Research Statement and Interests
One of my interests is research in the field of physical phenomena and obtaining an
analytical and exact answer to these phenomena.
Most physical and engineering events are expressed in terms of differential equations of
fractional type. The subject of fractional calculus and fractional derivatives deal with
investigation of integrals and derivatives of arbitrary real or complex order. Fractional
calculus is thus an extension of integer order calculus, which is applicable in many areas
of science and engineering, such as fluid flow, rheology, electric network, viscoelasticity,
electrochemistry of corrosion and etc.
In all my works, I use fractional derivatives since it has physical and engineering
interpretation.
My master’s thesis, under supervision Dr. Arman Aghili, was about “Fractional Calculus
of Variations”. Fractional calculus of variations, that has been developed in the last two
decades, is a combination of the calculus of variations and fractional calculus. This
combination is not artificial, it comes naturally from more accurate descriptions of
physical phenomena. It is a field of mathematical analysis that uses variations, which are
small changes in functions and functionals, to find maxima and minima of functional:
mappings from a set of functions to the real numbers. Functionals are often expressed as
definite integrals involving functions and their derivatives of fractional order. Functions
that maximize or minimize functionals may be found using the Euler–Lagrange equation
of the calculus of variations. Its development is also in direct connection with various
problems occurring in biology, chemistry, control theory, dynamics, economics,
engineering, physics, etc.
My PhD thesis, under supervision Dr. Arman Aghili, was around “methods for solving
fractional partial differential equations and fractional dynamical systems and
computing some difficult integrals” that couldn’t be solved by ordinary ways. Tools
such as Laplace transform and Fourier transform were very strong for finding solution of
fractional PDEs and solving such integrals.
After I received my PhD, I continued my researches on studying some equations and
their solving method that they aren’t easy to utilize.
My current research is around fractional PDEs and analytical methods for their solutions.
These fractional PDEs happen in engineering, mathematical physics, chemistry and
natural events. In most cases, it is very difficult to obtain the exact solution to these types
of equations. I use integral transform techniques such as the Laplace transform, Fourier
transform, Hankel transform, Mellin transforms and other transforms for obtaining an
exact solution of these types of equations. Furthermore, in most cases, I use special
functions such as Bessel, Hankel, Mittag-Leffler, H-functions and Airy functions and
more. Working with these functions and how they are used is one of my skills.
In my recent research, first paper in my CV, under supervision Dr. Alireza Ansari, a
fractional mixed boundary value problem is solved by combination of Cagniard-Hoop
method and Wiener-Hopf method. Using of these methods is so much hard, since in
Wiener-Hopf method, one should factorize a function as two parts that they be analytical
on semi-regions. It needs a strong knowledge of complex analysis and Wiener-Hopf
method. The complexity of the topic occurs when we use this method along with
Cagniard-Hoop method. The Cagniard–De Hoop method is a sophisticated mathematical
tool for solving a large class of wave and diffusive problems in horizontally layered
media. The method is based on the combination of a unilateral Laplace transformation
with real and positive transform parameter and the slowness field representation.
This project was conducted by the Shahrekord university and carried out in year 2018.
The title of this project is as
“Transient Thermal Mixed Boundary Value Problem of Distributed Order”
Furthermore, in year 2019, I have commenced another project around medical science. It
is a special PDE of fractional order that occurs in medical science.
The result of this project have published in an ISI journal (number 1 of the list of
publications). Furthermore, I am working another project with this university about
solving a special PDE of fractional order in medical science.
I am especially intrigued by the connections between the physical systems and neural
phenomena and methods for their solution.