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Boundary Value Problems for Fractional Differential Equations: Existence Theory and Numerical Solutions

Thesis Info

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Author

Rehman, Mujeeb Ur

Program

PhD

Institute

National University of Sciences & Technology

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2011

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2267/1/2508S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725654761

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Fractional calculus can be considered as supper set of conventional calculus in the sense that it extends the concepts of integer order differentiation and integration to an arbitrary (real or complex) order. This thesis aims at existence theory and numerical solutions to fractional differential equations. Particular focus of interest are the boundary value problems for fractional order differential equations. This thesis begins with the introduction to some basic concepts, notations and definitions from fractional calculus, functional analysis and the theory of wavelets. Existence and uniqueness results are established for boundary value problems that include, two–point, three–point and multi–point problems. Sufficient conditions for the existence of positive solutions and multiple positive solutions to scalar and systems of fractional differential equations are established using the Guo–Krasnoselskii cone expansion and compression theorems. Owning to the increasing use of fractional differential equations in basic sciences and engineering, there exists strong motivation to develop efficient, reliable numerical methods. In this work wavelets are used to develop a numerical scheme for solution of the boundary value problems for fractional ordinary and partial differential equations. Some new operational matrices are developed and used to reduce the boundary value problems to system of algebraic equations. Matlab programmes are developed to compute the operational matrices. The simplicity and efficiency of the wavelet method is demonstrated by aid of several examples and comparisons are made between exact and numerical solutions.
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علامہ راشد الخیری

علامہ راشد الخیری
ہم نے یہ خبر بھی دلی رنج و افسوس سے سنی کہ ۳؍ فروری کو مولانا راشد الخیری نے اس دار فانی کو خیر باد کہا اور ہندوستان کا طبقۂ نسواں اپنے ایک بڑے معلم اور اپنے حقوق کے ایک بڑے محافظ سے، اور ہندوستانی زبان اپنے ایک بڑے محسن کی خدمات سے ہمیشہ کے لئے محروم ہوگئی، مرحوم شمس العلماء ڈپٹی نذیر احمد کے عزیز قریب اور طرزِ تحریر میں ابتداً ان کے پیرو تھے، مرحوم نے اپنی ادبی زندگی رسالہ مخزن کی ادارت سے وابستہ ہوکر شروع کی، پھر ۲۷ سال گزرے کہ عصمت کے نام سے ایک زنانہ رسالہ جاری کیا، اور ساری عمر طبقہ نسواں میں بیداری پھیلانے اور انہیں تعلیم و تربیت کے زیور سے آراستہ کرنے میں گزاردی، ہندوستان کے طبقۂ نسواں کو بیدار کرنے کی خدمت دو بزرگوں شمس العلماء مولوی ممتاز علی (تہذیب نسواں) اور مولانا راشد الخیری (عصمت) نے انجام دی افسوس کہ یہ دونوں ۶ ماہ کے اندر ہم سے جدا ہوگئے، مولانا راشد الخیری کی تحریک نسواں کا یہ امتیازی وصف تھا کہ وہ حقوق نسواں کے سب سے بڑے حامی کے ساتھ شریعت اسلامی کی حدود کا پاس نگاہ میں رکھتے تھے اور حقوق نسواں کے حامیوں کے اس گروہ کے سب سے بڑے مخالف تھے جو عورتوں کو مغرب کی کورانہ تقلید کی طرف لے جانے والا ہے، وہ ہندوستانی زبان کے مشہور ادیب، اور صاحب طرز انشا پرداز تھے، انہیں لال قلعہ کی پاکیزہ زبان لکھنے پر قدرت حاصل تھی، وہ انسانی درد و مصیبت اور معاشرتی زندگی کا خاکہ کھینچنے میں کمال رکھتے تھے، ان کی صبح زندگی و شام زندگی، ڈپٹی نذیر احمد کی توبۃ النصوح وغیرہ کے پہلو میں رکھے جانے کے قابل ہیں، اور بلاشبہ ان کی کتابیں پڑھ کر آنسوؤں کا ضبط کرلینا دشوار ہے،...

Aristotle’s Virtue of Justice as an Ethical Solution to Political Corruption: Analysis and Reflection

This paper attempts to discuss Aristotle’s concept of justice as an ethical solution to political corruption. The goal of this paper is to present corruption as a form of injustice that deprives the majority from a common good. This paper is very important because it provides ethical solution to grand corruption whereby unscrupulous individuals divert funds meant for development of the entire society into private hands. Due to grand corruption the poor masses are deprived of quality basic needs. The paper focuses on corruption as an immoral act through the lens of Aristotle’s ethical and political insights. The virtue of justice in Aristotle’s ethical and political works are spelled out. The methodology used in this paper is analytical and applied. It is analytical because it analyses Aristotle’s understanding of virtue of justice in his moral theory especially in his book entitled Nicomachean Ethics and justice in his political theory. Finally, the paper makes subsequent applications of Aristotle’s virtue of justice in the realm of ethical solutions to the problem of corruption.

Antecedents of Brand Extension Success in Glocalization: A Mediation and Moderation Analysis

This research is focused on finding whether brand extension strategy can be used in this glocalized markets and what are the antecedents (Mediating and Moderating variables) of successful brand extension? Glocalization has become a serious concern for managers that they have to manage the global brands with local trends and local brands in the global trends. Brand extension is a strategy in which new products can be launched under existing brand name due to which multinational and local firms expects that their consumers will respond to the product either positively or negatively because they are already familiar of the existing brand name or the parent brand. Two actual real time brands one representing close extension and other an example of distance extension has been used in this research and the sample respondents have been selected in a way that it is a good representation of population. No such broad and empirical research on brand extension in Pakistani context has been carried out. The independent variables used in this study are (1) Consumer Perceived Brand Concept, (2) Brand Image (3) Brand loyalty (4) Brand Evangelist (5) Parent Brand Conviction (6) Perceived Risk of unknown (7) Marketing support (8) Parent Brand Experience (9) Consumer Perceived Fit and dependent variable was Successful brand extension evaluation. Self developed questionnaire with items adapted from various past studies was self administrated among the sample respondents by simple random sampling. For close (Sufi Frozen Food) extension 464 sample respondents filled questionnaire. For Distance (Guard Rice) extension 432 respondents filled out questionnaire. Questionnaire was distributed outside famous big retail grocery stores like Agha‟s , Naheed, Chaseup and Imtiaz store located in Clifton Schon Circle, Shaheed-e-Millat Road and KPT Interchange respectively. 27 hypotheses were formulated for Close and Distance extensions. Data screening was performed on the data collected like Missing values, Aberrant values, detecting outliers (Mahalanobis distance) from the respondents, Normality test and questionnaire reliability. Cronbach Alpha was taken out of 37 question items and instrument revealed reliability with score greater than 0.6. Hierarchical regression, Single Mediation, Multi mediation (Sobel Test, Preacher & Hayes test) and Moderation analysis were applied to test the hypotheses. After test run lot of new concepts have unfold which are important to know before one opts for brand extension strategy in the glocal market context. Results reveal previous parent brand image built does not affect the consumer evaluation and hence managers should not worry about the past positive or negative image while deciding to use brand extension strategy for close or distance extension for brand development. Perceived Risk of Unknown as moderator does not moderate the relationship between Brand Image and Success of Brand Extension Evaluation. It means that these days consumers in Pakistan are not worried to put their money on unknown brand.