Introduction to Fractional-Order Operators and Their Engineering Applications

Mihir Sen Department of Aerospace and Mechanical Engineering University of Notre Dame, Notre Dame, IN 46556

April 22, 2014 ii c Mihir Sen, 2014 ° Contents

Preface ...... v

1 Basics 1 1.1 Operators ...... 1 1.2 Matrices with fractional exponents ...... 3 1.3 Integrals of fractional order ...... 3 1.4 Derivatives of fractional order ...... 4 1.5 Example...... 5 1.6 Properties...... 7 1.7 Solutions of fractional-order diﬀerential equations ...... 7 1.8 Fractional vector calculus ...... 8 1.9 Physicalinterpretation...... 8 1.10 Taylor series and continued fractions ...... 8

2 Numerical aspects 11 2.1 Forwardshiftoperator ...... 11 2.2 Forward difference operator ...... 13 2.3 Backward Difference Operator ...... 17 2.4 Differentialoperator ...... 19 2.5 Integraloperators...... 23 2.6 Numerical implementation summary ...... 28 2.7 Binomial coefficients ...... 34

3 Applications 37 3.1 Networks ...... 37 3.2 Non-local eﬀects ...... 42 3.3 Probabilistic...... 45 3.4 Fractal and random media ...... 45 3.5 Fractionally-dependent component ...... 45 3.6 Applications to mechanics ...... 45 3.7 Applications to controls ...... 47 3.8 Applications to transport phenomena ...... 48 3.9 Fractional equations and chaos ...... 50 3.10 Experimental evidence ...... 50 3.11Futureideas...... 53

iii CONTENTS CONTENTS

A Appendix 59 A.1 Continuedfractions...... 59 A.2 Factorial and Gamma functions ...... 60 A.3 Cauchy’s formula for repeated integration ...... 61 A.4 Binomial coeﬃcients ...... 63 A.5 Laplacetransform ...... 63

Index 69

iv c Mihir Sen, 2014 ° Preface

This is a set of notes on fractional operators and their applications to engineering problems, concen- trating principally on the fractional-order derivative. Though these operators have been around for a long time, it is only in recent years that they have been included among the mainstream engineering tools such as those used in modeling or experimental data analysis. Though there are a number of good books available on the subject, the present notes intend to present a quick overview of the subject that may be of use to the modeler and the data analyst.

v Chapter 1

Basics

When l’Hôpital1 asked what would be the result of half-differentiating a function, Leibnitz (1695)2 replied that “It leads to a paradox, from which one day useful consequences will be drawn.” Heavi- side’s (1871)3 view was “There is universe of mathematics lying in between the complete differenti- ations and integrations, and that fractional operators push themselves forward sometimes, and are just as real as others.” The process by which we arrive at fractional operators is somewhat like what was done for numbers. First we had positive integers, and then followed the zero, fractions, irrational, negative, and complex numbers. A scalar α raised to a fractional power such as 1/2 is understood in the context of the law of exponents, αnαm = αn+m, where n and m are numbers. Although αn, where n is a positive integer, is defined by α being multiplied by itself (n 1) times, α1/2 is defined by α1/2α1/2 = 1. √2 is merely a notation, but α1/2 can be used with ease− in algebraic manipulations, and can participate in binary operations such as addition, subtraction, multiplication, division, and exponentiation. Though the use of fractional operators and derivatives is wide-spread, by choice we will restrict ourselves to the areas of engineering related to mechanical systems. For convenience we will also assume that the numbers and functions treated here are real, although generalizations to complex numbers exist.

1.1 Operators

Let T : or T(x) = y, be a map where x , y , and and are vector spaces. Sometimes,X →Y especially for matrices, scalar multiplication,∈X ⊆U and∈Y⊆V derivatives, Uwe willV follow the notation Tx = y, but there should be no confusion. More importantly, we should be able to recognize the operator T in the map. T is linear iﬀ

T(x + y) = T(x) + T(y), T(αx) = αT(x), where α is a scalar (i.e. belongs to a real ﬁeld).

1Guillaume de l’Hˆopital, French, 1661-1704. 2Gottfried Wilhelm Leibniz, German, 1646-1716. 3Oliver Heaviside, English, 1850-1925.

1 1.1. OPERATORS CHAPTER 1. BASICS

The most common linear operators that are used in engineering are the following. Scalar multiplication of a vector like, for example, αx. • Matrix A operating on a vector x to give another vector y. This can be written as Ax = y. • Of course, A and x must be compatible for the matrix multiplication to be possible. Derivative D operating on a function f(x) to give another function g(x). We write this as • Df(x) = g(x), where D = d/dx. Of course, we assume that f(x) is smooth enough for it to be differentiable. If we include distributions4 also, then we increase the range of possibilities for f(x). Integral I operating on a function f(x) to give another function g(x). We write this as • I(f(x)) = g(x), where I( ) = ( ) dx. This comes in several flavors. Thus, I can simply be an anti-derivative with an· additive· arbitrary constant C. Or, it can be an integral with lower R x limit a and upper limit x so that I( ) = a ( ) dy; this is an anti-derivative with C = I(a). A definite integral from a to b will, of course,· not· produce a function of x, and hence will not be included. R Linear combinations of the above, when possible. D + α and A + αI are examples. • 1.1.1 Operator with integer exponent If = , then the same operator can be repeatedly applied , and we can write U V Tn(x) = T T T T . . . T(x) , ³ ³ ³ ´´´ for an integer n > 0. To complete the non-negative integers, we can define T0 to be the identity operator I. It is obvious that the additive law of exponents

Tn+m = TnTm holds for non-negative integers. It is common to use this property to deﬁne an inverse T−1, for which

T−1T = T0, = I.

Repeated application of T−1 then extends Tn to all negative integers also, so that Tn is deﬁned for all n, positive, negative or zero. We ignore issues such as if T−1T = TT−1 or T−1T = TT−1, which can be easily answered for speciﬁc operators T. 6

1.1.2 Operator with fractional exponent We can conveniently deﬁne operators with exponents that are real numbers5 if we assume that the additive law of exponents holds for real n and m. Thus, for example

T = T1/2T1/2.

4L. Schwartz, L., 1951, Th´eorie des distributions 1–2, Hermann, Paris. 5Thus, strictly speaking, one should talk about an operator with a real exponent. One can also extend the deﬁnition to complex exponents, but that is beyond our scope here.

2 c Mihir Sen, 2014 ° CHAPTER 1. BASICS 1.2. MATRICES WITH FRACTIONAL EXPONENTS

This defines the square root of an operator . Similarly, any other nth root T1/n can be defined. Repeated application gives Tm/n, where m/n is a rational number. Of course, this a special case of a function of operators such as f(T). The latter is also widely used in engineering, but its inclusion here is outside our scope. Suffice it to say that the Cayley-Hamilton theorem about a square matrix satisfying its own characteristic equation is an example. Other matrix examples are eA and log A which are commonly used. A function such as (D + α)2 is a derivative example that is routinely used for the solution of ordinary differential equations.

1.2 Matrices with fractional exponents

The law of exponents enables us to define matrices elevated to a fractional power. B is the square root of A, written as B = √A = A1/2 if A = BB = B2. The square root may not exist, or the number of roots may be finite or infinite. Other nth roots may be similarly defined [20, 5, 14, 21]. For example, if

A1/n = B then

BBB . . . B = A. n times Furthermore, A1/n multiplied by itself (m| 1){z times} gives Am/n. − Exercises

1 4 5 2 33 24 1. Show that and are square roots of . 8 5 4 7 48 57 „ « „ « „ «

1.3 Integrals of fractional order

Starting from Cauchy’s6 formula for repeated integration 1 x Inf(x) = (x y)n−1f(y) dy (n 1)! − − Za we can generalize n to a real number α, and factorial to gamma function to get the left-handed Riemann-Liouville7 integral 1 x RLIαf(x) = (x y)α−1 f(y) dy, a x Γ(α) − Za α α α for α 0. Other notations for the operator, like aIRL, aIx , or simply I are common. Similarly, the right-handed≥ integral is

1 b RLIα = (y x)α−1 f(y) dy. x b Γ(α) − Zx 6Augustin-Louis Cauchy, French, 1789-1857. 7Bernhard Riemann, German, 1826-1866; Joseph Liouville, French, 1809-1882.

3 c Mihir Sen, 2014 ° 1.4. DERIVATIVES OF FRACTIONAL ORDER CHAPTER 1. BASICS

Exercises

Show that, if f(x)=(x a)β , then − Γ(β + 1) RLIαf(x)= (x a)α+β x a Γ(α + β + 1) − for β > 1 and α 0. For α = 1, this becomes (wrong!) − ≥ 1 RLI1f(x)= (x a)n+1 . a x n + 1 −

Other fractional integrals that are frequently used are: Liouville • 1 x LIαf(x) = (x y)α−1 f(y) dy. a x Γ(α) − Z−∞ Weyl • 1 ∞ W Iαf(x) = (y x)α−1 f(y) dy. a x Γ(α) − Zx Riesz • 1 ∞ RIαf(x) = y x α−1 f(y) dy. a x 2Γ(α) cos(απ/2) | − | Z−∞ 1.4 Derivatives of fractional order

There are several definitions that are commonly used. 1. Differentiating the Riemann-Liouville derivative gives dn RLDα = Iα f(x), a x dxn RL 1 dn x = (x y)α−1 f(y) dy. (1.1) Γ(α) dxn − Za sometimes written as Dα or dα/dxα. We can use a combined integro-differential operator notation

RL α a Dx if α > 0, RL α a Jx = 1 if α = 0, RL −α  a Ix if α < 0.

Writing α instead of n α in Eq. (1.1) gives − 1 dn x RLJ αf(x) = (x y)n−α−1f(y) dy, a x Γ(n α) dxn − − Za where n 1 α < n. − ≤ 4 c Mihir Sen, 2014 ° CHAPTER 1. BASICS 1.5. EXAMPLE

diﬀerentiation

3 −3 −2 −1 0 1 2

α integration

Figure 1.1: Integro-diﬀerential order line.

2. Another popular operator is the Caputo [8] derivative

1 x dnf(y) C Dαf(x) = (x t)n−α−1 dy a x Γ(n α) − dyn − Za

for n 1 α < n. − ≤ 3. For computational purposes, the Gr¨unwald-Letnikov derivative8

(x−a)/h GL α 1 m Γ(α + 1) a Dx f(x) = lim ( 1) f(x mh). h→0 hα − m! Γ(α m + 1) − m=0 X − is generally used. If we write

α ∆αf(x) = ( 1)m f(x + (α m)h), h − m − ≤ ∞ 0 Xm< µ ¶ then

α GL α ∆h f(x) a Dx f(x) = lim . h→0 hα

More about fractional derivative:

http://people.tuke.sk/igor.podlubny/fc.html http://www.cs.dartmouth.edu/farid/research/fracderiv.html http://en.wikipedia.org/wiki/Fractional_calculus

1.5 Example

2 α Some examples of fractional-order derivatives , i.e. function f(x) = x and its derivatives DRL, are shown in Fig. 1.2 (from Schumer et al. [61]). (a) α = 0.2, 0.4, 0.6, 0.8, 1.0. (b) α = 1, 1.2, 1.4, 1.6, 1.8, 2.0.

8Anton Karl Gr¨unwald, Czech, 1828-1920; Aleksey Vasilievich Letnikov, Russian, 1837-1888.

5 c Mihir Sen, 2014 ° 1.5. EXAMPLE CHAPTER 1. BASICS

Figure 1.2: Examples of fractional order derivatives of function f(x) = x2.

6 c Mihir Sen, 2014 ° CHAPTER 1. BASICS 1.6. PROPERTIES

1.6 Properties

• Dαf(x) = Dα1 Dα2 ...Dαn

where

α = αi i X The fractional derivative Dα is non-local if α is not an integer9. • Laplace transform of Dαf(t) is [f(t)] sαF (s), plus initial conditions. Examples are • L ∼ – Riemann-Liouville

[RLDαf(t)] = sαF (s) + . . . L 0 t – Caputo

n−1 k C α α α−k−1 d f [0 Dt f(t)] = s F (s) s L − dtk ¯ k=0 ¯t=0 X ¯ ¯ where n 1 α < n. ¯ − ≤ 1.7 Solutions of fractional-order diﬀerential equations

Using Laplace transforms • Two-parameter Mittag-Leﬄer10 function •

http://en.wikipedia.org/wiki/Mittag-Leffler_function

∞ xk E (x) = α,β Γ(αk + β) kX=0 Special case

x E1,1(x) = e

Series solutions • 9An exception is the Kolwankar-Gangal derivative, [31] Dα f(y) = lim Dα (f(x) f(y)), KG x→y RL − used to differentiate nowhere differentiable functions. 10Gösta Mittag-Leffler, Swedish, 1846-1927.

7 c Mihir Sen, 2014 ° 1.8. FRACTIONAL VECTOR CALCULUS CHAPTER 1. BASICS

¥ Example

Consider the semi-diﬀerential equation [49] D1/2y + y = 0. This can be simpliﬁed to −3/2 Dy y = C1x , − from which x x x −s −3/2 y(x)= C2e + e C1e s ds. Z0

1.8 Fractional vector calculus

Generalized Taylor’s series [48] is (y x)α (y x)2α F (y) = F (x) + DαF (x+) − + DαDαF (x+) − + ..., x Γ(α + 1) x x Γ(2α + 1) α where Dx is the Caputo fractional derivative of order 0 < α < 1. Spatial derivatives [42]

1.9 Physical interpretation

Probabilistic interpretation [35] and others [47, 58, 59, 66] (see references [27]–[40] in [35]).

1.10 Taylor series and continued fractions 1.10.1 Relation Question: What is the relation between fractional operators (actually functions of operators), Taylor series, continued fractions and operator trees?

Assume that any function of D, where D = d/dt, can be written as a Taylor series f ′′(0) f ′′′(0) f(D) = f(0) + f ′(0)D + D2 + D2 + . . . (1.2) 2! 3! Examples are D2 D3 eD = 1 + D + + + . . . 2! 3! D2 D3 ln(1 + D) = D + + . . . − 2 3 α(α 1) α(α 1)(α 2) (1 + D)α = 1 + αD + − D2 + − − D3 + . . . 2! 3! D D2 D3 (α + D)1/2 = α1/2 1 + + + . . . 2α − 8α2 16α3 µ ¶ D D2 D3 D D2 D3 (α + D)1/2 (β + D)1/2 = α1/2 1 + + + . . . β1/2 1 + + + . . . 2α − 8α2 16α3 2β − 8β2 16β3 µ ¶ µ ¶

8 c Mihir Sen, 2014 ° CHAPTER 1. BASICS 1.10. TAYLOR SERIES AND CONTINUED FRACTIONS

1.10.2 Operator tree Can every operator tree be written as a continued fraction? If true, then every function of an operator can be written as a tree.

¥ Example

Solve (1 + D)1/2y = 0.

D 1 ( 1 1) 1 ( 1 1)( 1 2) (1 + D)1/2y = 1+ + 2 2 − D2 + 2 2 − 2 − D3 . . . y, " 2 2! 3! # D D2 D3 5D4 = 1+ + + . . . y. 2 − 8 16 − 128 » – Let y = eλt, then λ λ2 λ3 5λ4 1+ + + . . . = 0. 2 − 8 16 − 128 Thus (1 + λ)1/2 = 0, from which λ = 1, so that − − y = e t.

9 c Mihir Sen, 2014 ° 1.10. TAYLOR SERIES AND CONTINUED FRACTIONS CHAPTER 1. BASICS

10 c Mihir Sen, 2014 ° Chapter 2

Numerical aspects

Written with Jason Mayes

Here we will discuss numerical aspects related to of shift , difference , differential , and integral operators. In the following h, α, β R; i,j,m,n N = 1, 2, 3,... ; s Z = ..., 2, 1, 0, 1, 2,... . f, g : R R are sufficiently smooth functions∈ of t. I∈is the{ identity operator.} ∈ { − − } 7→

2.1 Forward shift operator

2.1.1 Deﬁnitions

def Ehf = f(t + h) n times n def Eh f = Eh (Eh . . . (Ehf) . . .) n m def z n m}| { (Eh + Eh ) f = Eh f + Eh f

2.1.2 Properties

n n n Eh (f + g) = Eh f + Eh g n n Eh (αf) = αEh f

Eαhf = f(t + αh) n Eh f = f(t + nh)

Eh(fg) = Ehf Ehg 1 Eh = Eh

E0 = I n Eh = Enh n m m n n+m Eh Eh = Eh Eh = Eh m n mn (Eh ) = Eh

11 2.1. FORWARD SHIFT OPERATOR CHAPTER 2. NUMERICAL ASPECTS

j+1 j! (En + Em)j = En(i−1)Em(j−i+1) h h (i 1)!(j i + 1)! h h i=1 X − − where two operators are equivalent if, when operating on the same arbitrary function, the two operators produce the same result.

2.1.3 Extension to negative integers

A forward shift operator taken to a positive power, En, results in a forward shift. A forward shift operator taken to a negative power, Es, where s = n, results in a backwards shift. Thus, the backward shift operator of order m is merely the forward− shift operator raised to the power m. −

s def Ehf = f(t + sh) s Eh = Esh 0 Eh = I

2.1.4 Extension to reals

α def Eh f = f(t + αh) α Eh = Eαh α β α+β Eh Eh = Eh

2.1.5 Inverse operators

1 If Y is a function for which application of the shift operator Eh yields y(x), then we can write −1 Eh y = Y , i.e.,

1 −1 if EhY (x) = y(x), then Eh y(x) = Y (x).

β α β −α Then the inverse, Eh , of an operator Eh is deﬁned as Eh = Eh , where β = α, so that β α 0 − Eh Eh = Eh = I are all equivalent operators. α R The inverse of the Shift operator Eh , for α , always exists and is unique and can be easily shown to be E−α: ∈

if E−αf(x) = f(x αh), then E−αEαf(x) = E−α[f(x + αh)] = f(x). h − h h h α −α The inverse of the forward shift operator Eh is the backward shift operator Eh .

2.1.6 Null space of the shift operator

The null space, (L), of a linear operator L : X Y is the subset of X deﬁned by (L) = x N α →α N { ∈ X : Lx = 0 . Taking L = Eh , the only x (Eh ) is the origin of X, that is the trivial solution x = 0. For the} shift operator then, the null∈ space N is trivial, that is (Eα) = 0 . N h { } 12 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.2. FORWARD DIFFERENCE OPERATOR

2.1.7 Numerical implementation

s Numerical implementation of the forward shift operator Eh is straight forward. For the operator Es, where s Z and the function F (t) is sampled such that F = F (t + ih), then h ∈ i 0 s EhFi = F (t0 + ih + sh)

= F t0 + (i + s)h = Fi¡+s ¢

α Implementation of Eh requires that the function be sampled at intervals no greater than αh. Given a signal F (t) and the freedom to sample it at any time to produce the data set Fi = F (t0+iαh), where i = 1, 2, 3 . . . , then

α Eh Fi = F t0 + iαh + αh = F ¡t0 + (i + 1)αh¢ = Fi¡+1 ¢

2.1.8 Linear equations To ﬁnd f(t) given g and

Lf = g where

j αi L = pi(x)Eh , i=1 X and p (x) and g(x) are real valued functions and p (x) = 0 for all x x , then the initial conditions i i 6 ≥ 0 Eαi f = a , for i = 1, 2, ..., j 1, h i − are necessary for a unique solution. Furthermore, the solution procedure is similar to the solution of ordinary linear diﬀerential equations. The fundamental solution set can be found by substituting λn for x into the homogeneous equation and solving the resulting characteristic equation. Particular solutions can be found using an analog of the method of undetermined coeﬃcients or variation of parameters.

2.2 Forward diﬀerence operator

2.2.1 Deﬁnitions

∆ f def= f(t + h) f(t) h − n times n def ∆hf = ∆h (∆h . . . (∆hf) . . .) z }| { 13 c Mihir Sen, 2014 ° 2.2. FORWARD DIFFERENCE OPERATOR CHAPTER 2. NUMERICAL ASPECTS

2.2.2 Properties

n n n ∆h(f + g) = ∆hf + ∆hg n n ∆h(αf) = α∆hf ∆(fg) = f ∆g + Eg ∆f = Ef∆g + g∆f n m n m (∆h + ∆h ) f = ∆hf + ∆h f n m m+n ∆h∆mf = ∆h f ∆ = E E0 h h − h Eh∆h = ∆hEh 1 ∆h = ∆h ∆n = (E I)n h h − n n = ( 1)kEn−k k − kX=0 µ ¶ n n = ( 1)n−kEk k − kX=0 µ ¶ n n! = ( 1)n−k Ek − k!(n k)! h kX=0 − 2.2.3 Extension to negative integers

0 ∆h = I s 0 s ∆h = (Eh Eh) ∞ − s = ( 1)kEs−k k − k=0 µ ¶ X∞ s = ( 1)s−kEk k − k=0 µ ¶ X∞ s(s 1)(s 2) . . . (s k + 1) = − − − ( 1)s−kEk k! − h kX=0 2.2.4 Extension to reals

α 0 α ∆h = (Eh Eh) ∞ − α ∆α = ( 1)kEα−k h k − k=0 µ ¶ X∞ α = ( 1)α−kEk k − k=0 µ ¶ X∞ α(α 1)(α 2) . . . (α k + 1) = − − − ( 1)kEα−k k! − h kX=0

14 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.2. FORWARD DIFFERENCE OPERATOR

2.2.5 Null space of the forward diﬀerence operator

α α The null space of the forward diﬀerence operator ∆h includes every function f such that ∆h f = α N { } 0. Functions that exist in ∆h include every periodic function with period h and for α = n, all N { } α polynomials of degree n 1. This might not be a complete list, but the point is that ∆h is not trivial. For the sake of− this summary, p(x) is used to represent the set of functions thatN { exist} in α ∆h . N { } α α α Note that even though ∆h can be written in terms of the shift operator Eh and Eh is trivial, the null space of the forward diﬀerence operator is not trivial. N { }

2.2.6 Inverse operators If Y is a function whose first difference is the function y, then Y is called an indefinite sum of y and −1 is denoted by ∆h y, i.e.,

−1 if ∆hY = y(x), then ∆h y(x) = Y (x) + p(x). where p(x) ∆α . ∈ N { h } Finding an indefinite sum of y is the problem inverse to finding the difference of y. Instead of starting with y and differencing it, we instead seek another function which results in y upon being differenced. If s is a negative integer such that s = n, then the following is true: − n s ∆h∆hf(t) = f(t) s n ∆h∆hf(t) = f(t) + p(x)

Similarly,

α −α ∆h ∆h f(t) = f(t) −α α ∆h ∆h f(t) = f(t) + p(x)

2.2.7 Numerical implementation There are two possible formulas for implementing the forward diﬀerence operator numerically,

∞ α ∆α = ( 1)kEα−k h k − kX=0 µ ¶ and

∞ α ∆α = ( 1)α−kEk h k − kX=0 µ ¶ The first option results in real coefficients and fractional step sizes, while the second provides complex coefficients with integer order steps (if α N). While both are equally valid, the first will be used for 6∈ α convenience sake to avoid the use of complex coefficients. Using this formula, implementation of ∆h is straightforward for α 0, Q, although it has the disadvantage of requiring data at fractional N ≥ ∈ p step sizes if α . However, if α Q then α can be written as α = q and the problem of requiring 6∈ ∈ q data at fractional time steps can be avoided by taking the sampling rate to be f = h . Now given a

15 c Mihir Sen, 2014 ° 2.2. FORWARD DIFFERENCE OPERATOR CHAPTER 2. NUMERICAL ASPECTS

h signal F (t) and sampling at the appropriate frequency results in the data set Fi = F (to + i q ). The forward diﬀerence of order α at point Fi is now given by

∞ α ∆αF = ( 1)kEα−kF h i k − h i kX=0 µ ¶ where

α−k p E F = F − , α = h i i+p kq q

For α N, this is very simple as every α = 0 for k > α. However, for α 0 and ∈ k ≥ α N, α = 0 for k > α. Luckily, α 0 as k , so only a a few terms for k > α are needed to k k ¡ ¢ approximate6∈ 6 the sum. Also note that the→ data required→ ∞ to calculate the fractional forward difference ¡ ¢ ¡ ¢ at Fi requires data not only at fractional multiples of h, but also at points both before and after ti. For α Q, numerical implementation of the forward difference operator can only be approximated. 6∈ α Implementation of ∆h is more complicated for α < 0 because the binomial coefficient does not necessarily tend to zero as k goes to infinity. While it is straightforward, there is no guarantee the series will converge which decreases its usefulness. Implementing an infinite sum numerically is no easy task, and only taking a few terms is not always a good idea because the number of terms needed to be even somewhat accurate depends on α. However, a simple assumption can simplify the infinite sum to a finite sum. When dealing h with a discrete data set fi = f(a + i q ), there is always an initial and final data point, f0 = f(a) h α and fn = f(a + n q ). When considering ∆h Fi it is obvious to see data is required at points Fi+p−kq, which, as k , requires data before f = f(a), which does not exist, or you do not have. → ∞ 0 However, by making the assumption that Fi = 0 for i < 0 (F (a + t) is causal), the infinite sum can t−a be simplified to a finite sum from k = 0 to k = n, where n = h . Denoting this operator with the α causal assumption as a∆h , we can write

n α ∆αF = ( 1)kEα−kF a h i k − h i kX=0 µ ¶ where

α−k p E F = F − , α = h i i+p kq q

t−a and n = h .

2.2.8 Linear equations To ﬁnd f(t) given g and

j αi ∆h f = g i=1 X along with j independent conditions on f.

16 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.3. BACKWARD DIFFERENCE OPERATOR

2.3 Backward Diﬀerence Operator

2.3.1 Deﬁnitions

f def= f(t) f(t h) ∇h − − n times nf def= ( . . . ( f) . . .) ∇h ∇h ∇h ∇h z }| { 2.3.2 Properties

n(f + g) = nf + ng ∇h ∇h ∇h n(αf) = α nf ∇h ∇h (fg) = f g + E−1g f = E−1f g + g f ∇ ∇ ∇ ∇ ∇ ( n + m) f = nf + mf ∇h ∇h ∇h ∇h n mf = m+nf ∇h∇m ∇h = E0 E−1 ∇h h − h −1 = Eh ∆h E = E ∇h h h∇h = ∆h 1 = ∇h ∇h n = (I E−1)n ∇h − h n n = ( 1)n−kEk−n k − h kX=0 µ ¶ n n = ( 1)kE−k k − h kX=0 µ ¶ 2.3.3 Extension to negative integers

0 = I ∇h s −1 s h = (I Eh ) ∇ ∞− s = ( 1)s−kEk−s k − h k=0 µ ¶ X∞ s = ( 1)kE−k k − h kX=0 µ ¶ 2.3.4 Extension to reals

α = (I E−1)α ∇h − h 17 c Mihir Sen, 2014 ° 2.3. BACKWARD DIFFERENCE OPERATOR CHAPTER 2. NUMERICAL ASPECTS

∞ α = ( 1)α−kEk−α k − h k=0 µ ¶ X∞ α = ( 1)kE−k k − h k=0 µ ¶ X∞ α(α 1)(α 2) . . . (α k + 1) = − − − ( 1)kE−k k! − h kX=0 2.3.5 Null space of the backward diﬀerence operator

α The null space of the backward difference operator h includes every function f such that α α N {∇ } h f = 0. Functions that exist in h include every periodic function with period h, and for ∇ N {∇ } α α α = n, all polynomials of degree n 1. Not surprisingly, any function f h is also in ∆h , thus p(x) is again used to represent− the set of all functions that span ∈ Nα {∇. } N { } N {∇h } 2.3.6 Inverse operators If Y is a function whose first difference is the function y, then Y is called an indefinite sum of y and is denoted by −1y, i.e., ∇h if Y = y(x), then −1y(x) = Y (x) + p(x). ∇h ∇h where p(x) is the set of functions that span α . N {∇h } Finding an indefinite sum of y is the problem inverse to finding the difference of y. Instead of starting with y and differencing it, we instead seek another function which results in y upon being differenced. If s is a negative integer such that s = n, then the following is true: − n s f(t) = f(t) ∇h∇h s nf(t) = f(t) + p(x) ∇h∇h Likewise, for α > 0,

α −αf(t) = f(t) ∇h ∇h −α αf(t) = f(t) + p(x) ∇h ∇h 2.3.7 Numerical Implementation

α Numerical implementation of the backward diﬀerence operator h for α 0 is similar to that of α ∇ ≥ ∆h , but is even more straightforward. Given a function F (t) sampled such that the discrete data set is given by Fi = F (a + ih), then the backward diﬀerence of order α at Fi is given by

∞ α αF = ( 1)kE−kF ∇h i k − h i kX=0 µ ¶ where

−k Eh Fi = Fi−k

18 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.4. DIFFERENTIAL OPERATOR

Once again, for α N, this is very simple as every α = 0 for k > α. However, for α 0 and ∈ k ≥ α N, α = 0 for k > α. Luckily, α 0 as k , so only a a few terms for k > α are needed to k k ¡ ¢ approximate6∈ 6 the sum. Also note that to→ calculate→ the ∞ fractional backward difference at F requires ¡ ¢ ¡ ¢ i data only at or before ti and only at integer step sizes, which are the two main advantages of using the backward difference. When dealing with a discrete data set fi, there is always an initial data point f0 = f(a). α α From the formula above for h Fi, it is easy to see that the numerical implementation of h requires ∇α ∇ data at F − , so as k , requires data before f , which does not exist. However, by assuming i k → ∞ ∇h 0 Fi to be causal (Fi = 0 for i < 0), the formula can be simplified and the infinite sum becomes finite, α which can now be easily calculated. Denoting a h as the backward difference operator with Fi = 0 for i < 0, we can write ∇

n α αF = ( 1)kE−kF a∇h i k − h i kX=0 µ ¶ where

−k Eh Fi = Fi−k

t−a and n = h . α Implementation of h is more complicated for α < 0 because the binomial coefficient does not necessarily tend to zero∇ as k goes to infinity. While it is straightforward, there is no guarantee the series will converge which decreases its usefulness. Alternatively, as was the case with the forward difference, there is another way to implement the backward difference operator that has decided disadvantages when compared to the previous method. The backward difference of a function at point Fi can also be expressed as

∞ s αF = ( 1)α−kEk−αF ∇h i k − h i kX=0 µ ¶ Now, for α N, it is obvious that this implementation results in both imaginary coeﬃcients and fractional step6∈ sizes. The combination of the two make it very impractical.

2.4 Differential operator 2.4.1 Definitions The differential operator can be defined using both the forward and backward difference operators ∆ and . With the forward difference operator ∇ ∆ f Df def= lim h h→0 h µ ¶ n times Dnf def= D (D... (Df) . . .) z ∆n}|f { Dnf = lim h h→0 hn µ ¶

19 c Mihir Sen, 2014 ° 2.4. DIFFERENTIAL OPERATOR CHAPTER 2. NUMERICAL ASPECTS

Using the backward diﬀerence operator

def f Df = lim ∇h h→0 h µ ¶ n times Dnf def= D (D... (Df) . . .) n def z }|f { Dnf = lim ∇h h→0 hn µ ¶ 2.4.2 Properties Properties of the differential operator are independent of the choice of difference operator used to define it. The following properties are true for both definitions.

(Dn + Dm) f = Dnf + Dmf Dn(f + g) = Dnf + Dng Dn(αf) = αDnf D(fg) = f Dg + g Df D1 = D DmDnf = Dm+nf

2.4.3 Extension to reals Like the shift and difference operators, the differential operator can also be extended to include derivatives of arbitrary real order. Using the forward difference definition and α > 0:

∆αf Dαf = lim h h→0 hα µ ¶ For the backward diﬀerence deﬁnition: αf Dαf = lim ∇h h→0 hα µ ¶ Derivatives of order α 0 are not considered in this section, but will be discussed with integral operators. ≤

2.4.4 Null space of the differential operator The differential operator is another operator with a non-trivial null space. In general, Dα = f F : Dαf = 0 . For α = n where n is an integer, Dn = C(x), the set of polynomialN functions { } of{ order∈ n 1. If the} operator L is taken to be a linearN { combination} of differential operators such as L = D2 +−D1 + D0 , then the null space includes functions of the type f = eλx. The point, once again, is that the null space of the differential operator is non-trivial. ¡ ¢ When solving differential equations of the type Lf = g, the homogeneous solution (the solution to the equation Lf = 0) is exactly the set of functions (and using the superposition principle, the sum of all such functions) that lie in the null space of the operator L.

20 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.4. DIFFERENTIAL OPERATOR

2.4.5 Inverse operators If Y is a function that when operated upon by Dα (α > 0) results in the new function y, then Y is called the anti-derivative, or integral of y and is denoted by D−αy, i.e.,

if DαY (x) = y(x), then D−αy(x) = Y (x) + C

The inverse for a diﬀerential operator of order α is an integral, or anti-derivative, of the same order. However, this inverse is only unique to a constant if initial conditions or boundary conditions are not provided. Likewise, the inverse of an integral operator is a derivative of the same order. This inverse is unique. Mathematically, this can be expressed as

D1J 1f = f J 1D1f = f + C where the new operator J is used to denote an integral. More generally,

DαJ αf = f J αDαf = f + C(x) where J α is an integral of order α, deﬁned as J α = D−α. Alternatively,

DαD−αf = f D−αDαf = f + C(x) J αJ −αf = f + C(x) J −αJ αf = f

2.4.6 Numerical implementation Now on the discrete level, the derivative of any order (including fractional order) can be approximated using finite differences. The finite difference formula based on the forward differencing scheme for a derivative of order α for α > 0 can be derived from ∞ α 1 α k α−k Dh f = lim ( 1) Eh f h→0 hα k − ³ kX=0 µ ¶ ´ Similarly,

∞ α 1 α k −k Dh f = lim ( 1) Eh f h→0 hα k − ³ kX=0 µ ¶ ´ for the backward difference formulation of the differential operator. Implementation is thus slightly complicated because of the infinite upper limit on the summation. This infinite upper limit means, in a physical sense, that the derivative of a function f at time t = τ depends on the complete history t−a of the signal, for all < t < τ. This problem can be simplified by taking h = n where a and −∞ t−a t are the lower and upper limits of differentiation, respectively. This allows us to write n = h , so now as h 0, n . Now the derivative can be approximated by a finite difference formula assuming h is→ small.→ Alternatively, ∞ the use of terminals to convert the infinite sum to a finite sum can be considered to be an assumption that the signal is causal, or that the signal or function f = 0

21 c Mihir Sen, 2014 ° 2.4. DIFFERENTIAL OPERATOR CHAPTER 2. NUMERICAL ASPECTS

for t < a, which is, effectively, the case when dealing with a discrete data set that surely does not extend to t = . Now the fractional order finite difference approximation using the forward difference is given as−∞

1 n α t a Dαf ( 1)kEα−kf, where n = − a ≈ hα k − h h kX=0 µ ¶ Similarly,

1 n α t a Dαf ( 1)kE−kf, where n = − a ≈ hα k − h h kX=0 µ ¶ for the backward difference definition of the differential operator. As can be seen above, the formula using the backward differencing scheme has a distinct advantage over the forward differencing scheme in that it does not require data at fractional step sizes (the forward difference based model requires data at fractional time steps through the operator Eα−k). The forward difference scheme also has the added problem of requiring data at times both before and after the time step where the derivative or integral is being calculated. While this poses no real problem to calculating the derivative or integral given a discrete data set, it does pose a problem from a practical standpoint in that it causes the derivative to depend not only on the signal’s history, but its future as well.

Linear equations To ﬁnd f(t) given g and

j Dαi f = g i=1 X along with j independent conditions on f.

Eigenvalue problem 2.4.7 Derivatives and Integrals with Limits When performing the typical integration of a function, limits are necessary to eliminate the constant of integration. Limits, or terminals as we will call them, serve the same function when evaluating integrals and derivatives of non-integer order using the shift and difference operators as well as serving α another purpose. The differential operator defined in terms of the forward difference operator ∆h , where α < 0, is

∆αf Dαf = lim h h→0 hα µ ¶ n 1 α k α−k = lim ( 1) Eh f h→0 hα k − kX=0 µ ¶ which, for finite and fixed n, tends to the uninteresting limit Dαf = 0 as h 0. Only if, as seen before, this upper limit is set at infinity will an interesting answer be produced.→ However, if we take

22 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.5. INTEGRAL OPERATORS

t−a h = n where t and a are the upper and lower terminals of the integro-differentation, respectively, then n as h 0. This allows us to write → ∞ → α α ∆h f aDt f = lim α h→0 h nh=t−a µ ¶ n 1 α k α−k = lim α ( 1) Eh f h→0 h k − nh=t−a kX=0 µ ¶ which represents the derivative of order β if α = β and the β-fold integral if α = β. Following the same steps as above, the derivative of order β if α = β and the β-fold integral if −α = β using the backward difference definition of the differential operator yields −

α α h f aDt f = lim ∇ α h→0 h nh=t−a µ ¶ n 1 α k −k = lim α ( 1) Eh f h→0 h k − nh=t−a kX=0 µ ¶ 2.5 Integral operators

2.5.1 Indeﬁnite integrals The integral operator has been discussed in short in the previous section, but for the sake of com- pleteness, a thorough discussion will be included herein.

Definition The indefinite integral has already been defined in terms of the inverse of the derivative. If Y is a function that when operated upon by D1 results in the new function y, then Y is the integral, or anti-derivative, of y and is denoted by D−1y, i.e.,

if DY = y, then D−1y = Y + C

The integral can also be represented using the notation J n, which is read the n-fold integral, or the integral of order n. Using the deﬁnition of the derivative based on the forward diﬀerencing scheme,

Jf = D−1f + C ∆−1f = lim h + C h→0 h−1 J nf = D−n¡f + C(x¢) ∆−nf = lim h + C(x) h→0 h−n ¡ ¢ Using the backward diﬀerencing scheme,

Jf = D−1f + C

23 c Mihir Sen, 2014 ° 2.5. INTEGRAL OPERATORS CHAPTER 2. NUMERICAL ASPECTS

−1f = lim ∇h + C h→0 h−1 J nf = D−n¡f + C(x¢) −nf = lim ∇h + C(x) h→0 h−n ¡ ¢ In all of the above cases, C and C(x) are the set of functions or vectors that span the null space of the operators D−1 and D−n, respectively.

Properties

The integral shares many of the same properties of the derivative.

(J n + J m) f = J nf + J mf J n(f + g) = J nf + J ng J n(αf) = αJ nf J 1 = J J mJ nf = J m+nf

Extension to negative integers

In the previous section covering differential operators, the extension to negative powers was not made. By now, the reasons why are obvious. Now, before extending the definition of an integral operator to negative integer order, we must first relate integrals of integer order to derivative of negative integer order. The integral can be viewed as the extension of the derivative operator to negative powers. Thus extending the integral to a negative integer power is, in effect, the same as a derivative of integer order, which has already been discussed. Mathematically,

J n = D−nf + C(t) ∆−nf = lim h + C(t) h→0 h−n µ ¶ and

J n = D−nf + C(t) −nf = lim ∇h + C(t) h→0 h−n µ ¶ In both cases, C(t) exists in the null space of Dn. In similar fasion to before, an integral of order s = n is the same as a derivative of order n, −

J −n = Dn

which has been discussed previously.

24 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.5. INTEGRAL OPERATORS

Extension to reals Similarly, integration of order α is related to diﬀerentiation of order α through the following (for α 0): ≥ J α = D−α + C(t) J −α = Dα

2.5.2 Null space of the indefinite integral operator Generally, when considering the integral operator J α, only for values of α 0 is J α considered to be an integral. For α < 0, J α represents the differential operator. A discussion≥ of the differential operator’s null space, inverse, and implementation has already been included, so we will focus here on J α with α 0. The null≥ space of the integral operator (J α, α 0) is defined as J α = f F : J αf = 0 . While not proven here, J α = 0, meaning the equation≥ J αf = 0 hasN {only} the{ trivial∈ solution f}= 0, and thus the indefiniteN { integral} operator has a trivial null space.

Inverse operators Integration is defined as the inverse to a differential operator. As would be expected, the differential operator is the inverse to the integral operator J α. However, unlike before, the inverse to the integral operator is unique, which is also expected, as the null space of the integral operator is trivial.

DαJ αf = f

Alternatively, the inverse of the indefinite integral operator can be defined as follows. If y is a function that when operated upon by J α (α 0) results in the function Y + C(x), then y is called the derivative of Y + C(x) and is denoted by≥ Dα Y + C(x) or J −α Y + C(x) , i.e., { } { } if J αy(x) = Y (x) + C(x), then Dα Y + C(x) = y(x) { } where C(x) Dα . ∈ N { } Numerical implementation Implementing the indefinite integral numerically, as defined above, is not an easy task and requires the evaluation of an infinite sum and is still is not unique as any function in the null space of the corresponding differential operator can be added to to the result. For these reasons indefinite integration through discrete formulas is impractical. However, if the definition is changed slightly, implementation of the indefinite integral becomes simple.

2.5.3 Indefinite integration with terminals Indefinite and definite integration are two different operators related by the first fundamental theorem of calculus, which states that if f is continuous on the interval [a, b] and F is the anti-derivative (or indefinite integral) of f such that F = Jf, then

b f(x)dx = F (a) F (b) − Za

25 c Mihir Sen, 2014 ° 2.5. INTEGRAL OPERATORS CHAPTER 2. NUMERICAL ASPECTS

From this theorem, it is noteworthy that the indefinite integral operator J operates on a function and returns another function, J : F F. while the definite integral operator, aJb, where a and b are the upper and lower limits or terminals→ of the operation, operates on a function and returns a real number, aJb : F R. The second fundamental→ theorem of calculus holds for a function f, continuous on an open interval I, and a is any point in I, and states that if F is defined by x F (x) = f(t)dt Za then F ′(x) = f(x) at each point in I. This integral operator with an indefinite terminal, another form of indefinite integration, operates on a function and returns another function, aJ : F F. Using this form of indefinite integration allows us to numerically compute the indefinite integral→ using a data series.

Deﬁnition

Denoting the integral operator with an indeﬁnite terminal as aJ, the operator is deﬁned by the following. If Y is a function that when operated on by D1 results in the new function y, then Y is −1 the integral, or anti-derivative of y and is denoted by aJy or D , i.e.,

if DY = y, then aJy = Y Using this definition, the lower terminal a serves as an initial, or boundary condition to eliminate the unknown C(x) that was present before. In this way, aJ is defined as the inverse to the operator D, −1 aJf = D f −1 ∆h f = lim −1 h→0 h nh=t−a µ ¶ but unlike J, this operation is unique. The definition can be extended to positive integers as n −n aJ = D −n ∆h f = lim −n h→0 h nh=t−a µ ¶

Properties Extension to negative integers

The operator aJ can be extended to negative numbers in the same fashion as J. Using the deﬁnition as the inverse to the derivative, −n n aJ =a D n ∆hf = lim n h→0 h nh=t−a µ ¶ which is, obviously, just the derivative operator of order n.

26 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.5. INTEGRAL OPERATORS

Extension to reals Likewise, the extension to all real numbers is straight forward. For α 0, ≥ α −α aJ = D −α ∆h f = lim −α h→0 h nh=t−a µ ¶ −α α aJ = D α ∆h f = lim α h→0 h nh=t−a µ ¶

2.5.4 Null space of the integral operator with terminals Like the indeﬁnite integral form discussed previously, the null space of the integral operator with terminals is also trivial. That is aJ = 0, or, the only solution to the equation aJf = 0 is the trivial solution f = 0. N { }

Inverse operators Having been defined as the inverse to the derivative, it is, once again, not surprising to find the the inverse of the indefinite integral operator with terminals of order α is the differential operator of order α. The inverse of J α is unique and is given by Dα. Mathematically,

α α D aJ f = f

Also interesting to note is the fact that

α α aJ D f = f

α which is not the same as before. This is another way of saying that aJ is the unique inverse of the operator Dα.

Numerical implementation Now that the integral operation (deﬁned as the inverse to the derivative) is unique, it can be α implemented numerically. Previously, the operator aJ was given as

α −α aJ = D −α ∆h f = lim −α h→0 h nh=t−a µ ¶ or using the backward diﬀerence deﬁnition,

α −α aJ = D −α h f = lim ∇ −α h→0 h nh=t−a µ ¶

27 c Mihir Sen, 2014 ° 2.6. NUMERICAL IMPLEMENTATION SUMMARY CHAPTER 2. NUMERICAL ASPECTS

Now writing the diﬀerence operators as a sum gives

−α α ∆h f aJ f = lim −α h→0 h nh=t−a µ ¶ n α α k α−k = lim h ( 1) Eh f h→0 k − nh=t−a kX=0 µ ¶ and −α α h f aJ f = lim ∇ −α h→0 h nh=t−a µ ¶ n α α k −k = lim h ( 1) Eh f h→0 k − nh=t−a kX=0 µ ¶ The integral can now be approximated as

α α α aJ f = lim h ∆h f h→0 nh=t−a n α hα ( 1)kEα−kf ≈ k − h kX=0 µ ¶ and

α α α aJ f = lim h h f h→0 ∇ nh=t−a n α hα ( 1)kE−kf ≈ k − h kX=0 µ ¶ t−a where, for both cases, n = h . As before, the backward differencing result again has an obvious advantage over the forward difference in that data is only required at regular intervals. When implementing this numerically, then h is the step-size, n is the number of intervals between t and a (or n 1 is the number of points), and a is the starting point for the integration. − 2.5.5 Definite integrals Definite integration is exactly as its name implies: definite. A definite integral is calculated or evaluated over a given interval. A definite integral evaluated over the interval [a, b] is written aJb. The definite integral is merely a special case of the indefinite integral discussed previously, only with a definite upper and lower limit. All of the discussion in the previous section all applies here, but with the upper bound (previously x) replaced with the constant b. Now, as was discussed before, aJb : F R. One important difference between the definite and indefinite integral is that the null space of→ the definite integral is not trivial.

2.6 Numerical implementation summary

We have already briefly discussed the shift, forward difference, backward difference, differential, and integral operators. This section is just a brief summary of the results.

28 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.6. NUMERICAL IMPLEMENTATION SUMMARY

F(t) = t3 − t2 f(t) = F(t+α h) α 20 g = E F i h i

10 F,f and g

−5 0 0.5 1 1.5 2 2.5 3 t

Figure 2.1: The function f(t) = t3 t2, the function f(t + αh), and EαF . Shown for α = 0.5. − h i 2.6.1 Finite diﬀerence formulas These formulas can be used to numerically implement the following operators:

Shift Given the data f = f(a + ih), implementation of Esf where s Z is simply i h i ∈ s Ehfi = fi+s Given the data f = f(a + i h ), implementation of Eαf where α = p , R is simply i q h i q ∈ α Eh fi = fi+p Note that in order to shift in fractional steps of h, the data set must be sampled at a frequency q/h. s α Also note that the formula for Eh is simply a special case of Eh , with q = 1. Figure 2.2 shows the shift operator Eα applied to f = f(a + i h ) where f(t) = t3 t2 for two diﬀerent pairs of α and h. h i q − Forward diﬀerence

h α p Given the data fi = f(a + i q ), implementation of the operator a∆h where α = q , is given by n α ∆αf = ( 1)kEα−kf a h i k − h i k=0 µ ¶ Xn α k = ( 1) f − k − i+p kq kX=0 µ ¶ t−a where n = h . Figure 2.4 shows a function f(t) and its fractional forward diﬀerence or order α, calculated using the above formula for two values of α.

29 c Mihir Sen, 2014 ° 2.6. NUMERICAL IMPLEMENTATION SUMMARY CHAPTER 2. NUMERICAL ASPECTS

18 F(t) = t3 − t2 16 f(t) = F(t+α h) α E F h i 14

8 F,f and g 6

−2 0 0.5 1 1.5 2 2.5 3 t

3 2 α Figure 2.2: α = 4/3, h = 0.18. The function f(t) = t t , the function f(t + αh), and Eh Fi. Shown for α = 4/−3. −

F = x2.1 4.5 α f = ∆ F α g = ∆ F 4 a

3.5

2.5 F, f and g 2

1.5

0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Figure 2.3: α = 1/3. The function f(t) = t2.1, its forward diﬀerence of order α, and the forward diﬀerence of order α calculated numerically. Shown for both α = 1/3 and α = 1/2. −

30 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.6. NUMERICAL IMPLEMENTATION SUMMARY

F = x2.1 4.5 α f = ∆ F α g = ∆ F 4 a

3.5

2.5 F, f and g 2

1.5

0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Figure 2.4: α = 1/2. The function f(t) = t2.1, its forward diﬀerence of order α, and the forward diﬀerence of order− α calculated numerically. Shown for both α = 1/3 and α = 1/2. −

Backward diﬀerence Given the data f = f(a + ih), αf is given by i a∇h i n α αf = ( 1)kE−kf ∇h i k − h i k=0 µ ¶ Xn α k = ( 1) f − k − i k kX=0 µ ¶ t−a where n = h . Figure 2.6 shows a function f(t) and its fractional backward diﬀerence or order α, calculated using the above formula for two values of α.

Diﬀerential

Given the data fi = f(a + ih), the diﬀerential operator of order α can be implemented as

1 n α Dαf ( 1)kE−kf a i ≈ hα k − h i k=0 µ ¶ Xn 1 α k ( 1) f − ≈ hα k − i k kX=0 µ ¶ t−a where n = h is the number of intervals of step size h between t and a, or n + 1 is the number of data points between a and t. Figure 2.8 shows a function f(t) with both its analytical derivative of order α and its derivative of order α calculated using the above formula, for two diﬀerent values of α.

31 c Mihir Sen, 2014 ° 2.6. NUMERICAL IMPLEMENTATION SUMMARY CHAPTER 2. NUMERICAL ASPECTS

4.5

F = x2.1 4 α f = ∇ F α g = ∇ F a 3.5

2.5

F,f and g 2

1.5

0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Figure 2.5: α = 1/2. The function f(t) = t2.1, its backward diﬀerence of order α, and the backward diﬀerence of order α calculated numerically. Shown for both α = 0.5 and α = 2/3. −

180

F = x2.1 160 α f = ∇ F α g = ∇ F a 140

120

100

F,f and g 80

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Figure 2.6: α = 2/3. The function f(t) = t2.1, its backward diﬀerence of order α, and the backward diﬀerence of order− α calculated numerically. Shown for both α = 0.5 and α = 2/3. −

32 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.6. NUMERICAL IMPLEMENTATION SUMMARY

1.6

2.1 1.4 F = x α f = D Fα g = D a 1.2

0.8 F,f and g

0.6

0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t

Figure 2.7: α = 0.5. The function f(t) = t2.1, its analytical derivative of order α, and the derivative of order α calculated numerically. Shown for both α = 0.5 and α = 4/3.

2.5

F = x2.1 α f = D Fα 2 g = D F a

1.5 F,f and g 1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t

Figure 2.8: α = 4/3. The function f(t) = t2.1, its analytical derivative of order α, and the derivative of order α calculated numerically. Shown for both α = 0.5 and α = 4/3.

33 c Mihir Sen, 2014 ° 2.7. BINOMIAL COEFFICIENTS CHAPTER 2. NUMERICAL ASPECTS

0.9 F = x2.1 α f = D Fα 0.8 g = D F a

0.7

0.6

0.5 F,f and g 0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t

Figure 2.9: α = 0.5. The function f(t) = t2.1, its analytical integral of order α, and the integral of order α calculated numerically. Shown for both α = 0.5 and α = 4/3.

Integral

Given the data fi = f(a + ih), the integral operator of order α can be implemented as

n α J αf hα ( 1)kE−kf a i ≈ k − h i k=0 µ ¶ Xn α α k h ( 1) f − ≈ k − i k kX=0 µ ¶ t−a where n = h is the number of intervals of step size h between t and a, or n + 1 is the number of data points between a and t. Note that the discrete formula for the α-order integral is the same as the α-order derivative of order α, thus the same numerical formula is implemented. − Figure 2.10 shows a function f(t) with both its analytical integral of order α and its integral of order α calculated using the above formula, for two diﬀerent values of α.

2.7 Binomial coeﬃcients

In the following α R; n, k N = 1, 2, 3,... . In mathematics, the binomial theorem is an important formula giving∈ the expansion∈ { of powers} of sums. Its simplest version reads

n n (A + B)n = AkBn−k k kX=0 µ ¶

34 c Mihir Sen, 2014 ° CHAPTER 2. NUMERICAL ASPECTS 2.7. BINOMIAL COEFFICIENTS

F = x2.1 0.9 α f = D Fα g = D F 0.8 a

0.7

0.6

0.5 F,f and g 0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t

Figure 2.10: α = 4/3. The function f(t) = t2.1, its analytical integral of order α, and the integral of order α calculated numerically. Shown for both α = 0.5 and α = 4/3.

n where k are the binomial coefficients defined in the following section. When generalized for any real exponent, the result is ¡ ¢ ∞ α (A + B)α = AkBα−k k kX=0 µ ¶ The preceding formula is known as the generalized binomial theorem and is not only valid for α R, but for α C as well. ∈ ∈ In the following α R; n, k N = 1, 2, 3,... ; s Z = ..., 2, 1, 0, 1, 2,... . The n ∈ ∈ { } ∈ { − − } symbol k denotes a binomial coefficient and is commonly read ’n choose k.’ While the binomial coefficient is most commonly defined in terms of the factorial, which limits its usefulness to positive ¡ ¢ integers, its definition can be expanded to allow for both positive and negative non-integer arguments.

2.7.1 Properties The binomial coeﬃcient possesses the following properties:

n n = = 1 0 n µ ¶ µ ¶ n n = k n k µ ¶ µ − ¶ n k n 1 = ( 1)k − − k − k µ ¶ µ ¶ n n n k = − k k k + 1 µ ¶ µ ¶ 35 c Mihir Sen, 2014 ° 2.7. BINOMIAL COEFFICIENTS CHAPTER 2. NUMERICAL ASPECTS

n + 1 n n = + k k k 1 µ ¶ µ ¶ µ − ¶ 2.7.2 Positive integers For n, k N, ∈ n n! = k (n k)!k! µ ¶ − 2.7.3 Positive reals Writing the factorial as a gamma function Γ(n + 1) = n! allows the binomial coefficient to be generalized to non-integer arguments. Assuming α > 0, k N, ∈ α Γ(α + 1) = k k!Γ(α k + 1) µ ¶ − α α(α 1)(α 2) . . . (α k + 1) = − − − k k! µ ¶ 2.7.4 Negative integers While the Gamma function and factorial are not defined for negative integers, it is still possible to define the binomial coefficient. For s < 0, or s = n, and k N the following is true: − ∈ n (n + k 1)! − = ( 1)k − k − k!(n 1)! µ ¶ − s (k s 1)! = ( 1)k − − k − k!( s 1)! µ ¶ − − s s(s 1)(s 2) . . . (s k + 1) = − − − k k! µ ¶ 2.7.5 Negative reals For α R, α > 0, and k N ∈ ∈ α Γ(α + k) − = ( 1)k k − Γ(α)!k! µ ¶ α α( α 1)( α 2) . . . ( α k + 1) − = − − − − − − − k k! µ ¶ 2.7.6 All real numbers For any α R and k N the binomial coefficient can be generalized to ∈ ∈ α α(α 1)(α 2) . . . (α k + 1) = − − − k k! µ ¶ k α k + n = − n n=1 Y This final expression is known as the generalized binomial coefficient.

36 c Mihir Sen, 2014 ° Chapter 3

Applications

3.1 Networks

[28, 40]

3.1.1 Inﬁnite trees http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Mayes2011.pdf

An inﬁnite tree is a structure of the form shown in Fig. 3.1. For linear, unsteady potential- driven ﬂow, in each branch we have [39]

Tij (uij) = ∆φij where ∆φij is the driving potential in the branch, and uij is the resulting ﬂow. For the overall tree network,

TN (u) = ∆φ, where u and ∆φ are overall quantities. As an example, for laminar ﬂow in the pipe ij we have

du ij + a u = b ∆φ , dt ij ij ij ij where uij is the volume ﬂow rate, ∆φij is the pressure diﬀerence, and aij and bij are constants for this pipe. Thus

d T = + a ij dt ij

37 3.1. NETWORKS CHAPTER 3. APPLICATIONS

i = 1 i = 2 i = 3 i = n

Tn1 T31 T21 T32

T33 T11 T22 T34

φ + ∆φ T35 φ T12 T23 T36

T24 T37

T38 T2n

Figure 3.1: Tree with potential-driven ﬂow.

Symmetric tree [39] Replace Tij with T1 and T2 for j odd and even, respectively. The overall result is

T∞ = lim TN , N→∞ 1 = , 1 1 + 1 1 T + T + 1 1 1 2 1 1 + + T1 + . . . T2 + . . . T1 + . . . T2 + . . .

= T1T2 p The convergence is shown in the Laplace domain in Figs. 3.2. As N , TN T∞. It is not necessary to go beyond N = 10 to be able to calculate N . The reverse→ ∞ is also→ true: to calculate a tree for ﬁnite but large N, calculating the inﬁnite tree→ may ∞ be an easier approximation.

Capacitor-resistor tree (fractance) In the case of a time-dependent current i driven by a voltage diﬀerence ∆V

Laplace transform iR = R s0I (resistor) ⇒ di  L = L s1I (inductor) ∆V =  dt ⇒  t  1 1 i(t′) dt′ = s−1I (capacitor) C a ⇒ C  Z   38 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.1. NETWORKS

1.4

1.2

) 1 s ( N

T 0.8 n = 0.6 ∞n = 6 n = 4 0.4 n = 1 n = 2

0 1 2 3 4 5 6 7 8 9 10 s

(a) T1 = s, T2 = 1

30 n = 8

) 25 s

( n =

N 20 ∞ n = 4 T 15 n = 2 10

5 n = 1

0 1 2 3 4 5 6 7 8 9 10 s

2 (b) T1 = s + 1, T2 = s + 2s

Figure 3.2: Convergence in Laplace domain for two diﬀerent sets of T1 and T2, with T∞ and TN shown for diﬀerent values of N.

39 c Mihir Sen, 2014 ° 3.1. NETWORKS CHAPTER 3. APPLICATIONS

i = 1 i = 2 i = 3 i = N

(sC)−1 (sC)−1 −1 (sC) R −1 (sC)−1 R (sC) R Vin (sC)−1 Vout R −1 (sC) R − R (sC) 1 R R

Figure 3.3: Fractance tree (Nakagawa and Sorimachi [46]).

which can be written as

∆V = iZ where Z is the impedance. The fractance, which is the equivalent impedance of the alternating-current (AC) tree shown in Fig. 3.3, was ﬁrst investigated by Nakagawa and Sorimachi [46]. As a special case, the left branch is taken to be a capacitance with impedance (sC)−1 and the right with R, written in Laplace transform form. The equivalent circuit shown in Fig. 3.4 gives

1 1 1 = + Z 1 R + Z + Z sC from which 1 1 + Z (R + Z) = Z + R + 2Z , sC sC µ ¶ µ ¶ R Z H Z H + + RZH+ Z2 = + RZH+ 2Z2, sC sC sC R 1/2 Z = . sC µ ¶ Converting back from the Laplace domain,

∆V (t) = I1/2(i), (R/C)1/2 t i(τ) = dτ, Γ(1/2) (t τ)1/2 Z0 −

40 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.1. NETWORKS

(sC)−1

R Z

Figure 3.4: Equivalent to fractance tree.

3.1.2 Self-similar ladders Circuits A “domino” ladder is shown in Fig. 3.5. The total impedence Z can be calculated from

1 Z = R + , 1 sC + 1 R + 1 sC + R + . . . which is equivalent to

1 Z = R + , 1 sC + Z Z = R + , sCZ + 1 so that

(sCZ + 1) (Z R) = Z, − sCZ2 + Z sCRZ R = Z, − − R Z2 RZ = 0, − − sC from which

1 4R Z = R R2 + , 2 " ± r sC # of which we choose only the plus sign. For R 1/sC or sCR 1, ≪ ≪ 4R Z , ≈ rsC

41 c Mihir Sen, 2014 ° 3.2. NON-LOCAL EFFECTS CHAPTER 3. APPLICATIONS

R R R

C C C

Figure 3.5: Domino ladder.

Figure 3.6: Schematic of network.

3.1.3 Complex networks

St(φi) = uji, (nodal equation), j X T (u ) = k (φ φ ), (coupling equation). t ji ji j − i 3.2 Non-local effects 3.2.1 History effects In fluids Coimbra & Rangel, 2000: ”. . . temperature response of small particles subjected to diffusive and radiative heat transfer in a homogeneous medium . . . the history term . . . is . . . a Riemann-Liouville- Weyl half-derivative of the temperature potential between the free-stream and the particle surface.”

t ρpcpa dTp κ κm 1 d(Tp T˜m) ρmcma dT˜m + (Tp Tm) + − dτ = 3 dt a − √παm 0 √t τ dτ 3 dt ³ ´ Z − ³ ´

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Coimbra2000.pdf

42 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.2. NON-LOCAL EFFECTS

x x′

Figure 3.7: Non-local eﬀect in porous medium.

3.2.2 Spatially non-local eﬀects Spatially non-local porous medium See Fig. 3.7. Sen & Ramos, 2010

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Sen2012.pdf

∞ u(x) = f(x′,x) φ(x′) φ(x) dx′ { − } −∞Z where the flow rate at x is due to the pressure differences between x and all other points x′. f(x′,x) is a flow conductivity that relates the driving pressure difference to the resulting flow. Special cases: Local: If • f(x′,x) = k(x)δ′(x′ x) − we get Darcy’s law ∂φ u = k − ∂x which is a local relationship. Dirac delta distribution and its derivative

http://en.wikipedia.org/wiki/Dirac_delta_function

Almost local: One can use nascent delta functions δ (x′ x) for which • ǫ − ′ ′ lim δǫ(x x) = δ(x x). ǫ→0+ − − For example, the Gaussian nascent delta function is

1 x′ x 2 δ (x′ x) = exp − , ǫ − ǫ√π − ǫ ( µ ¶ ) 43 c Mihir Sen, 2014 ° 3.2. NON-LOCAL EFFECTS CHAPTER 3. APPLICATIONS

x Figure 3.8: Superposition of grains and tubes.

2(x′ x) x′ x 2 δ′ (x′ x) = − exp − . ǫ − − ǫ3√π − ǫ ( µ ¶ ) For a porous medium that is almost-local, we can take

f(x′,x) = k(x)δ′ (x′ x), ǫ − where ǫ is small but not zero, so that

∞ u(x) = k(x) δ′ (x′ x) φ(x′) φ(x) dx′. ǫ − { − } −∞Z

Fig. 3.8 shows the granularity of a porous medium which leads to almost-local behavior. Power-law non-local: If • − f(x′,x) = k (x x′)α 1 − then

α α ′ q(x) = kΓ(α) [−∞D + D ] p(x ) p(x) , x x ∞ { − } where x α 1 ′ α−1 ′ ′ −∞D g(x) = (x x ) g(x ) dx , x Γ(α) − −∞Z

44 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.3. PROBABILISTIC

∞ 1 Dα g(x) = (x x′)α−1 g(x′) dx′. x ∞ Γ(α) − Zx 3.3 Probabilistic

[36, 16]

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Machado2009.pdf

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Goodwine2014b.pdf

3.4 Fractal and random media 3.5 Fractionally-dependent component

[63] Bode plot [15]

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Dzielinski2010.pdf

3.6 Applications to mechanics 3.6.1 Vibrations and elasticity Non-locality [9], [34]

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/DiPaola2009.pdf

3.6.2 Viscoelasticity A viscoelastic material is schematically modeled in Fig. 3.9 [2, 19, 25]. The governing equations are

k(x x ) + µ(x ˙ x˙ ) = f(t), 1,1 − 0 1,2 − 0 k(x x ) + µ(x ˙ x˙ ) k(x x ) = 0, 2,1 − 1,1 2,2 − 1,1 − 1,1 − 0 k(x x ) + µ(x ˙ x˙ ) µ(x ˙ x˙ ) = 0, 2,3 − 1,2 2,4 − 1,2 − 1,2 − 0 k(x x ) + µ(x ˙ x˙ ) k(x x ) = 0, 3,1 − 2,1 3,2 − 2,1 − 2,1 − 1,1 k(x x ) + µ(x ˙ x˙ ) µ(x ˙ x˙ ) = 0, 3,3 − 2,2 3,4 − 2,2 − 2,2 − 1,1 k(x x ) + µ(x ˙ x˙ ) k(x x ) = 0, 3,5 − 2,3 3,6 − 2,3 − 2,3 − 1,2 k(x x ) + µ(x ˙ x˙ ) µ(x ˙ x˙ ) = 0, 3,7 − 2,4 3,8 − 2,4 − 2,4 − 1,2 k(x x ) + µ(x ˙ x˙ ) k(x x ) = 0, 4,1 − 3,1 4,2 − 3,1 − 3,1 − 2,1 45 c Mihir Sen, 2014 ° 3.6. APPLICATIONS TO MECHANICS CHAPTER 3. APPLICATIONS

k(x x ) + µ(x ˙ x˙ ) µ(x ˙ x˙ ) = 0, 4,3 − 3,2 4,4 − 3,2 − 3,2 − 2,1 k(x x ) + µ(x ˙ x˙ ) k(x x ) = 0, 4,5 − 3,3 4,6 − 3,3 − 3,3 − 2,2 k(x x ) + µ(x ˙ x˙ ) µ(x ˙ x˙ ) = 0, 4,7 − 3,4 4,8 − 3,4 − 3,4 − 2,2 k(x x ) + µ(x ˙ x˙ ) k(x x ) = 0, 4,9 − 3,5 4,10 − 3,5 − 3,5 − 2,3 k(x x ) + µ(x ˙ x˙ ) µ(x ˙ x˙ ) = 0, 4,11 − 3,6 4,12 − 3,6 − 3,6 − 2,3 k(x x ) + µ(x ˙ x˙ ) k(x x ) = 0, 4,13 − 3,7 4,14 − 3,7 − 3,7 − 2,4 k(x x ) + µ(x ˙ x˙ ) µ(x ˙ x˙ ) = 0, 4,15 − 3,8 4,16 − 3,8 − 3,8 − 2,4 . .

The Laplace transform is

k(X X ) + µs(X X ) = F (s), 1,1 − 0 1,2 − 0 k(X X ) + µs(X X ) k(X X ) = 0, 2,1 − 1,1 2,2 − 1,1 − 1,1 − 0 k(X X ) + µs(X X ) µs(X X ) = 0, 2,3 − 1,2 2,4 − 1,2 − 1,2 − 0 k(X X ) + µs(X X ) k(X X ) = 0, 3,1 − 2,1 3,2 − 2,1 − 2,1 − 1,1 k(X X ) + µs(X X ) µs(X X ) = 0, 3,3 − 2,2 3,4 − 2,2 − 2,2 − 1,1 k(X X ) + µs(X X ) k(X X ) = 0, 3,5 − 2,3 3,6 − 2,3 − 2,3 − 1,2 k(X X ) + µs(X X ) µs(X X ) = 0, 3,7 − 2,4 3,8 − 2,4 − 2,4 − 1,2 k(X X ) + µs(X X ) k(X X ) = 0, 4,1 − 3,1 4,2 − 3,1 − 3,1 − 2,1 k(X X ) + µs(X X ) µs(X X ) = 0, 4,3 − 3,2 4,4 − 3,2 − 3,2 − 2,1 k(X X ) + µs(X X ) k(X X ) = 0, 4,5 − 3,3 4,6 − 3,3 − 3,3 − 2,2 k(X X ) + µs(X X ) µs(X X ) = 0, 4,7 − 3,4 4,8 − 3,4 − 3,4 − 2,2 k(X X ) + µs(X X ) k(X X ) = 0, 4,9 − 3,5 4,10 − 3,5 − 3,5 − 2,3 k(X X ) + µs(X X ) µs(X X ) = 0, 4,11 − 3,6 4,12 − 3,6 − 3,6 − 2,3 k(X X ) + µs(X X ) k(X X ) = 0, 4,13 − 3,7 4,14 − 3,7 − 3,7 − 2,4 k(X X ) + µs(X X ) µs(X X ) = 0, 4,15 − 3,8 4,16 − 3,8 − 3,8 − 2,4 . .

This can be reduced to F (s) X(s) X = , − 0 1 1 + 1 1 1 1 + + k 1 1 µs 1 1 + + 1 1 1 1 + . . . + . . . + . . . + . . . k µs k µs 1 = s−1/2F (s). √kµ

In the time domain

1 1/2 x(t) x0 = I f(t), − √kµ

46 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.7. APPLICATIONS TO CONTROLS

x(t) x0 x2,1

x1,1

f(t)

x1,2

x2,4

Figure 3.9: Viscoelastic material.

(kµ)−1/2 t = (t τ)−3/2f(τ)dτ. Γ(1/2) − Za

Tissue

A ladder model to describe ultrasonic power law attenuation in mammalian tissue is shown in Fig. 3.10 [28]. The transfer function for stress-strain relationship

1 g(s) = ηs + 1 E−1 + 1 ηs + E−1 + . . . E ηs ηs ηs = + + 4 2 − E E E r · ηs ³ ´¸ ηsE for 1 ≈ E ≪ p Stress-strain relationship

4√ηE σ = D1/2ǫ 3

3.7 Applications to controls

[6, 13, 11]

47 c Mihir Sen, 2014 ° 3.8. APPLICATIONS TO TRANSPORT PHENOMENA CHAPTER 3. APPLICATIONS

Figure 3.10: Ladder model of mammalian tissue.

3.7.1 PIλDµ control Conventional PID controller

u(t) = a + I1 + D1 e(t)

Fractional PIλDµ controller ¡ ¢

u(t) = a + Iλ + Dµ e(t) ¡ ¢ 3.7.2 Robotic trees [17]

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Goodwine2014a.pdf

3.8 Applications to transport phenomena

[32, 33, 68, 70, 1, 53, 54, 62, 23, 56]

3.8.1 Conduction heat transfer http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Pineda2011.pdf

See Fig. 3.11. Podlubny

∂T ∂2T = α ∂t ∂x2

48 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.8. APPLICATIONS TO TRANSPORT PHENOMENA

Figure 3.11: Wall with thermocouple.

Figure 3.12: Mass transfer schematic.

Laplace transform in time

d2T sT = α dx2 b Heat ﬂux b ∂T q = k − ∂x k = D1/2T (t) √α wall

3.8.2 Mass diﬀusion Look at Fig. 3.12. If probability of a particle moving a given distance is a power law, then [61]

∂C ∂C + v = kDα(C) ∂t ∂x β and

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Schumer2001.pdf

49 c Mihir Sen, 2014 ° 3.9. FRACTIONAL EQUATIONS AND CHAOS CHAPTER 3. APPLICATIONS

3.8.3 Fractional time derivative Fractional model for solute spreading through heterogeneous, ﬂuid-saturated porous media with random porosity [27].

2 3/2 ∂ u ∂ 2 u = η∂ u th i − x h i t3/2 h i 3.9 Fractional equations and chaos

Seredy´nska and Hanyga (2004). Chaos in dynamical system with dimension less than 3. ”A nonlinear diﬀerential equation of fractional order µ between two and three is studied. . . . For µ = 3 the equation represents a chaotic dynamical system. As µ varies from 2 through 3 the dynamical system experiences period doublings and triplings and for two ranges of µ becomes chaotic.”

3.10 Experimental evidence 3.10.1 Visco-elasticity [38, 44]

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Makris1991.pdf

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Meral2010.pdf

3.10.2 Circuits

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Meral2010.pdf

3.10.3 Mass transfer See Fig. 3.13 [61, 4]

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Benson2001.pdf

3.10.4 Heat transfer Shell-and-tube heat exchanger, Mayes (2009) shown in Fig. 3.14. Regression results in Fig. 3.15. Approximations

1 ﬁrst-order c1D y(t) + c2y(t) = u(t) 2 1 second-order c1D y(t) + c2D y(t) + c3y(t) = u(t)

α 1 α c2 α fractional-order c1D y(t) + c2y(t) = u(t) [Sol. t Eα,α+1( t )] c1 −c1

50 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.10. EXPERIMENTAL EVIDENCE

Figure 3.13: Mass transfer results.

51 c Mihir Sen, 2014 ° 3.10. EXPERIMENTAL EVIDENCE CHAPTER 3. APPLICATIONS

Tc,o

Th,o

Th,i

Tc,i

Figure 3.14: Shell and tube heat exchanger.

0.8

0.6 ) t (

θ 0.4

0.2

0 0.2 0.4 0.6 0.8 1 τ

Figure 3.15: Regression ﬁts. dotted = data; dash = ﬁrst-order; dash-dot = second-order; continuous = fractional-order

52 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.11. FUTURE IDEAS

1.2 (b) (a) 1

0.8 θ 0.6 (c) 0.4

0.2

−0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 τ

Figure 3.16: Variation with time of air temperature for relaxation process. (a) exact; (b) D2θ + 1.9352Dθ + 0.8897θ = 0, (c) D1.2624θ + 3.9774 θ = 0.

2 Model c1 c2 c3 α error First-order 1.0401 0.028554 3.5284e-03 P Data Set #1 Second-order 0.097011 0.59086 0.89968 7.6902e-04 Fractional-order 0.22381 2.1700 1.8911 9.5609e-05 α = 1.8196 0.24855 2.0221 1.3355e-04

2 Model c1 c2 c3 α error First-order 1.2536 -0.32904 1.4135e-03 P Data Set #2 Second-order 0.12359 0.65039 0.85564 1.0973e-04 Fractional-order 0.33183 1.8782 1.7481 2.7757e-05 α = 1.8196 0.2976 2.0476 4.9404e-05

3.10.5 Another heat-transfer experiment Aoki, Sen & Paolucci (2007). Results are in Fig. 3.16.

http://www3.nd.edu/~msen/Teaching/FracDer/Papers/Aoki2008.pdf

3.11 Future ideas 3.11.1 Scaling What does the scaling property mean? [67] α α α Dx f(βx) = β Dβx

53 c Mihir Sen, 2014 ° 3.11. FUTURE IDEAS CHAPTER 3. APPLICATIONS

What relation do fractional diﬀerential equations have with power laws? With fractals? Both have similar scaling laws,

f(ax) = bf(x).

E.g. If f(x) = axn, then f(βx) = βnf(x).

3.11.2 Anomalous diﬀusion

What relation do fractional derivatives have with heavy-tailed probability distributions, anomalous diffusion and Lévy flights? With turbulence?

wikipedia.org/wiki/Stable_distribution

wikipedia.org/wiki/Heavy-tailed_distribution

wikipedia.org/wiki/Levy_flight

3.11.3 Generalized Taylor series

How to work with a generalized Taylor series, such as the following [26]?

∞ hαk f(x + h) = Dαk(x). Γ(1 + αk) kX=0 3.11.4 Fractional relaxation

http://en.wikipedia.org/wiki/Stretched_exponential_function

http://en.wikipedia.org/wiki/Fractional_relaxation

www.m-hikari.com/ams/forth/rahimyAMS49-52-2010.pdf

http://dea.iquanta.info/publications/2003a_hilfer.pdf

54 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.11. FUTURE IDEAS

mi ui

ci−1 f(t) ci

ki−1 ki

Figure 3.17: Schematic of inﬁnitely nested masses

3.11.5 Fractional oscillation 3.11.6 Inﬁnitely nested masses

The governing equation for the outermost mass 1 is

u˙ 1 = v1, m v˙ + c (v v ) + k (u u ) = f(t). 1 1 { 1 1 − 2 } { 1 1 − 2 } while for the other masses

u˙ i = vi,

m v˙ + c − (v v − ) + c (v v ) + k − (u u − ) + k (u u ) = 0. i i { i 1 i − i 1 i i − i+1 } { i 1 i − i 1 i i − i+1 } for i = 2, 3,.... These can be wrtten as

u˙ = v, Mv˙ + cv + ku = 0, where

u1 v1 u = u2 , v = v2 ,  .   .  . .         m1 0 0 . . . 0 m2 0 . . .   M = 0 0 m3 0 . . . ,  . .   . 0 0 ..     c c 0 0 . . . 1 − 1 c1 c1 + c2 c2 0 . . . c = −0 c c −+ c c 0 . . . , − 2 2 3 − 3  . .   . ..      55 c Mihir Sen, 2014 ° 3.11. FUTURE IDEAS CHAPTER 3. APPLICATIONS

k k 0 . . . 1 − 1 k1 k1 + k2 k2 0 . . . k = −0 k k −+ k k 0 . . . , − 2 2 3 − 3  . .   . ..     

Assume

i−1 mi = m/2 , i−1 ci = 2 c, i−1 ki = 2 k. then 1 0 0 . . . 0 2−1 0 . . . M = m 0 0 2−2 0 . . . , . .  . 0 0 ..    1 1 0 0 . . . 1 1+2− 1 21 0 . . . c = c −0 21 21−+ 22 22 0 . . . , − −  . .   . ..     1 1 0 0 . . .  1 1+2− 1 21 0 . . . k = k −0 21 21−+ 22 22 0 . . . − −  . .   . ..      Total mass m m m M = m + + + + ..., 2 4 8 1 m m = m + m + + + . . . , 2 2 4 Mh i = m + , 2 so that M = 2m. Question: If

α MDt (u1) = f(t),

α what is Dt ?

3.11.7 Self-similar networks Fig. 3.18.

56 c Mihir Sen, 2014 ° CHAPTER 3. APPLICATIONS 3.11. FUTURE IDEAS

Figure 3.18: Schematic of self-similar network.

57 c Mihir Sen, 2014 ° 3.11. FUTURE IDEAS CHAPTER 3. APPLICATIONS

58 c Mihir Sen, 2014 ° Appendix A

Appendix

A.1 Continued fractions

A continued fraction is 1 x = a + 0 1 a + 1 1 a + 2 1 a3 + a4 + . . . ¥ Example

Find x, where 1 x = . 1 1+ 1 1+ 1 1+ 1 1+ ··· The part in red is simply x, so that 1 x = , 1+ x x2 + x 1 = 0. − Thus 1+ √5 − golden mean 8 2 x = > > <> 1 √5 − − 2 > > :> A function f(x) can be expanded in a Taylor series

f ′′(0) f ′′′(0) f(x) = f(0) + f ′(0)x + x2 + x2 + ..., (A.1) 2! 3!

59 A.2. FACTORIAL AND GAMMA FUNCTIONS APPENDIX A. APPENDIX

or, can be written as

1 f(x) = a + , 0 1 a + 1 1 a2 + a3 + . . .

where

a0 = f(0). 1 a = . 1 f ′(0)x ′′ 1 1 f (0) 2 a2 = 2 x . −a1 − a1 − 2! . .

So, every Taylor series can be written as a continued fraction.

A.2 Factorial and Gamma functions

The factorial of a non-negative integer n is deﬁned as

n n! = k, iY=k and 0! = 1. Thus

n! = n (n 1)! − The gamma function is deﬁned as

∞ Γ(x) = yx−1e−y dy. Z0 for x 0. Integration by parts shows that ≥ ∞ Γ(x + 1) = yxe−y dy, 0 Z ∞ y= ∞ = yxe−y +x yx−1e−y dy, − ¯ 0 ¯y=0 Z ¯ =0¯ ¯ = x Γ(x). | {z } For integers then Γ(n) = (n 1)!. − 60 c Mihir Sen, 2014 ° APPENDIX A. APPENDIX A.3. CAUCHY’S FORMULA FOR REPEATED INTEGRATION

Figure A.1: Gamma function.

A.3 Cauchy’s formula for repeated integration

Starting from a function f(x), we repeatedly integrate as

x 1 I f(x) = f(y1) dy1, a Z x y1 2 I f(x) = f(y2) dy2 dy1, a a Z x ³ Z y1 y2 ´ 3 I f(x) = f(y3) dy3 dy2 dy1, Za Za Za . ³ ³ ´ ´ . so that

x y1 yn−1 n I f(x) = . . . f(yn) dyn ...dy1. Za Za Za n integrals Cauchy’s formula is that | {z } 1 x Inf(x) = (x y)n−1f(y) dy. (n 1)! − − Za The proof is by induction. For n = 1, this is x 1 I f(x) = f(y1) dy1, Za 61 c Mihir Sen, 2014 ° A.3. CAUCHY’S FORMULA FOR REPEATED INTEGRATION APPENDIX A. APPENDIX

which is true by deﬁnition. Then we should show that, if it is true for n, then it must be true for n + 1.

http://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration

¥ Example

For f(x)= x, we have

I0f(x)= x, x 1 I f(x)= y1 dy1, Za 1 = x2 a2 , 2 − x 2 ` 1 2 ´ 2 I f(x)= y2 a dy2 2 − Za 1 ` ´ = x3 a3 a2 (x a) , 6 − − − x 3 ` 1 3´ 3 2 I f(x)= y3 a a (y3 a) dy3, a 6 − − − Z “ ” 1 ` ´a3 a2 = x4 a4 (x a) x2 a2 + a3 (x a) , 24 − − 6 − − 2 − − and so on. ` ´ ` ´

The algebra is simpler if, without any loss of generality, we move our x-coordinate to make a = 0. Then

I0f(x)= x, x2 I1f(x)= , 2 x3 I2f(x)= , 6 x4 I3f(x)= . 24

Cauchy’s formula for repeated integration for a = 0 and n = 1, 2, 3,... becomes x I1f(x)= f(y) dy, Z0 x2 = , 2 x I2f(x)= (x y)y dy, 0 − Z y=x y=x y2 y3 = x , 2 ˛ − 3 ˛ ˛y=0 ˛y=0 ˛ ˛ x3 ˛ ˛ = , ˛ ˛ 6 1 x I3f(x)= (x y)2y dy, 2 − Z0 1 x = (x2 2xy + y2)y dy, 2 − Z0 y=x y=x y=x 1 y2 y3 y4 = x2 2x + , 2 2 2 ˛ − 3 ˛ 4 ˛ 3 ˛y=0 ˛y=0 ˛y=0 ˛ ˛ ˛ 4 ˛ ˛ ˛ 5 ˛ ˛ ˛ 62 c Mihir Sen, 2014 ° APPENDIX A. APPENDIX A.4. BINOMIAL COEFFICIENTS

x4 = , 24 . .

A.4 Binomial coeﬃcients

n For any set containing n elements, k , which is read as n choose k, is the number of distinct k-element subsets of it that can be formed. It also occurs in the binomial theorem ¡ ¢ n n (x + y)n = xn−kyk, k kX=0 µ ¶ where n n! = for 0 k n. k k! (n k)! ≤ ≤ µ ¶ − A.5 Laplace transform

The Laplace transform F (s) of the function f(t) is given by

∞ f˜(s) = e−st f(t) dt. Z0 For integer n, the Laplace transform of dnf/dtn is

sn + initial conditions ∼

63 c Mihir Sen, 2014 ° A.5. LAPLACE TRANSFORM APPENDIX A. APPENDIX

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