Fractal Dimensions and Approximations of α-Fractal Interpolation Functions Md Nasim Akhtar ...

189 Pages·2017·1.15 MB·English

Fractal Dimensions and Approximations of α-Fractal
Interpolation Functions
by
Md Nasim Akhtar
DEPARTMENT OF MATHEMATICS
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI
GUWAHATI-781039, INDIA
November, 2016
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Fractal Dimensions and Approximations of α-Fractal
Interpolation Functions
A thesis submitted
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
by
Md Nasim Akhtar
(Roll No.: 11612310)
DEPARTMENT OF MATHEMATICS
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI
GUWAHATI-781039, INDIA
November, 2016
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CERTIFICATE
It is certified that the work contained in the thesis titled “Fractal Dimensions
and Approximations of α-Fractal Interpolation Functions” by Md Nasim
Akhtar (11612310), a student in the Department of Mathematics, Indian Institute
of Technology Guwahati for the award of the degree of Doctor of Philosophy has been
carried out under my supervision and this work has not been submitted elsewhere for a
degree.
November, 2016
Dr. M. Guru Prem Prasad
Professor
Department of Mathematics
Indian Institute of Technology Guwahati
Assam, India
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Dedicated
To
My Parents
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Acknowledgements
Thanks and Praise to Allah.
There is no way I can thank everybody who helped me in this doctoral work, but this
thesis would not have been possible with out the help and support of the kind peoples
around me.
In the first place, I want to express my deepest gratitude towards my thesis super-
visor Prof. M. Guru Prem Prasad. He carefully guided me through this research period
in IIT Guwahati, discussed all ideas in this thesis, read all the words and provided the
foundation to explore the subject with great freedom. He is such a generous and well
behaved person. I wish, if I could learn his manners a little. He has put a tremendous
amount of effort in this thesis from his busy schedule of different administrative respon-
sibilities. One could not wish for a better and friendlier supervisor. Thank You very
much, Sir!
My gratitude towards Prof. M. A. Navascu´es, University of Zaragoza, Spain is
beyond words. I can not sum up it in words. My research work is mostly influenced by
her research and valuable suggestions. Whenever I contacted her through e-mail, she
was always readily available although she was going through her tough times in personal
life as well as her busy schedule. Thank You Madam, is a small word in front of your
immense support in my research.
I am very much thankful to Prof. G. P. Kapoor, IIT Kanpur, India. In my re-
search period, I got a chance to spend few months with him in IIT Kanpur. In this
period, he taught me the possible area to explore the subject Fractals. His continuous
encouragement and generosity during discussions is always influential.
I would like to acknowledge the financial, academic and technical support given by
the Department of Mathematics, Indian Institute of Technology Guwahati. My sincere
thanks to Prof. R. K. Sinha and Prof. B. K. Sarma (former Head of the Departments)
and Prof. S. N. Bora, the Head of the Department, for providing with all necessary
facilities and support.
I appreciate the feedback or advise given by my doctoral committee members Dr. A.
K. Chakrabarty, Dr. K. V. Srikanth and Dr. J. Swain. I admire Dr. A. K Chakrabarty,
the way he introduces a subject to students. I am lucky to have some courses taught by
Dr. A. K. Chakrabarty, Dr. K. V. Srikanth, Dr. S. Bora, Prof. M. G. P. Prasad, Prof.
D. C. Dalal, Dr. P. A. S. S. Krishna during my course work in PhD program.
I owe many thanks to Dr. S. R. Ahamed, Department of Electronics and Electrical
Engineering, IIT Guwahati and his family for their hospitality and support. By observ-
ing him, one can learn many good lessons in life. I had many healthy discussions with
him on academics as well as on personal aspects. I also thanks to Prof. R. Radha, IIT
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Madras, Prof. A. T. Khan, Prof. Natesan S and Prof. R. Alam, IIT Guwahati for their
love and care in different tenure.
I was supported financially and also motivated by few excellent teachers throughout
my academic life: Dr. A. Malek (School chemistry teacher), J. Ali (School mathematics
teacher), Dr. A. Mokid and Dr. A. Pal (College mathematics teachers). It was their
kindness towards me though I was not eligible for it. My humble gratitude towards
them and prayer for their good health.
I am lucky to have few tremendous friends in my life. Some individuals are worth
mentioning. From my school days, Cezar, Suman, Pasha, Apple, Kuntal da, Alauddin
da, Rajesh da. From my college days, Rahul and Hafiz who always encouraged me and
we had many discussion which helped me to learn the subject Mathematics. Special
thanks to Kamal Hassan who supported me by providing study materials during my
Bachelor degree. My stay in IIT Madras during my Masters degree, I was very much
cared by Faizan vai which goes on till date. It will not be easy staying in the beautiful
campus of IIT Guwahati with out these companion namely Debopam, Swarup, Gayatri,
Neelam di, Punit da, Saloni, Arnab da, Kalyan da, Chitralekha di, Niyas vai, Zayud vai,
A. Gaffar vai, Zaheer vai, Sameer vai, Palash vai, Kobirul, Madhu, Anirban, Debasish,
Ranjan, Zibrail vai, few to name. My special thanks to Bala S and Abhisek D for their
help in Matlab programming.
I am deeply grateful to my relatives. My three uncle and their family for their belief
in me. My three aunts who always treat me as their son. My friend cum cousin Kabir
da and his spouse my sister-in-law for their love. Special thanks to Habibor R Khan and
his family for their love and hospitality during my stay at Rampurhat and Guwahati. I
wish I could show Jishan and Juin how much I love and care them. With them I have so
many memories during my stay at Rampurhat. My younger brother Ashif, who always
make my stay comfortable wherever I am with him. My sister who is one of my best
friend and criticizer whenever I am wrong. Her advice has always a great impact in
my decision making. My brother-in-law Imam with whom I can share everything with
belief. Thanks to all of them from my deep inside. My lil one niece Inaya, whose one
single word ‘mama’ takes away all the pain, frustration and tiredness. My blessings for
her.
No one can ever repay the sacrifices done by parents for their children. I am blessed
to have those two people in my life as my Ammi and Abbu. They are like light in my
eyes. Till date, I am dependent on them like a small child. This Thesis is dedicated to
them. I only can pray to Allah for their peaceful and healthy life. “Oh Allah! Bestow
on them Your Mercy as they did bring me up when I was small.”
November, 2016 Md Nasim Akhtar
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