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Hadamard K-Fractional Integral and its Application

Thesis Info

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Author

Farid, Ghulam

Program

PhD

Institute

Global Institute Lahore

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2015

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/13035/1/Thesis%20Ghulam%20Farid%20%20%28HEC%29.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726355199

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The Fractional Calculus has been attractive and hot topic among the researchers since 18th century, because of its extensive application in differential and integral equations and other disciplines of mathematics, physics and economics. The motivation of this thesis is to extend the fractional integrals and derivatives, particularly Hadamard fractional integral, and to establish basic properties of the extended fractional integral operators. The application of the extended operators involving the formation of the fractional integral inequalities and solutions of fractional integral equations is focussed in the work. The first chapter includes the introductory background of the fractional calculus. The appropriate literature pertaining to the fractional calculus, involving the theoretical and practical aspects of fractional differential and fractional integral operators has been reviewed. In the second chapter, we have listed symbols, notations and the basic results that are used throughout the dissertation. A number of inequalities involving the Holder’s inequality and AM-GM inequality have been presented. We have defined an extended form of Hadamard fractional integral and have called it Hadamard k-fractional integral. We have also discussed a number of properties of the extended integral operator. In the third chapter, we have established numerous fractional integral inequalities involving the inequalities of Chebyshev functional using the notion of synchronous functions, asynchronous functions, and the like. In the fourth chapter, we have presented some inequalities involving the rearrangement inequalities. On the basis of AM-GM, Holder and the rearrangement inequalities, we have established many fractional integral inequalities related to the extended operator. In the fifth chapter, we have introduced a number of extensions of the fractional integral operators involving the Hadamard type fractional operators. We have discussed the properties of the extended operators involving the semigroup property and commutative law. We have also considered the Mellin transforms and boundedness of some of the extended operators. In the sixth chapter, we have introduced extended fractional derivatives related to the extended fractional integral operators and have discussed their compositions. In the seventh chapter, we have presented some integral equations and have found their solutions using some of the extended fractional integral operators. We have also illustrated the use of some of the extended fractional calculus operators in finding solutions of fractional differential equations.
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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]

Mīthāq Al-Madīnah: A Universal Charter of Peace an Analytical Study in the Modern Socio-Political Context

Different religious, political and social leaders tried their best to establish peace and prosperity in the society in different phases of the human history. An influential effort out of these efforts is that of the Prophet (r) Muḥammad. The Prophet (r) faced a pluralistic society of different faiths and religions in Madīnah. So, to make a better relationship and establish peace between the Muslims and the other communities of Madīnah, an agreement, which is called Mīthāq al-Madīnah, was made. Mīthāq al-Madīnah was not only a deed, but it presents all those principles and regulations, which were mandatory for peace building in a state or society. The excellent aspect of this charter is that the recipients of this charter were not the Muslims only, but a pluralistic society of different faiths. These communities were bound to establish peace with an agreement. This charter is an excellent model of peace, prosperity, freedom and human rights. According to this charter, all the parties were free with their religious beliefs and social interests. It was also the constitution of Madīnah. The renowned Muslim scholars are unanimously agreed that it was the first written constitution of the world. This agreement provides all other communities of Madīnah (the Jews, the Christians, and the polytheists) equal rights and freedom. Consequently, the charter of Madīnah can become a base for enduring peace and peaceful coexistence in a pluralistic world for the sake of the welfare of the human beings.

Evaluation of Natural Gums As Potential Precursors for the Synthesis of Gold and Silver Nanoparticles Their Characterization and Applications

Herbal medicines are recognized as multi-component remedies for the treatment of complicated diseases because of their better therapeutic value and fewer side effects compared to modern medicine. The work presented in this thesis consists of “Evaluation of natural gums as potential precursors for synthesis of gold and silver nanoparticles: their characterization and applications” which includes the synthesis, optimization, characterization, applications and bioactivity evaluation of gold and silver nanoparticles stabilized with different gums. Four different gums of plants origin and one gum of microbial origin were selected for the synthesis of gold and silver nanoparticles. Gums of plants origin were collected from three different plant families i.e. family Fabaceae (genus: Acacia), family Rosaceae (genus: Prunus), family Bixaceae (genus: Cochlospermum), while the microbial “Xanthan gum” was obtained from Xanthomonas campestris. In this study gold and silver nanoparticles (Au- and Ag-NPs) were biosynthesized by using the aqueous extracts of different gums. Synthesized nanoparticles were characterized with different spectroscopic and microscopic techniques such as UV–Vis spectroscopy, Fourier Transform Infrared (FTIR), Scanning electron microscopy (SEM), energy dispersive X-ray (EDX), X-ray diffraction (XRD) and atomic absorption (AA). The effects of gum, metal ions concentration, reaction temperature and time on the synthetic stability of nanoparticles was studied along with their post synthetic stability against varying pH and salt concentrations, long term storage and extreme of temperature. Formation of Au- and Ag-NPs was confirmed from the surface Plasmon resonance in the range of 500-560 and 400-450 nm respectively and was further corroborated from the results of FTIR, EDX, XRD and Atomic absorption. SEM analysis showed that Au- and Ag-NPs stabilized with different gums were mostly spherical and in the size range of 5–50 nm. FTIR analysis revealed that the abundance of hydroxyl, carboxylate, amide and acetyl groups of the gum biopolymers are involved in the reduction and stabilization of metal cations. The EDX analysis confirmed the presence of Au and Ag in the respective gum stabilized nanoparticles. No peaks were observed in the XRD spectrum of pure gums, thus indicating their non crystalline nature; while the XRD pattern of gum stabilized Au- and Ag-NPs have characteristic peaks, and so are crystalline in nature. Atomic absorption analysis showed almost complete reduction of the metal ions into the respective nanoparticles. The TGA spectrum further confirmed their thermal stability, where three successive weight losses were observed in the temperature range of 50–800 oC. It was observed that reaction temperature of 80oC and reaction time of 4 h was most suitable for the efficient synthesis of Au- and Ag-NPs. The results of post-synthetic stability revealed that Au- and Ag-NPs were stable in different concentrations of NaCl (1–3 M), neutral to acidic pH (7–2) and without any long term storage (eight months) or thermal (100 oC) induced degradable changes. Gold and silver nanoparticles stabilized with different gums were evaluated for their prospective anticancer, antibacterial, urease inhibition, anti-inflammatory and analgesic properties. Gold and silver nanoparticles possessed mild to moderate antibacterial activity and can observed from their zone of inhibition against Staphylococcus aureus, Escherichia coli and Pseudomonas aeruginosa. Gold nanoparticles significantly alleviated the acetic acid induced writhes at much lower doses of 40 mg/kg (P <0.01) and 80 mg/kg (P < 0.001) compared to that of gum extract at 200 and 400 mg/kg (P <0.001). At similar doses, Au-NPs also significantly inhibited the carrageenan induced paw edema during the 1st h (P < 0.05) and 2–5 h (P <0.001) of the study duration. Different gums stabilized Au- and Ag-NPs were tested to evaluate their cytotoxic potential against human cervical cancer cells (HeLa). Potential anticancer effect was showed by the respective gold and silver nanoparticles. Biosynthesized Au- and Ag-NPs were also able to inhibit urease with better affinity.