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Fuzzy Fractional Differential Equations

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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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مولاناحاجی معین الدین ندوی

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مصنف خلفائے راشدین
افسوس کہ ۵؍ ربیع الثانی ۱۳۶۰؁ھ کو ہماری جماعت کے ایک لائق فرد مولانا حاجی معین الدین صاحب ندوی صدر مدرس مدرسۂ اسلامیہ شمس الہدیٰ پٹنہ نے تقریباً پچاس ۵۰ برس کی عمر میں وفات پائی انہوں نے اپنی پوری تعلیم دارالعلوم ندوۃ العلماء میں حاصل کی اور ۱۹۱۲؁ء میں درجہ تکمیل سے فراغت پائی، ۱۹۱۴؁ء کے آخر میں دارالمصنفین کے قیام پر وہ دارالمصنفین کے رفیق منتخب ہوئے اور سلسلہ سیرالصحابہ کی پہلی اور دوسری جلد خلفائے راشدین اور مہاجرین حصہ اول لکھی، ایک سال کے بعد یہاں سے کتب خانہ ندوہ کی ترتیب کے لئے لکھنؤ گئے اور اس کام سے ان کو ایسی دلچسپی ہوئی کہ بوہار امپریل لائبریری کلکتہ میں ترتیب فہرست کے کام پر مقرر ہوگئے اور وہاں سے اورینٹل لائبریری بانکی پور پٹنہ کی عربی کتابوں کی ترتیب فہرست کے کام پر لگائے گئے، اور کئی جلدیں بڑی قابلیت سے انگریزی میں مرتب کیں اور گورنمنٹ کی طرف سے چھپیں، اس جگہ کی تخفیف ہونے پر دائرۃ المعارف حیدرآباد میں قدیم ہندوستانی تاریخی مقامات کا ایک جغرافیہ عربی زبان میں ترتیب دیا جو دائرہ کی طرف سے چھپا ہے، یہاں سے نکل کر وہ چند روز رامپور کی سرکاری لائبریری میں مقرر ہوئے اور آخر صوبہ بہار کی مشہور سرکاری عربی درسگاہ مدرسہ شمس الہدیٰ کے پرنسپل مقرر ہوئے اور اسی خدمت پر وفات پائی۔
وہ نہایت خاموش طبیعت ملنسار متواضع اور نیک دل تھے، وطن صوبہ بہار کے دو مشہور گاؤں گیلانی اور استھاواں میں تھا، نوجوانی ہی میں جب وہ دارالعلوم میں پڑھتے تھے، حج سے شرف ہوئے تھے، اسی لئے وہ ہماری جماعت میں حاجی صاحب کے نام سے ایسے مشہور و متعارف تھے کہ یہ ان کے اصلی نام کا جز بن گیا تھا، انگریزی تعلیم صرف ندوہ میں...

علم مختلف الحدیث پر امام شافعی کے تفکرات کا تحقیقی مطالعہ

After the era of Prophet Muhammad (S.A.W), there were dissimilarities in interpretations of contradictive revelations by Muhammad (S.A.W) thereby questioning their authentication while being considered a basic source of extracting the Sharia acts. In this context, Imam Shāfʿī studies all the dissimilarities and contradictions and devised finest principles which performed very positive role in rationalizing the conflicting Ahadiths. He was first to discuss these matters in an innovative manner. It is therefore, essential for investigator to refer to Shāfʿīs principles to rationalize any Hadith lying under contradiction. Imam Shāfʿī stated that Hadith is not itself contradicting but its apparent meaning leads the investigator into misconception. Consequently, he has stated three methods to remove this conflict. First is the combination (جمع) which recommends working on both Ahadiths at the same time rather than leaving either of them. Second is the abrogation (تنسیخ) which requires extensive knowledge of Quran and History before inferring the final decision of Muhammad (S.A.W) and scholars widely recommend Imam Shāfiʿī in recognizing abrogated Ahadiths. Third is preference (ترجيح) which is based upon peripheral attributes.

Study of Polymeric Hydrogels As Novel Biotechnological Materials

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