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’Hˆopital1 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.