57. Al-Hadeed/The Iron
I/We begin by the Blessed Name of Allah
The Immensely Merciful to all, The Infinitely Compassionate to everyone.
57:01
a. Whatever is within the celestial realm and the terrestrial world is Glorifying Allah – The One and Only God,
b. as HE is The Almighty, The All-Wise.
57:02
a. For HIM is the Sovereignty of the celestial realm and the terrestrial world.
b. HE gives life as well as death.
c. And HE Manifests Sovereignty over all existence.
57:03
a. HE is The First and HE is The Last, i.e. without a beginning and without an end, and
b. HE is The Apparent and HE is The Hidden, and
c. HE is The Knowledgeable about everything.
57:04
a. HE created the celestial realm and the terrestrial world in six days/ time span,
b. then established HIMSELF on the Throne of Almightiness.
c. HE Knows whatever enters the earth, and whatever comes out of it,
d. and whatever descends from the celestial realm, and whatever ascends to it.
e. And HE is with you wherever you may be.
f. Allah Watches whatever you do.
57:05
a. To HIM belongs the Sovereignty of the celestial realm and the terrestrial world,
b. and to Allah will return all matters for judgment and award.
57:06
a. HE makes the nighttime to pass into the daytime,
b. and HE makes the daytime to pass into the nighttime.
c. And HE knows whatever is within the hearts of people.
57:07
a. Believe in Allah and HIS Messenger, and
b. spend out in the Cause of Allah of that wealth, possessions, and knowledge which...
Spiritual/physical cleanliness/purification has always been man's concern and it is more so with religions. Concepts and procedures of the same have always been derived and framed according to the basic teachings of these religions. Islam and Hinduism, basically two different religions, have different theoretical assumptions about this issue. The study below critically examines these concepts as put forward by the two religions.
Inverse Problems for Some Fractional Differential Equations Fractional Calculus(FC) is about the investigation of arbitrary order derivatives, integrals, special functions and equations involving these operators. This subject has its roots back to late seventeenth century. In recent years scientists and engineers are using it rigorously as it provides an efficient method to model many well known physical phenomenon when compared with their counterpart (integer order calculus). For example, fractional order diffusion/transport equation has been used to explain anomalies in diffusion/transport process which occurs in many physical situations such as transport in heterogenous or porous media. For a physical process scientists are interested in the investigations of causes and effectsofthatphysicalprocess. Theeffectsofanyphysicalprocess(usuallyknown as direct problems) are easier to study then the causes that forces the system to behave in a particular manner. The mathematical models in which we study the causes are termed as inverse problems(IPs). The field of IPs investigates how to convert measurements into information about a physical process. The field of IPs is of great interest as it has many applications just to mention a few are in medical imaging, acoustic, heat conduction, source identification in a stream, shape optimization etc. In this thesis, we have studied time, space as well as space-time fractional differential equations. Through out our research investigation we have used fractional derivatives defined in the sense of Hilfer and Caputo. It is to be noted that Hilfer fractional derivative (HFD) interpolates both Riemann-Liouville(RLFD) and Caputo fractional derivatives(CFD) for particular choices of parameters. For a fourth order time fractional differential equation(TFDE) with nonlocal boundary conditions(knownasSmaraskii-Ionkinboundaryconditions)involvingHilferfractionalderivative(HFD),twoinversesourceproblems(ISPs)areconsidered. ISPof determining a space dependent source term for a TFDE in two space dimensions is also considered. Existence, uniqueness and stability results for the ISPs are proved under certain regularity conditions on the given data. For a multi-term TFDE involving HFDs ISP of recovering a time dependent source term is studied by using Heaviside-Mikusinski’s operational calculus approach. The spectral problem is non-self-adjoint and a bi-orthogonal system of functions(BSFs) is used toconstructtheseriessolutionoftheISPs. Foraspace-timefractionaldifferential equation(STFDE)withDirichletzeroboundaryconditionsalongwithappropriate over-specified conditions two ISPs of recovering space and time dependent sources are considered. In the last research problem of this thesis inverse coefficient problem(ICP)foraspacefractionaldifferentialequation(SFDE)isstudied. Weproved existence, uniqueness and stability results for the solution of the considered IPs by imposing certain regularity conditions on the given datum.