مُلاّ جان محمد
افسوس ہے گذشتہ مہینہ ہمارے دومخلص قومی کارکن ملاجان محمدکلکتہ اورمولانا محمد عاقل الٰہ آباد انتقال کرگئے۔ملا صاحب کااصل وطن پشاور تھالیکن عرصہ دراز سے کلکتہ میں آبسے تھے اوراب سچ مچ وہی ان کاوطن تھا۔نہایت پُرجوش،جری اوربیباک انسان تھے۔گذشتہ نصف صدی میں کوئی قومی اور ملی تحریک ایسی نہیں ہے جس میں انہوں نے بڑھ چڑھ کراورولولہ و عزم کے ساتھ حصہ نہ لیا ہو۔ ان کی علمی زندگی کاآغاز تحریک خلافت سے ہوا اوراختتام مجلس مشاورت پر۔کلکتہ میں شاید ہی کوئی مسلم ادارہ(یہاں تک کہ محمڈن اسپورٹنگ کلب بھی) ایسا ہوجس میں ملاصاحب نے نمایاں حصہ نہ لیاہو۔اسی وجہ سے وہ کلکتہ کے لوگوں میں بے حد مقبول تھے۔ بڑے بے غرض،بے لوث اورنہایت سادہ اورمخلص مسلمان تھے۔۱۹۵۰ء میں کلکتہ کے فساد میں لوگوں نے ان کو بچوں کی طرح چیختے اور روتے دیکھا ہے۔ عمر۸۵ سال کے قریب تھی۔ [نومبر ۱۹۷۲]
Ibn e Khaldun (1406 C.E.) has been an imminent scholar and well known for his work in the study of civilization. His vision regarding Civilization holds the significant place according to the philosophers of history. Arnold J. Toynbee (1975 C.E.) is one such prominent thinker who not only applauded the thoughts of Ibn e Khaldun but was influenced by Ibn e Khaldun’s views as it can be seen in Toynbee’s book: “A Study of History”. As a philosopher of history, he has much contribution in the field; He interlinks History with civilization. He presented a thoughtful book surrounding his civilizational vision; which explains the causes of world’s ups and downs. Although he presented a quality research about the division of the civilizations in the light of religion, many aspects of his work need to be reassessed. As per his understanding of world civilizations, he represents twenty-one civilizations, but with the passage of time, the number reduces and now only five are left in the contemporary epoch. According to him, religion has played significant role in the rise and fall of civilizations through their various stages. This research will highlight his thought about Islam through a comparison between Toynbee and Ibn e Khaldun’s Islamic civilizational thoughts. The study will also mention several problems in his approach to the Islamic Civilization. Furthermore, along with due importance of both scholars in the subject of history, their authoritative status will be stated. This research aims to discuss some misconceptions of the West that are based on Toynbee's understating of Islamic civilization and history. And, consequently, it intends to improve relations between people of the west and east.
FractionalCalculus(FC)isthestudyofintegralsandderivativesofarbitraryorder, this subject is as old as integer order calculus and is supposed to be initiated from the question of L’Hôpital to Leibniz when the notion of nth order derivative was coinedfortwo n timesdifferentiablefunctions. FromlastfewdecadesFChasbeen consideredbymanyresearchersduetoitsapplicationsindiversefieldsofsciences, not to mention all some are in Physics, Chemistry, Viscoelasticity, Biology etc. Due to these applications the integral or differential operators of arbitrary order and equations involving these operators are considered by many researchers for mathematical investigations. We intend to consider some Fractional Differential Equations (FDEs) in this dissertation. Indeed, in one part of this dissertation we have considered diffusion equations with fractional derivative in time only. Let us mention that in many physical phenomena, the data obtained from field as well as lab experiments is not in agreement with the integer order Partial Differential Equations (PDEs). The phenomena is usually known as anomalous diffusion/transport. Among several techniques to explain these anomalies one is by considering fractional order operators instead of integer order operators in PDEs. It is important to mention that throughout this dissertation, we have considered the fractional derivatives defined in the sense of Riemann-Liouville, Caputo or Hilfer. The Hilfer fractional derivative is a generalization of the Riemann-Liouville and the Caputo fractional derivatives. The particular choices of the parameters involved in Hilfer fractional derivative give us Riemann-Liouville and Caputo fractional derivatives. We considered direct as well as inverse source problems for FDEs involving time fractionalderivativewithnonlocalboundaryconditions. Theeigenfunctionexpansion method has been used and the spectral problem obtained is non-self-adjoint. The problems considered have initial conditions as in case of integer order deriva x tivesasweconsideredfractionalderivativedefinedinthesenseofCaputo. Forthe case of Hilfer fractional derivative rather than taking a nonlocal initial condition in terms of fractional integral two local conditions are considered. Under certain regularity conditions on the given data, we obtained existence, uniqueness and stability results for the problems. For a space-time fractional diffusion equation with Dirichlet boundary conditions, some inverse problems are also discussed. The spectral problem is generalization of the regular Sturm-Liouville operator. Several properties of the eigenvalues and eigenfunctions of the fractional order Sturm-Liouville operator are used to prove the existence results for the solution of the inverse problems. Some special cases of the inverse problems in the case of space-time differential equations are discussed and results are deduced from the generalized results. In the last part of the dissertation a nonlinear system of fractional differential equations are considered. The results about existence of finite time blowing-up solutions is proved.