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Applications of Orthogonal Polynomials for the Numerical Solutions of Higher Order Boundary Value Problems

Thesis Info

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Author

Elahi, Zaffer

Program

PhD

Institute

University of the Punjab

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10389/1/zaffer%20Elahi_Mathematics_UoP%20lahore_2018_2542019.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725524310

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A differential equation along with a set of additional constraints (called boundary conditions) form a boundary value problem. Boundary value problems (linear or nonlinear) are found mostly in engineering, applied mathematics and physical sciences as well. Although, few numerical algorithms are found in solving the higher order boundary value problems using orthogonal polynomials. Since polynomials play a vital role in computing the numerical solutions of the differential equations therefore, the research, conducted during my PhD programme, concerns with the study of the numerical solutions of higher order linear boundary value problems. The main objectives of the research are to develop Galerkin technique for solving special eighth, tenth and twelfth order linear boundary value problems using Legendre polynomials, and numerical approximations for solving the system of Fredholm integro–differential equations and the system of differential equations using Laguerre polynomials. The numerical schemes, developed, have been compared with the existing methods, which shows the higher accuracy of the schemes.
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مولانا شوکت علی

مولانا شوکت علی مرحوم
اسی مہینہ کادوسرا المناک سانحہ مولانا شوکت علی خادم کعبہ کی وفات ہے، شوکت علی مرحوم ہندوستان کے اُن پرچند مسلمانوں میں ایک تھے جن کی شہرت نہ صر ف ہندوستان تک محدود ہے بلکہ دنیائے اسلام کے دوردراز گوشوں تک ان کانام عزت واحترام کے ساتھ لیاجاتاہے اوریہ واقعہ ہے کہ مرحوم بجا طور پر اس شہرت و احترام کے مستحق تھے، پچھلے چند برسوں کوچھوڑ کر بلاخوف تردید کہا جا سکتاہے کہ مرحوم کی زندگی قربانی، ایثار، ولولہ اورجوش عمل کے اعتبار سے مسلمانوں کے لیے قابل تقلید نمونہ تھی،جنگ طرابلس اورجنگ بلقان سے لے کراب تک ہندوستانی مسلمانوں کی اجتماعی اورسیاسی زندگی کے جتنے دور گذرے ہیں مرحوم کی خدمات اُن تمام دوروں میں اس قدر نمایاں اوراس قدر روشن ہیں جنہیں کسی طرح فراموش نہیں کیا جاسکتا’’علی برادران‘‘ہندوستان کی دوشخصیتوں سے مرکب ایک ایسی حقیقت کانام ہے جس کے زبان پرآتے ہی کرداروعمل اورشجاعت وبسالت کاایک سبق آموز نقشہ آنکھوں کے سامنے آجاتاہے۔
صد حسرت وافسوس کہ ہندوستان اپنے ایک جانباز،بہادر سپاہی اورپرانے خادم سے ہمیشہ کے لیے محروم ہوگیا۔انا ﷲ واناالیہ راجعون۔ حق تعالیٰ مرحوم کی خدمات کوقبول فرمائے اوردامانِ رحمت میں جگہ دے۔
[دسمبر۱۹۳۸ء]

دعوت کے میدان میں تبلیغی جماعت کے مساعی و مشکلات کا تحقیقی جائزہ

Muhammad (PBUH) was the last Prophet. Almighty Allah sent Him message to preach the people, but the Kufar made His enemies. Instead the situation of opposition and hatred from his enemies how He treated them with patience and love. And how He would be able to establish Islamic states and invites the people of other areas towards Islam. Muhammad (PBUH) was succeeded from the reformers of his ummah and among these reformers were the founder of tableghi jammat, Molana Muhammad Ilyas Kandhalwi. This research paper will compare the difficulties of Muhammad's (PBUH) Da'wah with tableghi Jammat.                       

Socio-Cultural Discourse on Female Reproductive Rights A Case Study of Pothohar Village, District Attock

This thesis mainly explores the association between the socio-cultural context and women‘s perception and practice of their reproductive rights. Meeting reproductive needs of rural women living in a traditional culture of Pakistan is no less than a challenge where sexuality is a taboo subject. There is little known about the sexual and reproductive practices and behaviour of women. The issues pertaining to female‘s reproductive rights remain unacknowledged at the policy level and lack implementation. Studies that were conducted In Pakistan before had mostly focused on the issues of maternal health from the service delivery aspect and very few attempts have been made to empirically assess linkages between socio-cultural factors and Reproductive Rights. In order to have a clear picture of the state of females reproductive rights, this study was conducted in village Choha Shah Ghareeb of District Attock, in the Province of Punjab with the prime objective to understand the perception and practices of reproductive rights among females of the reproductive age group (15-49 years). The study employed anthropological methods including participant observation, in-depth interviews, illness narratives to provide a comprehensive analysis of reproductive with an emic perspective. The study explored their participation in Fertility Decision Making and utilization of reproductive health services such as antenatal care, delivery and postnatal care. Furthermore, it investigated those traditional practices which affect female‘s reproductive rights such as early marriages, inability to access modern health care facilities, nutritional taboos, consequences of infertility, son preferences and unsafe induce abortion etc. The study elaborated key aspects linked to the notion of female sexuality and fertility. The findings show that socio-cultural norms inhibit discussion related to sexuality and reproductive health and young girls‘ receive little information about puberty prior to menarche which becomes the foundation for their incapacity to safeguard their future reproductive health. Marriage is considered a religious duty and an inevitable act. Field findings reflect the importance of marriage to protect the honor of women and also for the formation of a family unit. The right to marry and form a family was more recognized vii in terms of parental responsibility and obligation to arrange timely marriage of their children. Among married women interviewed in this study, 45% were asked to give their opinion in spouse selection and more than one third of respondents were married to their cousins. Women were also aware of their right to seek divorce yet the majority of them (47%) disagreed and urged the necessity to avoid dissolution of marriage due to the fear of social disgrace, family pressure, being separated from children and due to the limited opportunities to marry again. Sexuality for married women is an area of compromise, obligation, and lack of control. Sexual obedience is considered an essential characteristic of a good and successful wife. The majority of the respondents agreed that they would never like to refuse their husbands and perceived sex as a ‗wife‘s duty & religious obligation‖. Furthermore, good women are expected to be ignorant and passive in sexual matters and sexual subservience is used as a strategy to gain love and respect from their husbands and also to secure their present and future well-being. Women‘s understanding of their reproductive health comes from their socialization pattern, configuration of gender roles and sense of motherhood and identity based on their norms and traditions. Women placed safe motherhood as an important element of their expressed meaning of reproductive rights and strongly emphasized their right to have good reproductive health care particularly during pregnancy and post-partum period. But practically, the majority of them considered conception and the act of giving birth as ‗natural‘ and received antenatal care mostly in case having some health problem. There was a clear preference for home delivery and hospitals were chosen in case of emergency or health risk or in the absence of family member to assist home delivery. Women perceived childlessness as a curse because children strengthened the marital bond and raised their status in the family. Having a quick pregnancy meant compliance with social norms and family expectations. Successful childbearing gives them respect and decision making authority in reproductive matters. Women also discussed the deteriorating effect of excessive births for their reproductive health and the majority of viii them acknowledged their right to space births and determine family size. Many women mentioned their limited decision making authority with regards to keeping their family size smaller and also accepted the covert use of contraceptives and induced abortion without consent of their husbands in case of having closely spaced pregnancies, health concerns, or under extreme economic misery etc. The majority of women had knowledge of modern contraceptives and apart from them they also knew and practiced traditional methods such as withdrawal. Family planning methods and their perceived viability is at the heart of the discourse surrounding women‘s fertility control behavior. The findings demonstrate that women‘s lower social position and economic dependence on their husbands and families limit their ability to practice their reproductive rights by not allowing them to make independent decisions to regulate their reproductive lives. Fertility decisions are made and experienced within their cultural context whether it is early marriage, compulsory motherhood, home based delivery, lack of fertility control, closely spaced pregnancies, unsafe abortions or utilization of reproductive health care services. The study concludes that women situated their reproductive rights within the broader spectrum of their socio- economic, familial and cultural context. Having a good family life, securing social and economic protection and attaining social respect were key elements of the meaning of reproductive rights for them. Women, in the desire to keep their marriage successful or for the sake of their children, prefer to compromise for the sake of this affiliation, which ultimately provides them socio-economic and emotional support.