Scientific Study of Balance (Al-Mīzān) in the Light of Sūrah Al-Rahmān

This article is an attempt to elaborate the phenomenon of equilibrium prevailing everywhere in the microcosmic and macrocosmic systems associated with universal stability and rhythm. It has been accentuated what is beautifully illuminated in the beginning verses of Sūrah Al-Rahmān successively regarding balance that invites one’s thought towards the well controlled cosmic system as well as with the concept of balance by means of different manners as identified by exegetes and scholars concerning daily life i.e. From simple and common to that of complex issues and intricacies. Qualitative method has been employed for this research while some aspects have also been handled in accordance with quantitative approach. Article also emphasized the series of cause and effect nexus may be declared as self explanatory episode an indication towards a Supreme Force whose widespread control and interference can’t be denied rationally. Sūrah beautifully pictured the physical and metaphysical sketch of balance having many dimensions from man to universe and vice versa as well as from both towards Ultimate Reality whose dominion is limitless where one has to follow peaceful living and to put oneself aside indulging any disruption from ordinary clash to that of nuclear war to be waged.

On the Fractional Initial-Boundary Value Problems

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.