Fractional Calculus

Chapter 1

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Fractional Calculus

Definition and Applications

1.1

History

Definition

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operators

and the integration operator J. (Usually J is used in favor of I to avoid confusion with other I-like glyphs and identities) [1] Fractional calculus comes out of generalizing the differential/integral operator, applying it at non-integer order (example, a "half derivative" operation such that taking the half derivative of a function twice yields the first derivative The traditional integral and derivative are, to say the least, a staple for the technology professional, essential as a means of understanding and working with natural and artificial systems. [7] 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. [7]

1.1.1

Physical Meaning

Consider the physical meaning of the exponent. According to our primary school teachers exponents provide a short notation for what is essentially a repeated multiplication of a numerical value. This concept in itself is easy to grasp and straight forward. However, this physical definition can clearly become confused when considering exponents of non integer value. While almost anyone can verify that x 3 = x. x. x, how might. [7] We describe the physical meaning of x 3.4 , or moreover the transcendental exponent x π . We cannot conceive 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

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Fractional Calculus

Definition and Applications

History

value x, verifiable by infinite series expansion, or more practically, by calculator. 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. Once mastered, the idea of completing numerous of these operations, integrations or differentiations follows naturally. Given the satisfaction of a very few restrictions (e.g. function continuity) completing n integrations can become as methodical as multiplication. [7] But the curious mind can not be restrained from asking the question what if n were not restricted to an integer value? Again, at first glance, the physical meaning can become convoluted (pun intended), but as this report will show, fractional calculus flows quite naturally from our traditional definitions. And just as fractional exponents such as the square root may find their way into innumerable equations and applications, it will become apparent that integrations of order 1/ 2 and beyond can find practical use in many modern problems. [7]

1.2 Historical Survey Most authors on this topic will cite a particular date as the birthday of so called 'Fractional Calculus'. In a letter dated September 30th, 1695 L’Hospital wrote to Leibniz asking him about a particular notation he had used in his publications for the nth-derivative of the Dn x linear function f(x) = x, . L’Hospital posed the question to Leibniz, Dx n what would the result be if n = 1/ 2. Leibniz's response: "An apparent paradox, from which one day useful consequences will be drawn." In these words fractional calculus was born. [7]

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Leibniz

Following L’Hospital and Leibniz’s first inquisition, fractional calculus was primarily a study reserved for the best minds in mathematics. Fourier, Euler, Laplace are among the many that dabbled 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. [7] 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. While the shear numbers of actual Definitions are no doubt as numerous as the men and women that study this field, they are for the most part variations on the themes of these two and so are addressed in detail in this document. [7] Most of the mathematical theory applicable to the study of fractional calculus was developed prior to the turn of the 20th century. However it is in the past 100 years that the most intriguing leaps in engineering and scientific application have been found. The

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Definition and Applications

History

mathematics has in some cases had to change to meet the requirements of physical reality. [7] Caputo reformulated the more 'classic' definition of the RiemannLiouville fractional derivative in order to use integer order initial conditions to solve his fractional order differential equations. As recently as 1996, Kolowankar reformulated again, the RiemannLiouville fractional derivative in order to differentiate no-where differentiable fractal functions. [7] Leibniz's response, based on studies over the intervening 300 years, has proven at least half right. It is clear that within the 20th century especially numerous applications and physical manifestations of fractional calculus have been found. However, these applications and the mathematical background surrounding fractional calculus are far from paradoxical. While the physical meaning is difficult (arguably impossible) to grasp, the definitions themselves are no more rigorous than those of their integer order counterparts. [7] Understanding of definitions and use of fractional calculus will be made clearer by quickly discussing some necessary but relatively simple mathematical definitions that will arise in the study of these concepts. These are The Gamma Function, The Beta Function, The Laplace Transform, and the Mittag-Leffler Function and are addressed in the second and third chapter of this thesis. [7]

1.3 Application of Fractional Calculus In the last decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics and notably control theory and signal and image processing. In these last three fields, some important considerations such as modeling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observables and robustness are now linked to long-range dependence phenomena.[9]

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Fractional Calculus

Definition and Applications

History

Fractional calculus concerns the generalization of differentiation and integration to non-integer (fractional) orders. The subject has a long mathematical history being discussed for the first time already in the correspondence of G.W. Leibniz around 1690. Over the centuries many mathematicians have built up a large body of mathematical knowledge on fractional integrals and derivatives. Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. One reason could be that, until recently, the basic facts were not readily accessible even in the mathematical literature. [10] Fractional calculus provides novel mathematical tools for modeling physical and biological processes. The bioheat equation is often used as a first order model of heat transfer in biological systems. The solution to the resulting fractional order partial differential equation reflects the interaction of the system with the dynamics of its response to surface or volume heating.. In the future we hope to interpret the physical basis of fractional derivatives using Constructal Theory, according to which, the geometry biological structures evolve as a result of the optimization process.[11] About a decade ago we noticed the possibility to employ fractional calculus in statistical physics. More precisely, Ehrenfests classification of equilibrium phase transitions was generalized using fractional derivatives. It transpired that fractional calculus may also be useful for problems of non equilibrium phenomena. In the meantime fractional calculus has found many new applications in physics, particularly to diffusion and relaxation phenomena. During the period of this report we have prepared and edited extensive tutorial reviews of the mathematical theory and its applications in physics. [12] Most of our work in the area of fractional calculus centers around equilibrium and non equilibrium critical phenomena and scaling theory. The concept of universality classes pervades much of the research on equilibrium and non equilibrium phenomena, and it illustrates the necessity of classifying the large variety of microscopic theories [12]

1.4

Fractional Derivative

1.4.1

History

Chapter 1

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Fractional Calculus

Definition and Applications

History

Differentiation and integration are usually regarded as discrete operations, in the sense that we differentiate or integrate a function once, twice, or any whole number of times. However, in some circumstances it’s useful to evaluate a fractional derivative. In a letter to L’Hospital in 1695, Leibniz raised the possibility of generalizing the operation of differentiation to non-integer orders, and L’Hospital asked what would be the result of half-differentiating x. Leibniz replied “It leads to a paradox, from which one day useful consequences will be drawn”. The paradoxical aspects are due to the fact that there are several different ways of generalizing the differentiation operator to non-integer powers, leading to in equivalent results [5] 1.4.2

Notation [1]

In this context powers refer to iterative application, in the same sense that f2(x) = f (f(x)). For example, one may pose the question of interpreting meaningfully D D

1 2

as a square root of the differentiation operator (an operator half iterate), i.e., an expression for some operator that when applied twice to a function will have the same effect as differentiation. More generally, one can look at the question of defining Ds for real-number values of s in such a way that when s takes an integer value n, the usual power of n-fold differentiation is recovered for n > 0, and the −nth power of J when n < 0.[1]

As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832. The fractional derivative of a function to order a is often now defined by means of the Fourier or Laplace integral

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Fractional Calculus

Definition and Applications

History

transforms. An important point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integral cases we cannot say that the fractional derivative at x of a function f depends only on the graph of f very near x, in the way that integer-power derivatives certainly do. Therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out. [1]

1.4.3

Definition 1 [1]

The first derivative of x k is as usual f (x) 

dx k  kx k 1 dx

Repeating this gives the more general result that daxk k!  x ka a dx (k  a) Which, after replacing the factorials with the Gamma function, leads us to daxk (k  1)  x k a a dx (k  a  1) Or daxk dx a

=

k! x k a (k  a  1)

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Definition History

and Applications

1.4.4

Alternate Derivation [2]

The fractional derivative of the function x k is

D a x k  D m [ D ( ma ) x k ] 

 (k  1) x k ma   ( k  m  a  1) 

 Dm 

1.4.5

=

(k  1)(k  a  m)(k  a  m  1)........(k  a  1) k  a x (1  m  k  a )

=

(k  1) k  a x (k  a  1)

=

k! x k a (k  a  1)

Semi derivative

A fractional derivative of order ½ of the function x k is given by

D1/ 2 x k  x k 1/ 2

(k  1) (k  1/ 2)

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Fractional Calculus

Definition History

and Applications

1.4.6

Definition 2 [6]

The definition 1 in [1.4.1] is restricted to only monomial function or you can say algebraic function, there is another definition for general functions in terms of integral is given by

t

1 dm f ( ) d D f (t)  m  (m   ) dt 0 (t   ) 1m 

where, m  1    m and m  ¥

If m  0 The above equation reduces to

Chapter 1

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Fractional Calculus

Definition History

and Applications

t

1 f ( ) d D f (t)   ( ) 0 (t   ) 1 

1.4.8

1.4.7

Properties of fractional derivative

1. D   f ( x)  D D  f ( x)    2. D  af ( x)  bg ( x )   aD f ( x )  bD g ( x ) Fractional derivatives satisfy quite well all the properties that one could expect from them, despite some of them are only characteristic of integer order differentiation and some other have restrictions. For instance, the property of linearity is fulfilled, while that of the iteration has some restrictions in the cases that positive differentiation orders are present. Some properties that include summatories can be generalized changing the summatories into integrals. One such property is the expansion in Taylor series [4]

f ( z) 







a

Dzt f ( z0 ) ( z  z0 )t dt (t  1)

1.4.8.1

and other is the Leibniz rule

a

D f ( z) g ( z)  t z



(  1)

 (t  1)(  t  1)

a

Dz t f ( z ) a Dzt g ( z )

1.4.8.2



These and other generalized properties can be applied to the study of special functions, which often can be expressed in terms of simple formulas involving fractional derivatives. [4]

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Fractional Calculus

Definition History

and Applications

1.4.9

Left fractional derivative

Let x be a function defined on ¡ ,   0 , n be the smallest integer greater thank  (n  1    n) and   n   . Then the left fractional derivative of order  is defined to be [8]

D  x (t ) 

  Dt x (t ) 

dn dt n

  Dt x (t ) 

1.4.10 Right fractional derivative

1 dn ( ) dt n

t

 (t  s)



 1

x( s )ds

Chapter 1

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Fractional Calculus

Definition History

and Applications

Let x be a function defined on ¡ ,   0 , n be the smallest integer greater thank  (n  1    n) and   n   . Then the right fractional derivative of order  is defined to be

D x(t )  t D x (t )  (1) n

1.4.11



dn (1) n d n  D x ( t )  (t  s )  1 x ( s)ds [8] n t  n  dt ( ) dt t

Some important semi derivative [28]

f

1

d2f 1

dx 2 0

0

C

C x

x0  1

1 x

x

Chapter 1

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Fractional Calculus

Definition History

and Applications

2

x2

1

x2 1

x2

ex

Sin(x)

Cos(x)

ln(x)

x  3 2

8x 3 

 2 0

1  e x erf ( x ) x

  Sin(x  )  2 gres *  4 

2x    

1   2x   Cos(x  )  2 fres *   4 x   

ln(4 x) x

Chapter 1

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Fractional Calculus

Definition and Applications

History

The functions we denote fres() and gres() are those auxiliary Fresnel integrals *Fresnel integrals, S(x) and C(x), are two transcendental functions named after Augustin-Jean Fresnel that are used in optics. x

S(x)   S in(t2 )dt 0

x

C(x)   Cos(t 2 )dt 0

1.5 The Fractional Integral 1.5.1

Definition 1 [7]

It was touched upon in the introduction that the formulation of the concept for fractional integrals and derivatives was a natural outgrowth of integer order integrals and derivatives in much the same way that the fractional exponent follows from the more traditional integer order exponent. For the latter, it is the notation that makes the jump seems obvious. While one can not imagine the multiplication of a quantity a fractional number of times, there seems no practical restriction to placing a non-integer into the exponential position. Similarly, the common formulation for the fractional integral can be derived directly from a traditional expression of the repeated integration of a function. This approach is commonly referred to as the Riemann-Liouville approach. (1.5.1) Demonstrates the formula usually attributed to Cauchy for evaluating the n th integration of the function f (t).

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Definition History

and Applications



t

...... f (τ )dτ =

t

1 (t − τ ) n −1 f (τ )dτ (n − 1)! 0

1.5.1.1

For the abbreviated representation of this formula, we introduce the operator J n such as shown in (1.5.2). J n f (t ) = f n (t ) =

t 1 (t − τ ) n −1 f (τ )dτ ( n − 1)! ∫0

1.5.1.2

Often, we will also find another operator, D − n used in place of J n . While they represent the same formulation of the repeated integral function, and can be seen as interchangeable, we will find that the use of D − n may become misleading, especially when multiple operators are used in combination. For direct use in (1.5.1), n is restricted to be an integer. The primary restriction is the use of the factorial which in essence has no meaning for non-integer values. The gamma function is however an analytic expansion of the factorial for all reals, and thus can be used in place of the factorial. Hence, by replacing the factorial expression for its gamma function equivalent, we can generalize (1.5.2) for all α ∈ R+ as shown in (1.5.3). J α f (t ) = f α (t ) =

1.5.2

1 t (t − τ ) n −1 f (τ )dτ ∫ Γ(α ) 0

1.5.1.3

Definition 2 [1]

A fairly natural question to ask is, does there exist an operator H, or halfderivative, such that H 2 f (x) = Df (x) =

d f (x) = f ′(x) dx

1.5.2.1

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Definition History

and Applications

Or to put it another way,

d ny is well-defined for all real values of n > 0. A similar result dx n

applies to integration To delve into a little detail, start with the Gamma function  , which extends factorials to non-integer values. This is defined such that (n  1)  n !

1.5.2.2

Assuming a function f(x) that is well defined where x > 0, we can form the definite integral from 0 to x. Let's call this Repeating this process gives x

x

t

0

0 0

( J 2 f (x))   ( Jf )(t)dt   ( f (s)ds)dt

1.5.2.3

and this can be extended arbitrarily The Cauchy formula for repeated integration, namely x

1 ( J f )(x)  (x  t)n 1 f (t)dt  (n  1)! 0 n

1.5.2.4

Leads to a straightforward way to a generalization for real n. Simply using the Gamma function to remove the discrete nature of the factorial function (recalling that (n  1)  n ! , or equivalently (n)  (n  1)! gives us a natural candidate for fractional applications of the integral operator. ( J  f )(x)  f (x) 

1.5.3

x

1 (x  t)n 1f (t)d ( ) 0

Definition 3 [3] Denote the th derivative D n and the -fold integral D  n . Then

1.5.2.5

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Fractional Calculus

Definition History

and Applications

x

D 1 f ( x)   f (t )dt

1.5.3.1

0

Now, if the equation D  n f ( x) 

x

1 ( x  t ) n 1 f (t )dt  (n  1)! 0

1.5.3.2

for the multiple integral is true for , then D

 ( n 1)

 1 f (t )  D  ( x  t ) n 1 f (t )dt    ( n  1)! 0  1



x

 1 n 1 ( y  t ) f ( t ) dt =   dy (n  1)! 0 0   x



1.5.3.3

y

1.5.3.4

Interchanging the order of integration gives

x

D

 ( n 1)

1 n 1 f (t ) =  ( x  t ) f (t )dt n! 0

1.5.3.5

But [1.5.3.3] is true for n  1 , so it is also true for all by induction. The fractional integral of f (t ) of order   0 can then be defined by D  f (t )  Or it can be written as

x

1 ( x  t ) n 1 f (t )dt ( ) 0

1.5.3.6

Chapter 1

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Fractional Calculus

Definition History

and Applications

x

1 d  ( x  t ) 1 f (t ) dt f ( x )    ( ) 0 dx

1.5.3.7

where ( ) is the gamma function. More generally, the Riemann-Liouville operator of fractional integration is defined as  a Dx f (t ) 

x

1 ( x  t ) 1 f (t )dt  ( ) 0

1.5.3.8

0 Where   0 with a Dx f (t )  f (t ) [3]

1.5.4

Semi Integral [5]

The fractional integral of order 1/2 is called a semi-integral d



dx

=

1 2



1 2

x

1  1 ( x  t ) 2 f (t )dt ( x)  1 0 ( ) 2

4 32 x 3 

1.5.4.1

1.5.4.2

The convergence properties of this formula are best when n has a value between 0 and 1. There are two different ways in which this formula might be applied. For example, if we wish to find the (7/3) rd derivative of a function, we could begin by differentiating the function three whole times, and then apply the above formula with n = 2/3 to

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Fractional Calculus

Definition History

and Applications

“deduct” two thirds of a differentiation, or alternatively we could begin by applying the above formula with n = 2/3 and then differentiate the resulting function three whole times. [5] These two alternatives are called the Right Hand and the Left Hand Definitions respectively. Although these two definitions give the same result in many circumstances, they are not entirely equivalent, because (for example) the half-derivative of a constant is zero by the Right Hand Definition, whereas the Left Hand Definition gives for the halfderivative of a constant the result given previously as equation [5]

1.6

Some important semi Anti-Derivative [28]

f



d f dx

0

C

x0  1

x

1 2



0

2C

2

x 

x 

1 2

Chapter 1

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Fractional Calculus

Definition History

and Applications

3

4x 2 3  x2

x

5

16 x 2 15 

1 2

x 2

1



x2

ex

e x erf ( x )

Sin(x)

 Sin(x  ) 4

Cos(x)

 Cos(x  ) 4

ln(x)

2

x  ln 4 x  2 

Chapter 1

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Fractional Calculus

Definition and Applications

History