Physical and Numerical Applications of Fractional Calculus Ahmed Soliman Omer Bakheet A ...

55 Pages·2017·1.57 MB·English

Physical and Numerical Applications of Fractional Calculus
Ahmed Soliman Omer Bakheet
B.Sc.(Honour) in Faculty of Education , University of Gezira (2017)
A Dissertation
Submitted to the University of Gezira in Partial
Fulfilment of the Requirements for the Award of the
Degree of Master of Science in Mmathematics
Department of Mathematics and Physics
Faculty of Education

February, 2017

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Physical and Numerical Applications of Fractional Calculus
Ahmed Suleiman Omer Bakheet
Master of Science in Mathematics – February 2017
Department of Mathematics and Physics
Faculty of Education – Hantoub
University of Gazira
Abstract
Using ideas of ordinary calculus, we can differentiate any function, say 𝑓(𝑥) = 𝑥,
to the 1st or 2nd order. We can also establish a meaning or some potential applications
of the results. But, can we differentiate the same function to the ½th order or 0.3 , 0.7?
Better still, can we establish a meaning or some potential applications of the results? We
may not achieve that through ordinary calculus, but we may through fractional
calculus more generalized form of calculus. Fractional calculus is a branch
of mathematical analysis that studies the possibility of taking real number powers
or complex number powers of the differentiation operator and the integration operator.
This thesis, aims to defined Fractional calculus from beginning and how it developed,
explaining the concept of Fractional Differential Equations (FDE), using Laplace
transform to solve the (FDE), and how to use the Matrix Exponential Function to solve
the Decay Growth Problem. This study follows the extrapolative procedure. Some results
are reached: some natural phenomena's it is not any mathematical model to describe it,
the focus to study the Fractional calculus can be extend the domain of the mathematical
models, also we cannot find until now the geometrical interpretation for some fractional
integrals and derivatives. Lastly I think it must be to attention for the Fractional
calculus because it will be the future of the mathematic and it can be describe the
smallest and tenuous changes that can be effective in many applications، and I entrust to
teaching the Fractional calculus on graduate studies.
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يرسكلا نابسحلا يف ةيددعلاو ةيئايزيفلا تاقيبطتلا
تيخب رمع ناميلس دمحا
2017 رياربف – تايضايرلا يف مولعلا ريتسجام
ءايزيفلاو تايضايرلا مسق
بوتنح – ةيبرتلا ةيلك
ةريزجلا ةعماج
ةساردلا صخلم
𝑓(𝑥) = 𝑥 ةلادلا لاثم ،لاودلا ةلماكمو قاقتشا نم اننكمت يديلقتلا نابسحلا يف ةمدختسملا راكفلاا نإ
يتلا ةنكمملا تاقيبطتلا حيضوت عم ةلوهسب اهل لماكتلا داجيا اضياو اذكهو ةيناثلا و ىلولاا ةبترلا ىلا اهقاقتشا عيطتسن
1
اننكمي لهو ؟ 0.7 وا 0.3 وا ةبترلا ىلا 𝑓(𝑥) = 𝑥 ةلادلا قتشن نا اندرا ول اذام نكلو .جئاتنلا هذه اهنع ربعت
2
لضافتلا موهفم للاخ نم كلذ عيطتسن نل اننا حضاولا نم ؟ تاقتشملا هذهل يسدنهلا وا يئايزيفلا ينعملا حيضوت
ةساردلا تفده .نابسحلا موهفمل ميمعت وه يذلاو يرسكلا لماكتلاو لضافتلا ةركف تأدب انه نمو .يديلقتلا لماكتلاو
عم لماعتي وه و يضايرلا ليلحتلا نم اءزج ضعبلا هدعي يذلاو يرسكلا لماكتلا و لضافتلا ملعب فيرعتلا ىلإ
يلأ ام )ةلاد( نارتقلا ةقتشملا ميمعتب متهي لاجملا اذه و ،ةيرايتخلاا بترلا ةلاح يف قاقتشلاا و لماكتلا تاقيبطت
اهلح ةيفيكو ةيرسكلا ةيلضافتلا تلاداعملا موهفم حيضوتل اضيا ةساردلا فدهتو .ةحيحص ريغ ةبتر تاذ ةقتشم
جهنملا ةساردلا تعبتا. Decay growth problem ةلأسم لحل ىوقلا ةفوفصم مادختسا مث ،سلابلا رثؤم مادختساب
اهل دجوي لا ةيعيبطلا رهاوظلا ضعب كلانه :اهمهأ جئاتن ةدع ىلا لصوتلا مت ثحبلاو ةساردلا للاخ نمو .يئارقتسلاا
لا هنأ امك ،كلذ لحل ثحبلا لاجم عسوي يرسكلا لماكتلاو لضافتلا ىلع زيكرتلا كلذل اهنع ربعي يضاير جذومنأ
يرسكلا نابسحلا نيب قيثولا طابترلاا ،ةيرسكلا تلاماكتلاو تاقتشملا نم ريثكل يسدنه ريسفت نلاا ىتح دجوي
ةساردلا يصوت. ةيقيبطتلا لئاسملا ةجذمنل ةسيئرلا ةباوبلا يه يتلاو ةيئزجلاو ةيداعلا ةيرسكلا ةيلضافتلا تلاداعملاو
يف ةرثؤم يهو ادج ةقيقدو ةريغص تاريغتل هفصول لماكتلاو لضافتلا لبقتسم هنلأ ملعلا اذهب مامتهلاا بوجوب
.يلاعلا ميلعتلا ىوتسم ىلع هتاقيبطتبو هب فيرعتلاو يرسكلا نابسحلا سيردتب اضيا يصوتو ، ةديدع تاقيبطت

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Chapter One
Basic Concepts of the Fractional calculus
1.1 Introduction
The notion of integration and differentiation is habitual to all who have studied
primary calculus. We know, for example, that if 𝑓(𝑥) = 𝑥2 then integrating f(x) to the
1st order results in ∫𝑓(𝑥)𝑑𝑥 = 1 𝑥3 + 𝑐 and integrating the same function to the 2nd
1
3
order results in ∫[ ∫𝑓(𝑥)𝑑𝑥] = 1 𝑥4 +𝑐 𝑥+𝑐 . Similarly, 𝑑 𝑓(𝑥) = 2𝑥 and
1 2
12 𝑑𝑥
𝑑2 𝑓(𝑥) = 2 . However, what if we wanted to integrate our function f(x) to the ½th order,
𝑑𝑥2
or find its ½th order derivative? How could we define our operations? would our results
have a meaning or an application comparable to that of the familiar integer order
operations?
The traditional integral and derivative are to say the least a staple for the
technology professional, needful as a means of understanding and working with natural
and artificial systems. Fractional Calculus is a field of mathematic study that grows out of
the traditional definitions of the calculus integral and derivative operators in much the
same way fractional exponents is an outgrowth of exponents with integer value.
According to our primary school teachers that have explain a short notation for what is a
repeated multiplication of a numerical value. This concept in itself is easy to grasp.
However, this physical definition can clearly become mixed up when considering
exponents of non-integer value. While almost anyone can verify that x3 = x . x . x, how
might one describe the physical meaning of x3.4, or moreover the transcendental exponent
xπ. One cannot imagine what it might be like to multiply a number or quantity by itself
3.4 times, or π times, and yet these expressions have a definite value for any value x,
verifiable by infinite series expansion, or more practically, by calculator [9].
Now, in the same way consider the integral and derivative. Although they are
indeed concepts of a higher complexity by nature, it is still fairly easy to physically
represent their meaning. Fractional calculus flows quite naturally from our traditional
definitions. And just as fractional exponents such as the square root may find their way
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into innumerable equations and applications. It will become apparent that integrations of
order ½ and beyond can find practical use in many modern problems.
1.2 Brief History
Most authors on this topic will remember a specific date as the birthday of so
called 'Fractional Calculus'. In a letter dated September 30th, 1695 L'Hopital wrote to
Leibniz asking him about a particular notation he had used in his publications for the nth-
𝑫𝒏𝒙
derivative of the linear function f(x) = x, . L'Hopital's posed the question to Leibniz,
𝑫𝒙𝒏
what would the result be if n = ½. Leibniz's answer: "An apparent paradox, from which
one day useful consequences will be drawn." In these words fractional calculus was born.
Following L'Hopital's and Liebniz's first asking, fractional calculus was primarily a study
limited for the best minds in mathematics. Fourier, Euler, Laplace are among work with
fractional calculus and the mathematical consequences. Many found, using their own
notation and methodology, definitions that fit the concept of a non-integer order integral
or derivative. The most famous of these definitions that have been popularized in the
world of fractional calculus (not yet the world as a whole) are the Riemann-Liouville and
Grunwald-Letnikov definition.
In his 700-page long book on calculus be issued in 1819 Lacroix developed the formula
for the nth derivative of y = x m, m is a positive integer:

𝑚!
𝐷𝑛𝑦 = 𝑥𝑚−𝑛 , (1.1)
(𝑚−𝑛)!
where (n ≤ m) is an integer. By substitution the factorial symbol in the gamma function,
he also obtained the formula for the fractional derivative:

Г(𝛽+1)
𝐷𝛼𝑥𝛽 = 𝑥𝛽−𝛼 (1.2)
Г(𝛽−𝛼 +1)
where α and β are fractional numbers. In particular, he compute

Г(2)
𝐷½x = 𝑥½ = 2√𝑥 (1.3)
Г(3) 𝜋
2
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On the other hand, in 1832, Joseph Liouville (1809-1882) formally expands the formula
for the derivative of integral order n:

𝐷𝑛 𝑒𝑎𝑥 = 𝑎𝑛 𝑒𝑎𝑥 (1.4)
to the derivative of arbitrary order
𝐷𝛼 𝑒𝑎𝑥 = 𝑎𝛼 𝑒𝑎𝑥 (1.5)
Using the series expansion of a function f (x), Liouville derived the formula

∞
𝐷𝛼𝑓(𝑥) = ∑𝑐 𝑎𝛼𝑒𝑎𝑛𝛼 (1.6)
𝑛 𝑛
𝑛=0
Where
𝑛
𝑓(𝑥) = ∑𝑐 exp(𝑎 𝑥) , 𝑅𝑒 𝑎 > 0 (1.7)
𝑛 𝑛 𝑛
𝑘=0
Formula (1.6) is referred to as Liouville’s first formula for fractional derivative [5].
However, it can only be used for functions of the form (1.7). In order to extend his first
definition (1.6). Liouville formulated another definition of a fractional derivative based
on the gamma function:
∞
Г(𝛽)𝑥−𝛽 = ∫ 𝑡𝛽−1 𝑒−𝑥𝑡 𝑑𝑡, 𝛽 > 0 (1.8)
0
Г(𝛼 +𝛽)
𝐷𝛼 𝑥−𝛽 = (−1)𝛼 𝑥−𝛼−𝛽 , 𝛽 > 0 (1.9)
Г(𝛽)
This is called the Liouville’s second definition of fractional derivative. He successfully
applied both his definitions to problems in potential theory [5].
In 1822, Fourier obtained the following integral representations for f (x) and its
derivatives:
1 ∞ ∞
𝑓(𝑥) = ∫ 𝑓(𝜉)𝑑𝜉 ∫ cos 𝑡(𝑥− 𝜉)𝑑𝜉 (1.10)
2𝜋
−∞ −∞
and
1 ∞ ∞ 𝑛𝜋
𝐷𝑛𝑓(𝑥) = ∫ 𝑓(𝜉)𝑑𝜉 ∫ 𝑡𝑛cos(𝑡(𝑥− 𝜉)+ )𝑑𝑡, (1.11)
2𝜋 2
−∞ −∞

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Replacing integer n by arbitrary real α yields formally:
1 ∞ ∞ 𝛼𝜋
𝐷𝛼𝑓(𝑥) = ∫ 𝑓(𝜉)𝑑𝜉 ∫ 𝑡𝛼cos(𝑡(𝑥− 𝜉)+ )𝑑𝑡,(1.12)
2𝜋 2
−∞ −∞
Greer (1858-1859) derived formulas for the fractional derivatives of trigonometric
functions based on (1.5) in the form:
𝐷𝛼 𝑒𝑖𝑎𝑥 = 𝑖𝛼𝑎𝛼 𝑒𝑖𝑎𝑥 = 𝑖𝛼𝑎𝛼(cos𝑎𝑥 +𝑖sin𝑎𝑥 )
= 𝑎𝛼 (𝑐𝑜𝑠𝜋𝛼+𝑖 𝑠𝑖𝑛𝜋𝛼)(cos𝑎𝑥 +𝑖sin𝑎𝑥 ) (1.13)
2 2
so that the fractional derivatives of trigonometric functions are given by:
𝛼𝜋 𝛼𝜋
𝐷𝛼(cos𝑎𝑥) = 𝑎𝛼(𝑐𝑜𝑠 cos𝑎𝑥−sin sin𝑎𝑥)
2 2
𝛼𝜋
= 𝑎𝛼cos(𝑎𝑥+ ) (1.14)
2
𝛼𝜋 𝛼𝜋
𝐷𝛼(sin𝑎𝑥) = 𝑎𝛼(𝑠𝑖𝑛 cos𝑎𝑥+cos sin𝑎𝑥)
2 2
𝛼𝜋
= 𝑎𝛼sin(𝑎𝑥+ ) (1.15)
2
When α= ½ and a =1, Greer’s formulas are as follows:
𝜋 𝜋
𝐷½cos𝑥 = cos(𝑥+ ), 𝐷½sinx = cos(𝑥 + ), (1.16)
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Over the years, many mathematicians, using their own notation and approach,
have found different definitions that suitable for the idea of a non-integer order integral or
derivative. One version that has been generalized in the world of fractional calculus is the
Riemann-Liouville definition. It is interesting to note that the Riemann-Liouville
definition of a fractional derivative gives the same result as that obtained by Lacroix in
equation (1.1). Since most of the other definitions of fractional calculus are largely
variations of the Riemann - Liuoville version, so this version that will be mostly
addressed in this subject matter.
1.3 Useful Mathematical Functions
Before entering at the definition of the Riemann-Liouville Fractional Integral or
Derivative, we will first discuss some useful mathematical definitions that are inherently
bound to fractional calculus. These contain the Gamma function, the Beta function, the
Error function, the Mittag-Leffler function.

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1.3.1 The Gamma Function
The most basic interpretation of the Gamma function is simply the generalization
of the factorial for all real numbers.
Its definition is given by:
∞
Г(𝑥) = ∫ 𝑒−𝑡 𝑡𝑥−1 𝑑𝑡 , 𝑥 ∈ ℝ+ (1.17)
0
The Gamma function has some unique properties. By using its repeating relations we can
obtain formulas:
Г(𝑥+1) = 𝑥Г(𝑥) , 𝑥 ∈ ℝ+ (1.18)
Г(𝑥) = (𝑥−1)! , 𝑥 ∈ ℕ
The incomplete Gamma function is a closely related function defined as

1 𝑡
Г∗(𝑣,𝑡) = ∫ 𝑒−𝑥𝑥𝑣−1 𝑑𝑥 𝑅𝑒 𝑣 > 0 (1.19)
Г(𝑣)𝑡𝑣
0

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