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Generalization of Fractional Calculus Operators With Applications

Thesis Info

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Author

Muhammad Khurshid Azam

Program

PhD

Institute

University of Management and Technology

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/123456789/11039/1/Thesis%20Final.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726257327

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Generalized forms of fractional calculus operators (integrals and derivatives) are introduced. Caputo -fractional derivative and Hadamard Caputo type -fractional derivative are discussed and their results with some applications are presented. Extensions of Weyl -fractional integral and Hadamard -fractional integral are also introduced. Boundedness of the extended Hadamard -fractional integral inspaces is determined. The generalized -fractional derivative and generalized Caputo type -fractional derivative are introduced and their properties and results are found. Finaly, the generalized type -fractional integral (unifying eleven existing fractional integrals) is introduced and its boundedness inspaces is also determined.Further, integral transforms of - fractional and extended -fractional operators are found. Proofs of properties including semigroup, commutative and some other results for -Weyl fractional integral are given. Moreover, some inequalities for -Weyl fractional integral are discussed and examples are also given to illustrate the results. Relationship between these new generalized forms of fractional calculus operators with the existing fractional operators are discussed by substituting the different values of involved parameters. Integral transforms of new fractional operators and their applications are also given.
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اردو کے اہم مدونین (رشید حسن خان)

اردو کے اہم مدونین (رشید حسن خان)
پیدائش و ابتدائی تعلیم:
رشید حسن خان اتر پردیش کے ضلع شاہ جہاں میں 1925ء میں پیدا ہوئیا و ر وہاں ابتدائی تعلیم مدرسہ بحر العلوم سے حاصل کی۔آپ کے والد کا نام امیر حسن اور دادا کا نام علی حسن تھا۔ اعلی تعلیم سے فارغ ہونے کے بعد آپ نے اپنی تمام زندگی تحقیق و تدوین میں صرف کر دی۔ انہوں نے اپنی تمام ترخواہشوں اور خاندانی رشتوں کو بھی تدوین پر قربان کر دیا۔بہت سے امراض لاحق ہوتے ہوئے بھی تحقیق و تدوین کے کام کو جاری رکھا۔
ملازمت:
16 اگست 1959ء دہلی یونیورسٹی کے شعبہ اردو سے وابستہ ہوئے اور تادم مرگ تحقیق و تالیف کے سلسلے کو جاری رکھا۔ان کی خواہش تھی کہ آنے والی نسلیں ان کی تدوین کردہ کتابوں سے مستفید ہوں اور وہ ان سے متاثر ہو کر تحقیق و تدوین کی روایت کو آگے بڑھائیں۔
تدوینی معیار:
رشید حسن خان کی تحقیقی تداوین جدید تدوینی اصولوں پر مبنی ہوتی ہیں۔یہی وجہ ہے کہ اردو ادب کے کلاسیکی مدون رشید حسن خان کو گیان چند نے خدائے تدوین کہا ہے۔
تحقیق و تدوین کا ذوق و شوق:
رشید حسن خاں کو تحقیق و تدوین کا اس درجہ شوق تھا،کہ وہ جس شہر میں جاتے تھے سب سے پہلے وہ اس شہر کے کتب خانوں کا دورہ کرتے تھے۔وہ ان کے کیٹ لاگ دیکھتے ،قدیم نسخوں کے اندراج نوٹ کرتے، اور وقت ضرورت مختلف کتابوں سے عبارتیں نقل کرتے تھے۔وہ جب بھی کسی دوست کو خط لکھتے تھے تو اس میں بھی کتابوں کے تعلق سے کچھ نا کچھ ضرور رقم کرتے تھے۔اس طرح سے اپنی تلاش و جستجو جار ی رکھتے تھے۔
تدوینی کتب:
رشید حسن خان کی تحقیق و تدوین کے معیار کا اندازہ ان کی مدون شدہ کتب سے لگایا جا...

تأثر الأدب العربي من تعليمات النبي ﷺ دراسة و تحقيقا

It is estimated by studying the history that the imagination of life was limited before the appearance of Islam. A new era started after the arrival of Islam. Revolution came in thoughts and ideas. Every department was effected even poetry, literature and language pleasantly effected. A revolution created in the Arabic literature after the revelation of the Holy Quran even it taught the rituals of representation of emotions along with facial and spiritual beauty to the Arabic literature. Arabic language is full of knowledge and thoughts of whole world today and the axis of Arabic language and literature is the Holy Quran. The resources of ignorant literature which we get today was collected to save and understand the language of the Holy Quran. For example to eliminate the linguistic flaws, grammar science came into being and rhetoric science came into being to prove Quranic miracle and language and literature came into being to explain the poor words, and Hadith, tafseer, fiqah and other sciences came into being for religious laws. The Holy Quran changed the direction of literature towards justice, service to humanity and support of right and truth and chastity and modesty and God-worship. It gave appropriate dignified styles to explain every topic and invited to work by using reasons and thoughts. Arabic language is effected by the Holy Quran in such a way that it softened the hard and ruthless hearts of Arabs and made the surface wisdom heavy and solid by entering in it.  Could not get effected by Holy Quran as the level which prose got benefit. The prose got more shine in the time of Khulafa-e-rashidin when victories increased, boundaries of Islamic state expanded and political and developmental issues increased. It is a fact that Arabic prose got too high as compared to the Arabic poetry due to the Holy Quran.  

Plasma Processing of Materials Using Plasma Focus

The work presented in this thesis addresses the parametric study of ion beams emitted from Mather type plasma focus devices and their flourishing utilization in materials processing. Experiments have been performed by using two different plasma focus devices; a conventional 2.3 kJ plasma focus device developed under the joint venture of the United Nations University (UNU) and the Abdus Salam International Centre for Theoretical Physics (ICTP) designated as the UNU/ICTP device operational at the GC University Lahore and a modified version called the Nanyang X-ray source-2 designated as the NX2 device (a repetitive plasma focus) operational at the National Institute of Education (NIE), Nanyang Technological University (NTU), Singapore. The measurements of ion parameters such as energy, energy distribution, number density and current density are carried out in the ambient gas pressure by employing a BPX65 photodiode and a Faraday cup (FC) using time of flight technique. A major motivation is to establish the optimum processing conditions for ion nitriding, surface modification, phase changes and carburizing of materials of industrial interest like Ti, AlFe 1.8 Zn 0.8 alloy and SS-321 in plasma environment. The processed samples are characterized for structural and morphological changes, compositional profile and surface hardness by employing X-ray diffraction (XRD) at GC University Lahore, scanning electron microscopy (SEM) at University of Peshawar, field emission SEM (FESEM) and energy dispersive X-ray spectroscopy (EDX) at the NIE NTU Singapore, X-ray photoelectron spectroscopy (XPS) at the National University of Singapore (NUS) Singapore, Raman spectroscopy and Vickers microhardness test at Quaid-i-Azam University Islamabad, Pakistan. The SRIM code and microindentation measurements are used to estimate the depth profile of the modified layers. Nanocrystalline spatially uniform TiN thin films with petal like features are developed on Ti substrates exposed to 30 focus shots at various axial positions. The surface roughness and the relative proportion of the TiN films are strongly influenced by the ion beam energy flux. The film acquires eminent appearance with maximum relative proportion of nitrogen at 7 cm axial position. The probable energy of the ions reaching this position is 64 keV with the maximum ion number density of 5.9 ́10 13 cm -3 . The corresponding energy flux and current density are 2.69 ́10 13 keV cm -3 nsec -1 and 1142 A cm -2 viiirespectively. The grain size of the film is estimated to be about 90 nm while the compound layer thickness is about 0.66 μm. The surface microhardness is also maximum at this axial position with typical value of 7650±10 MPa. The SEM images of a typical microcracked TiN thin film and the SRIM code estimations of ion penetration help in understanding the growth mechanism of the film in terms of ion dose. The granular nanostructures appearing on the substrate surface are grown from nucleates of a few nm size developed by the energetic ions induced collision cascades. The predeposited nitride layer or nitrogen ions interstitially implanted into the substrate surface are also redistributed by the successive pulses of the ion beams leading to layer densification along with possible resputtering. Moreover, the temperature evolution during the DPF ions irradiation also enhances the reactivity of the nitrogen already introduced during the preceding pulses. The residual tensile stresses on the sample surface are transformed to the compressive stresses after DPF ion irradiation. Nitrogen ions induced surface changes in AlFe 1.8 Zn 0.8 alloy are investigated as functions of axial and angular positions for 30 shots. The expanded fcc phase of Al is evolved owing to the incorporation of nitrogen along with Fe and Zn into the Al lattice. A comparatively smooth and crack free nitride layer is formed on the sample treated at 7 cm axial and 10 0 angular position with 4- to 5-fold increase in Vickers hardness. TiN 0.9 and (Fe,Cr) 2 N are deposited on SS-321 along with formation of non-stoichiometric (Fe,Cr) x N phase by exposing the samples to multiple focus shots in nitrogen plasma at different axial and radial positions. The transformation from (Fe,Cr) x N to (Fe,Cr) 2 N is attributed to an increased nitrogen ion dose. The point-like structures of flakes reveal the nucleation of crystal growth with the increased ion doses. The nitride layer is golden in colour and is spatially uniform with improved surface hardness. Multiphase nanocrystalline titanium oxycarbide TiC x O y thin films composed of TiC 2 , TiO 0.325 , Ti 2 O 3 and carbon phases are deposited on titanium substrate in CH 4 discharges by the UNU/ICTP and the NX2 devices. The nanocomposite films are non-porous and microcrack-free with grain-like surface morphology having spatially uniform carbon distribution. XRD, Raman and XPS results reveal the favorable evolution of multiphase coatings having a stoichiometric TiC 2 phase and graphitic carbon adsorbates along with ixthe residual oxide (TiO 0.325 , Ti 2 O 3 ) phases with the lower energy flux and lower repetition rate in the UNU/ICTP treatment. Whereas, the deposition of carbon and a non- stoichiometric TiO 0.325 phase is favored due to the improved oxide removal and enhanced disorder in the substrate surface during the NX2 treatment. In addition, TiC 2 phase is also suppressed, possibly due to the enhanced substrate temperature caused by the higher energy flux of the ion beams and the higher repetition rate. The granular profile of the films attains a definite coagulation pattern. The energy flux of the ion beam and the repetition rate are found to be critical parameters which influence the preferred evolution of a particular phase during the restructuring of various phases.