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Existense and Approximation of Solutions of Differential Equations

Thesis Info

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Author

Talib, Imran

Program

PhD

Institute

University of Management and Technology

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2017

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/9784/1/Imran_Talib_Maths_HSR_2017_UMT_lahore_06.08.2018.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726153315

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Nonlinear coupled boundary value problems (BVPs) have very important and interesting aspects in the kingdom of Nonlinear Analysis due to not only the theoretical aspects but also the applications which they have in almost everyeld of science. Problems with coupled boundary conditions (BCs) appear while studying mathematical biology, Sturm-Liouville problems, reaction di usion phenomena, chemical systems, and Lotka-Volterra models. This thesis has two parts. In therst part, the existence results are established for therst{order and the second{order nonlinear coupled BVPs subject to nonlinear coupled BCs. Also in the same part, the existence results are established for the second{order nonlinear coupled BVPs when the nonlinear functions have dependence on therst-order derivative. Multiple approaches are available in the literature to investigate the existence of solutions of nonlinear BVPs, but lower and upper solutions (LUSs) approach is one of the strongest. In this approach the original problem is modi ed logically to a new problem, known as the modi ed problem, then the theory of di erential inequalities with the combination of well-known existence results are applied to establish the existence of solution of the modi ed problem. Finally the solution of the modi ed problem leads to the solution of the original problem. Moreover in therst part of the thesis the treatment of the many di erentrst-order and the second-order nonlinear BVPs are uni ed by developing the idea of coupled LUSs. Under this idea, some monotonicity assumptions are imposed on the arguments of the nonlinear BCs in the presence of the existence of a lower solution and an upper solution to unify the classical existence results for very important types of BVPs, like periodic, anti-periodic, Dirichlet, and Neumann. Several examples are discussed to support the theoretical results. The subject fractional calculus being a generalization of integer-order calculus has numerous applications in almost everyeld of science. Due to the intensive use of fractional order di erential problems (FODPs) in almost everyeld of science including, but not limited to, uid dynamics, physics, aerodynamics, chemistry, mathematical biology, image processing, and psychology, there is a strong motivation for the researchers to develop reliable and e cient numerical methods tond the approximate solutions of FODPs. xii xiii In the second part of the thesis, we consider a generalized class of multi{terms fractional order partial di erential equations (FOPDEs) and their coupled systems. We develop a new numerical method and generalize the corresponding Jacobi operational matrices of integrals and derivatives considered on a rectangular plane. By means of the operational matrices, the considered problem of fractional order is reduced to an algebraic one. Being easily solvable, the associated algebraic system leads tonding the solution of the considered problem of fractional order. Validity of the method is established by comparing our simulation results obtained by using MATLAB softwares with the exact solutions in the literature yielding negligible errors.
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پروفیسر عبدالقوی دسنوی

پروفیسر عبدالقوی دسنوی مرحوم
افسوس، پروفیسر عبدالقوی دسنوی اس دنیا سے رخصت ہوئے، اِناﷲ وَاِنا اِلَیہ رَاجِعُون۔ ۷؍ جولائی کو وہ بھوپال کی خاک میں آسودہ خواب ہوئے تو اردو کی دنیا صرف ایک شریف، متین اور نشیط صاحب قلم سے ہی محروم نہیں ہوئی، ایسے مخلص، فعال بلک مجاہد انسان کو کھوبیٹھی جس کا تعلق اس جنس سے تھا جس کو اب نایاب ہی کہا جاسکتا ہے۔
دیسنہ ان کا وطن تھا، سادات کا یہ گاؤں بہار کیا پورے ہندوستان میں ممتاز اور منفرد تھا کہ اکثر باشندوں نے تعلیم قدیم و جدید دونوں کو یکساں اہمیت دی۔ کم ایسی بستیاں ہیں جن کی خاک سے مولانا سید سلیمان ندوی، سید ابوظفر ندوی، سید نجیب اشرف ندوی اور سید صباح الدین عبدالرحمن جیسی شخصیتیں اٹھیں اور علم و قلم کے بادل بن کر ملک کے مختلف خطوں کو سیراب کرگئیں۔
پروفیسر عبدالقوی بھی اسی سلسلہ ابروسحاب کا ایک حصہ تھے۔ بچپن، وطن میں ضرور گزرا جہاں مدرسۃ الاصلاح سے تعلیم کا آغاز ہوا، ۴۲ء میں وہ آرہ کے ہائی اسکول کی آٹھویں جماعت میں تھے لیکن پھر وہ بمبئی آگئے جہاں ان کے والد پروفیسر سید سعید رضا ندوی درس و تدریس کی ذمہ داریاں انجام دے رہے تھے، قوی صاحب نے بمبئی کے مشہور سینٹ زیویر اسکول میں اعلیٰ تعلیم حاصل کی، بمبئی گویا وطن ثانی تھا لیکن قدرت نے ان کے لیے بھوپال دارالاقبال کی سرزمین اس طرح مقدر کی کہ ۶۱ء میں سیفیہ کالج کے شعبہ اردو کی خالی جگہ کے لیے ایک درخواست اور اس کی قبولیت نے ان کو ’’بھوپال والے عبدالقوی دسنوی‘‘ بنادیا، بمبئی کے ہنگامہ خیز اور پرشور ماحول سے نکلے تو کوہساروں، بیابانوں کے نشیب و فراز اور ان کے دامن میں بکھرے زندگی کے تازہ و شاداب جلوؤں اور پرسکون ماحول کو پاکر محسوس کیا...

انسانی دودھ کے بینک: تعارفی و فقہی جائزہ

Human Milk banks have been established in many parts of the world. The main purpose of these banks is to save the babies’ lives and to ensure that the newborn babies’ rights to breast milk are fulfilled. Especially for the infants whose Mothers could not feed them due to illness, lack of milk or lack of time. The problem is that milk bank practices in the Western countries contradict with the Islamic law whereby it may result in the possibility of overlapping of the progeny (nasab) and selling the organ of human, etc. The Muslim countries have been not participating in these milk sharing activities because of these religious issues. However, due to a critical need of breast milk in hospitals, this article addresses these issues and the different opinions of Islamic scholars and suggests ways to formulate a proper model of milk bank that is compatible with the Islamic law and to avoid further problems of nasab. This study has two main objectives: firstly, to introduce Milk Bank, causes of its’ existence, method of collection and storage the milk, the benefits of breastfeeding and the unpleasant effects of Milk Banks. Secondly, to review some authoritative legal scholars’ opinions on the issue of milk bank and to recommend a proposal on how to develop a milk bank in accordance with the Islamic law. The study is expected to be able to recognize the issues of Milk Bank, to make the people become aware of its side effects and religious problem.

Genetic Diversity in Chickpea Based on Morphometric and Molecular Markers

The exploration of genetically variable accessions is the key source of germplasm conservation and potential breeding material for the future. The more diverse group of cultivars can provide an ample opportunity to breeders for releasing new and superior varieties, considering their quality traits for direct commercial utilization. In advanced research studies the polymerase chain reaction (PCR) based molecular markers have a great contribution in genome analysis and marker-assisted selection. In this study, the genetic diversity of Cicer arietinum L. twenty four indigenous and forty six exotic accessions were assessed, obtained from plant genetic resource institute (PGRI), national agriculture research centre, Islamabad, Pakistan. These accessions were planted under field conditions at research area of University of Malakand, Chakdara, Khyber Pakhtunkhwa. The genetic diversity among seventy chickpea indigenous and exotic accessions was estimated using morphological, biochemical; sodium dodecyle sulphate polyachryl amide gel elecctrophoresis (SDS-PAGE) and molecular markers; random amplified polymorphic DNA (RAPD) and simple sequence repeat (SSR) markers. Based on qualitative and quantitative morphological traits, the average coefficient of variation (%) was calculated 44.8% and 56.8% respectively with significant correlation among yield traits. The analysis revealed that the accessions 1898, 2819, 3022, 3037, 3040, 3043, 3054, 3059 and 3063 were best in performance with a total of 12% environmental error. The statistical analysis showed that 100 seeds weight was significantly correlated with seed size quantitatively. The majority of accessions of USA origin were observed with maximum100 seed weight (30-57gm) and medium to large (7.2- 9.9mm) size seeds including one of the Pakistani accession 2562 also with large size seeds. The wilt incidence (%) was observed to be comparatively higher (30 - 42.85% ) at both growth stages in field screening of the germplasm than that of greenhouse conditions; reduced up to 8.57% at seedling stage and 24.28% at reproductive stage. The t-test however, indicated that chickpea both from indigenous and exotic origin showed a significant variation at alpha ≤ 0.050 at seedling and reproductive stage. The cluster analysis based on protein data indicated 50% genetic diversity among the accessions. The clustering pattern did not reveal any grouping that could be attributed to either the geographic distribution or the field performance. For molecular characterization of xiv germplasm twenty random amplified polymorphic DNA (RAPD) and twenty simple sequence repeat (SSR) polymerase chain reaction (PCR) based markers were screened for estimation of genetic variability. In the markers, five random amplified polymorphic DNA (RAPD) and fifteen simple sequence repeat (SSR) were polymorphic and showed significant level of coefficient of variation. The data of molecular markers were scored by the presence (1) and absence (0) of allele and subjected to statistical analysis. The analysis was based on coefficient of molecular similarity using un-weighted pairs group mean average (UPGMA) resulted in 37% and 55% genetic diversity among the total germplasm using random amplified polymorphic DNA (RAPD) and simple sequence repeat (SSR) markers respectively. For marker trait association analysis, twenty random amplified polymorphic DNA (RAPD) and twenty simple sequence repeat (SSR) makers were utilized to find correlation of markers with yield contributing components and chickpea Fusarium wilt resistant genes. None of the random amplified polymorphic DNA (RAPD) primers were linked to seed size and seed weight while, simple sequence repeat (SSR) markers TA72 and TA130 showed association at linkage distance 0.4 with seed weight and seed size. Based on which the high yielding accessions among chickpea germplasm were identified. Hence, the association of these makers is helpful for the plant breeders to select lines on the basis of yield contributing traits. Among the total used random amplified polymorphic DNA (RAPD) and simple sequence repeat (SSR) primers, TA194 (SSR marker) was linked to the disease response with 85% probability level. This association or correlation of the marker was reconfirmed by receiver operating characteristic curve (ROC). Hence, the use of the sorted wilt resistant genotypes through simple sequence repeat (SSR) marker TA194 can make available ample prospect in marker assisted breeding for yield improvement of chickpea in Pakistan.