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Fuzzy Fractional Differential Equations

Thesis Info

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Author

Sadia Arshad

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2008

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/2234

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726249737

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The concept of fuzzy fractional differential equation (FFDE) was introduced by Agar- wal, Lakshmikantham and Nieto [1]. We develop this concept of fuzzy fractional dif- ferential equation and obtain some results about existence and uniqueness of solution of FFDE. In the second chapter we recall some basic knowledge of fuzzy calculus and frac- tional calculus. In third chapter we introduce the concept of Riemann-Liouville integral and Riemann-Liouville derivative for fuzzy functions. Fuzzy derivative is consider in the Seikkala sense. Moreover we give the new results about the properties of fuzzy frac- tional integral and fuzzy fractional derivative. Further, we study the existence and uniqueness of the solution for a class of fractional differential equation with fuzzy ini- tial value. The fractional derivatives are considered in the Riemann-Liouville sense. In fourth chapter we establish that fuzzy fractional differential equation is equiv- alent to the fuzzy integral equation and using this equivalence we prove the existence and uniqueness of solution of fuzzy fractional differential equation. Fuzzy derivative is considered in the Goetschel-Voxman sense. Also the applications of the existence and uniqueness theorem has been given. In fifth chapter we present results regarding the existence of the solutions of fuzzy fractional integral equations (FFIE). We prove the existence of the solutions of FFIE under compactness type conditions using the Hausdorff measure of noncompactness. Also we prove an existence result without using noncompactness measure. For this we use a variant of the Schauder fixed point theorem in fuzzy metric space.
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صلاح الدین اویسی

جناب صلاح الدین اویسی مرحوم
جناب غلام محمود بنات والا کی فرقت کا صدمہ ابھی کم نہ ہوا تھا کہ مجلس اتحاد المسلمین کے قائد، فرزند دکن اور سالار ملت، سلطان صلاح الدین اویسی بھی اس دنیا سے رخصت ہوگئے، امت ایک اور ہوش مند، جرأت مند اور حوصلہ مند راہبر سے محروم ہوگئی، وہ عرصے سے صاحب فراش تھے، عمر مستعار کم نہ تھی، وقت موعود آیا اور رمضان المبارک کے نہایت مبارک اور آخری عشرے میں یعنی ۲۸؍ رمضان کو وہ اپنے مالک حقیقی کی رحمتوں سے جاملے ، اناﷲ وانا الیہ راجعون۔
قریب ساٹھ ستر سال پہلے جب مجلس اتحاد المسلمین کے بانی نواب بہادر یار جنگ کا ۱۹۴۴؁ء میں انتقال ہوا تھا تو اس وقت مولانا سید سلیمان ندوی نے معارف میں ان کا ماتم کرتے ہوئے لکھا تھا کہ ’’ان کا سا آدمی صدیوں میں پیدا ہوتا ہے اور جب پیدا ہوتا ہے تو انقلاب انگیز ہوتا ہے، ان کی ذات سے امت اسلامیہ کو بڑی بڑی امیدیں قائم تھیں اور خصوصیت کے ساتھ دکن کے مسلمانوں کے حق میں ان کا وجود آب حیات کا حکم رکھتا تھا‘‘۔
۱۹۴۴؁ء اور ۲۰۰۸؁ء کا زمانی فرق، تغیرات احوال کے لحاظ سے زمین آسمان کا فرق ہے لیکن صلاح الدین اویسی مرحوم کی ہستی کے لیے ان الفاظ کی حقیقت میں ذرا فرق نہیں۔
سلطنت آصفیہ اسلامیہ میں مجلس اتحاد المسلمین کا قیام، قومی تخیل اور سیاسی جذبوں کو نئی زندگی دینے کے ساتھ مسلمانوں کی تعلیمی اور معاشرتی ترقی کے لیے ہوا تھا، نصب العین تو ’’وسیع تر، عظیم تر، بیرونی و اندرونی مداخلتوں سے پاک اور کامل آزاد حیدرآباد تھا‘‘ اس وقت اس مجلس کے حوصلوں کو دیکھ کر کہا گیا تھا کہ دکن کے مسلمانوں نے صدیوں کے آرام کے بعد کروٹ لی ہے لیکن چند برسوں کے بعد تد اول...

Unveiling Self-Cursing Patterns in Sensitive Individuals: A Qualitative Inquiry from a Pakistani Context

Being sensitive means being overwhelmed and easily affected by little things happening in the environment. Sensitivity is usually considered negative trait in humans and sensitive people are being treated negatively and harshly in society, such individuals are usually observed to curse themselves. This qualitative study was designed to explore the meanings and perception of sensitivity. Triangulation of data sources was selected to find the results. This study recruited 21 participants for interview and 56 documents for data collection. The participants in this study were inducted purposefully having ages between 18 to 29 years. WhatsApp interviews, aligned best with the objectives of the research study that lasted from 20- 30 minutes. Data collection was started after the formal approval of research protocol from the departmental research committee. Three veterans of qualitative research were involved in the recording and transcription of data; memos and code book were also incorporated along with the data. Phenomenological analysis for interviews and content analysis for documents were used to analyze the data through the lens of Rubin (2021). Methodological Integrity was maintained through triangulation of data sources. The findings exhibit six main themes; intensive feelings, personality traits, expectations, prioritizing relations, emotions and extreme reactions. Researchers faced an increment in sensitivity during data collection. To deal with this problem few pauses were taken during data collection, so it took longer than usual to collect data. Implications of the study have been discussed.

A Natural Language Based Retrieval System for Data Warehouse Using Multi-Dimensional Entity Relationship Model

Natural Language Interfaces to Databases (NLIDB) is an area of research that deals with the representation of users request to database in their native language. Currently, data warehouses are widely used by enterprises for decision making. It is important to mention that characteristics of decision-making systems are inherently different from transactional systems. The users of decision-making systems are top management (executives) who are normally non-technical having less knowledge of the data warehouse schema and about writing database technical queries. In fact, ad-hoc query is the information need of user that may not be fulfilled with front-end tools having predefined capabilities e.g. Reporting, OLAP or Data mining tools etc. Such information needs are not easy to express in technical query language. This motivated us to propose a Natural Language Based Retrieval System for Data Warehouse (NLRSDW) to support users especially in the ad-hoc query development.A Logical Schema-based Mapping (LSM) technique has been developed. Using this technique, targeted search is performed efficiently in the data instances. For targeted search, a LSM oriented mechanism has been presented. In addition, 3 searching strategies are elaborated which include 1) Identified elements searching 2) Proximal elements searching and 3) Level-wise searching to retrieve the matching instances for each data value. The retrieved instances are ranked with 5 criterions based on which an algorithm has been developed. Furthermore, solution to identify the aggregation constructs (i.e. aggregation function, measure, level and grouping attributes) accurately has been presented. Data Warehouses maintain aggregated computations to efficiently answer queries on large volume of data. It is very challenging task to interpret accurate aggregation constructs from the keyword-based query written on the Natural Language Interface to Data Warehouse. Later, a semi-automatic approach to build the Data Warehouse logical Schema-based Domain Thesaurus is proposed.This approach takes the Data Warehouse logical Schema as input and generates Domain Thesaurus using multiple sources containing Schema, Data Instances, WordNet, WWW and Domain Repository. The Thesaurus evolution approach is also presented which shows how Thesaurus can be technically expanded at user query time.An in-depth experimental evaluation has been carried out in comparison to existing systems. The results are encouraging. Using NLRSDW, non-technical users can easily write any ad-hoc information need in natural language. As a result, executives do not have to take support of IT staff and time to develop query is negligible.