical verification and comparison of each graph is also included in this chapter. In chapter five, we have studied the global stability of the positive solutions and periodic character of the difference equation 0 1 2 3 1 0 1 2 4 5 6 7 n t n l n m n p n n n k n s n t n l n m n p b z b z b z b z z z z z b z b z b z b z - - - - + - - - - - - + + + =a +a +a + + + + x With non-negative initial conditions 1 1 0 , ,..., , z z z z -m -m+ - where { } max , , , , , k s t l m pm= and the coefficients 0 1 2 0 1 , , , , , bba a a 2 3 4 , , , b b b 5 6 7 ,, b b b R+ Î . Numerical examples are also given to confirm the obtained results.
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ہم زندہ قوم ہیں

ہم زندہ قوم ہیں
نحمدہ ونصلی علی رسولہ الکریم امّا بعد فاعوذ بااللہ من الشیطن الرجیم
بسم اللہ الرحمن الرحیم
معزز اسا تذہ کرام اور میرے ہم مکتب شاہینو!
آج مجھے جس موضوع پر لب کشائی کا موقع مل رہا ہے وہ ہے:’’ہم زندہ قوم ہیں‘‘
صدرِذی وقار!
زندگی جہاں بھی ہو حرکت کی علامت ہے۔ حرکت ہے تو زندگی ہے، حیات ہے، زیست ہے، زندگی کی اپنی برکات ہوتی ہیں، جن کی اپنی رعنائیاں ہوتی ہیں، زندگی کے اپنے نشیب و فراز ہوتے ہیں، زندگی کا تصور ہی دلفریب مناظر کا داعی ہوتا ہے۔
جنابِ صدر!
گل ٔپژ مردہ باعث نفرین ہوتا ہے، عروق مردہ کو حیات ِنو بخشنے کے لیے کافی تگ و دو کی ضرورت ہوتی ہے، مفلوج زدہ عضو کی بحالی کے لیے نظر یں کسی مسیحا کی متلاشی ہوتی ہیں، خزاں رسیدہ شجر پر بہار کی آمد نوید مسرت سے کم نہیں ہوتی۔
جنابِ صدر!
زندگی رات کی تنہائیوں میں آئے تو بدر کامل کا احساس دلاتی ہے، زندگی فلک کی رفعتوں میں پہنچے تو آفتاب بن کر چمکتی ہے، زندگی بحرِ ظلمات میں غواّصی کرے تو نا خدا کا وجود بن جاتی ہے، زندگی اقوام میں اپنے وجود کا احساس دلائے تو اس گیتی کے ماتھے کا جھومر ثابت ہوتی ہے۔
صدرِذی وقار!
اللہ کا شکر ہے کہ ہم زندہ قوم ہیں ، ہمارے افکار زندہ ہیں، ہمارے اطوار زندہ ہیں، ہماری سوچیںمثبت ہیں، ہمارے نوجوان حیات وزیست کا مجسمہ ہیں، ہمارے اسلاف کی نصائح زندگی کو تابندگی فراہم کرتی ہیں، ہمارے شعور علم وآگا ہی کانمونہ ثابت ہوتے ہیں۔
اس قوم کو شمشیر کی حاجت نہیں رہتی
ہو جس کے جوانوں کی خودی صورتِ فولاد
جنابِ صدر!
ہمارا مجاہد سرحدوں پر زندہ ہے، ہمارا منصف انصاف کی کرسی پر زندہ ہے، ہمارا کسان...

خصائصِ نبویﷺ : اردوتفسیری ادب کاتجزیاتی مطالعہ

There is no doubt that the personality of messenger of Allah peace be upon him is irreproachable from all kind of diminutions and barrier of salient bigness and glory. Not only historically biographers have written about the  greatness of Prophet peace be upon him, But the Muhadithin have collected  a large part of the collection of Hadiths on  prophet peace be upon him greatness and  honour. Urdu commentators are not far behind in this field. We have come to the conclusion after observing that while interpreting the Qur'anic verses relating to the greatness and character of the Prophet (peace be upon him) The commentators were very eloquent and described the characteristics of the Prophet (peace and blessings of Allah be upon him) in a very eloquent and loving manner. In this article, the detailed material written by the commentators on the characteristics of the Prophet (peace be upon him) in Urdu commentary literature will be presented.

Rheological and Release Characteristics of Drugs of Different Solubilities from Semisolid Matrix Filled in Hard Gelatin Capsule

The re-introduced liquid/semisolid matrix technology, in recent past encompasses filling of hard gelatin capsules for thixotropic system at ambient and thermosoftened system at elevated temperature. Formulations in both of the above systems may contain dissolved or dispersed drugs. In the present work thixotropic gels were prepared using a series of hydrophilic poloxamers (Pluronic/Synperonic L31, L43, L62, L64, L92) of different viscosities and composition of polyoxyethhylene and polyoxypropylenein co-polymers thickened with hydrophilic silicon dioxide (Aerosil®A200) to form gel structure. Model drugs of varying solubilities, isoniazid, metronidazole, paracetamol and mefenamic acid were dispersed in gels of above poloxamers. The gel formulations were filled into hard gelatin capsules via syringe. The rheological characteristics, dispersion stability, and FTIR of the gels with and without drugs were investigated. The gels with drugs were additionally characterized for their release pattern. In preliminary studies, twenty separate gels of poloxamers L31, L64 were thickened with concentrations 1 to 10% w/w of Aerosil® A200 to incorporate isoniazid (unsifted, 10% w/w). Different concentration of Aerosil® A200 affected dispersion stability as well as release of the drug incorporated into gels. All gels were thixotropic and shear thinning. A little change in apparent viscosity on storage upto 30 days was noted. Disperse phase sedimentation was not observed with more viscous poloxamer and with higher concentration of silicon dioxide. Higher concentration of Aerosil® A200 resulted into stronger gel structure between silicon dioxide and poloxamers and thereby, hindered release of drug from a gel. Rheograms of poloxamers, L31 and L64 confirmed their Newtonian behaviour, in contrast to the thixotropic behaviour of all gels made with Aerosil® A200 concentrations (1-10% w/w) and 10% w/w isoniazid. Therefore, ii in-vitro release of highly water soluble drug, isoniazid could be controlled by poloxamer/A200 thixotropic gel system which depended upon concentration of Aerosil® A200, viscosity of poloxamer and the aqueous solubility of drug. In the further work, the effect of selected concentration of Aerosil® A200 (8% w/w) and five liquid poloxamers of different viscosities on release of drugs with characterized particle size (180-250 μm) and of different from semi solid matrix/ thixotropic gel filled into hard gelatin capsules was tested by in-vitro dissolution test. The drugs with different solublities used in this study were isoniazid, metronidazole, paracetamol and mefenamic acid. The interaction of drug in gel formulations was also determined by Fourier transform infrared (FTIR) spectroscopy. The release of drugs from different formulations was related to their aqueous solubility, viscosity of poloxamers, and poloxamer’s polyoxyethylene and polyoxypropylene proportions. In most of the formulations drug remains dispersed over a period of one month depending upon the viscosity of gels. Most of the gels showed little change in apparent viscosity on storage. The apparent viscosity for each gel was found to be dependent on the viscosity of poloxamers from which the gel was prepared. Like in preliminary work, rheograms of five pure poloxamers showed Newtonian behaviour whereas each gel exhibited thixotropic and shear thinning. In-vitro dissolution data of drugs from their respective stable gels followed first order kinetics. No chemical interaction or incompatibilities between drugs and polymers were noted by FTIR. Long term stability studies at International conference on harmonization (ICH) recommended conditions of temperature/relative humidity (30°C/65% RH) over one year revealed that almost all the gels retained their rheological, dispersion stability, release profile and chemical integrity. Thus, the hydrophilic semisolid matrix (gels) made by mixing poloxamer and Aerosil® (A200) was suitable for filling into hard gelatin capsules because of their rheological characteristics. The release of drugs with different solubilities could be modified by using poloxamers of different viscosity. Overall present research study demonstrates the effect of silicon dioxide on the microstructure of the gel prepared with various poloxamers which is evident from the rheological studies, FTIR and release pattern of the drugs