Introduction to fractional calculus (Based on lectures by R. Goren‡o, F. Mainardi and I. Podlubny) R. Vilela Mendes July 2008 () July2008 1/44 Contents - Historical origins of fractional calculus - Fractional integral according to Riemann-Liouville - Caputo fractional derivative - Riesz-Feller fractional derivative - Grünwal-Letnikov - Integral equations - Relaxation and oscillation equations - Fractional di¤usion equation - A nonlinear fractional di¤erential equation. Stochastic solution - Geometrical interpretation of fractional integration () July2008 2/44 Fractional Calculuswas born in1695 n fd dt n What if the order will be n = ½? It will lead to a paradox, from which one day useful consequences will be Start drawn. JJ II G.F.A. de L’Hôpital G.W. Leibniz J I 10 / 90 (1661–1704) (1646–1716) Back Full screen Close End G. W. Leibniz (1695{1697) In the letters to J. Wallis and J. Bernulli (in 1697) Leibniz mentioned the possible approach to fractional-order differ- entiation in that sense, that for non-integer values of n the definition could be the following: dnemx = mnemx; n dx Start JJ II J I 11 / 90 Back Full screen Close End L. Euler (1730) n m d x m n = m(m 1) ::: (m n + 1)x − dxn − − Γ(m + 1) = m(m 1) ::: (m n + 1) Γ(m n + 1) n m − − − d x Γ(m + 1) m n = x − : dxn Γ(m n + 1) − Euler suggested to use this relationship also for negative or 1 non-integer (rational) values of n. Taking m = 1 and n = 2, Euler obtained: Start d1=2x 4x 2 JJ II = r = x1=2 dx1=2 π pπ J I 12 / 90 Back Full screen Close End S. F. Lacroix adopted Euler's derivation for his success- ful textbook ( Trait´edu Calcul Diff´erentiel et du Calcul Int´egral, Courcier, Paris, t. 3, 1819; pp. 409{410). Start JJ II J I 13 / 90 Back Full screen Close End J. B. J. Fourier (1820{1822) The first step to generalization of the notion of differentia- tion for arbitrary functions was done by J. B. J. Fourier (Th´eorieAnalytique de la Chaleur, Didot, Paris, 1822; pp. 499{508). After introducing his famous formula 1 1 1 f(x) = Z f(z)dz Z cos (px pz)dp; 2π − −∞ −∞ Fourier made a remark that Start dnf(x) 1 1 1 π = Z f(z)dzZ cos (px pz + n )dp; JJ II dxn 2π − 2 J I 14 / 90 −∞ −∞ Back and this relationship could serve as a definition of the n-th Full screen order derivative for non-integer n. Close End RiemannಥLiouville definition t d n fτ d τ Dα f t = 1 § · () a t () ¨ ¸ α−n + Γ()n − α dt ³ t − τ 1 © ¹ a () (n− 1 ≤ α < n ) Start JJ II G.F.B. Riemann J. Liouville J I (1826–1866) (1809–1882) 21 / 90 Back Full screen Close End Extend to any positive real value by using the Gamma function, (n 1)! = Γ(n) Fractional Integral of order α>0 (right-sided) t α 1 α 1 Ja+ f (t) := (t τ) f (τ) dτ , α R (2) Γ (α) a Z 2 0 0 De…ne Ja+ := I , Ja+ f (t) = f (t) Fractional integral according to Riemann-Liouville According to Riemann-Liouville the notion of fractional integral of order α (α > 0) for a function f (t), is a natural consequence of the well known formula (Cauchy-Dirichlet ?), that reduces the calculation of the n fold primitive of a function f (t) to a single integral of convolution type t n 1 n 1 Ja+f (t) := (t τ) f (τ) dτ , n N (1) (n 1)! a 2 Z vanishes at t = a with its derivatives of order 1, 2, ... , n 1 . Require n f (t) and Ja+f (t) to be causal functions, that is, vanishing for t < 0 . () July2008 3/44 Fractional Integral of order α>0 (right-sided) t α 1 α 1 Ja+ f (t) := (t τ) f (τ) dτ , α R (2) Γ (α) a Z 2 0 0 De…ne Ja+ := I , Ja+ f (t) = f (t) Fractional integral according to Riemann-Liouville According to Riemann-Liouville the notion of fractional integral of order α (α > 0) for a function f (t), is a natural consequence of the well known formula (Cauchy-Dirichlet ?), that reduces the calculation of the n fold primitive of a function f (t) to a single integral of convolution type t n 1 n 1 Ja+f (t) := (t τ) f (τ) dτ , n N (1) (n 1)! a 2 Z vanishes at t = a with its derivatives of order 1, 2, ... , n 1 . Require n f (t) and Ja+f (t) to be causal functions, that is, vanishing for t < 0 . Extend to any positive real value by using the Gamma function, (n 1)! = Γ(n) () July2008 3/44 Fractional integral according to Riemann-Liouville According to Riemann-Liouville the notion of fractional integral of order α (α > 0) for a function f (t), is a natural consequence of the well known formula (Cauchy-Dirichlet ?), that reduces the calculation of the n fold primitive of a function f (t) to a single integral of convolution type t n 1 n 1 Ja+f (t) := (t τ) f (τ) dτ , n N (1) (n 1)! a 2 Z vanishes at t = a with its derivatives of order 1, 2, ... , n 1 . Require n f (t) and Ja+f (t) to be causal functions, that is, vanishing for t < 0 . Extend to any positive real value by using the Gamma function, (n 1)! = Γ(n) Fractional Integral of order α>0 (right-sided) t α 1 α 1 Ja+ f (t) := (t τ) f (τ) dτ , α R (2) Γ (α) a Z 2 0 0 De…ne Ja+ := I , Ja+ f (t) = f (t) () July2008 3/44 Let α α J := J0+ Semigroup properties JαJ β = Jα+β , α , β 0 Commutative property J βJα = JαJ β E¤ect on power functions Jαtγ = Γ(γ+1) tγ+α , α > 0 , γ > 1 , t > 0 Γ(γ+1+α) (Natural generalization of the positive integer properties). Introduce the following causal function (vanishing for t < 0 ) α 1 t Φ (t) := + , α > 0 α Γ(α) Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) b α 1 α 1 Jb f (t) := (τ t) f (τ) dτ , α R Γ (α) t Z 2 (a = 0, b = +∞) Riemann (a = ∞, b = +∞) Liouville () July2008 4/44 Introduce the following causal function (vanishing for t < 0 ) α 1 t Φ (t) := + , α > 0 α Γ(α) Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) b α 1 α 1 Jb f (t) := (τ t) f (τ) dτ , α R Γ (α) t Z 2 (a = 0, b = +∞) Riemann (a = ∞, b = +∞) Liouville Let α α J := J0+ Semigroup properties JαJ β = Jα+β , α , β 0 Commutative property J βJα = JαJ β E¤ect on power functions Jαtγ = Γ(γ+1) tγ+α , α > 0 , γ > 1 , t > 0 Γ(γ+1+α) (Natural generalization of the positive integer properties). () July2008 4/44 Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) b α 1 α 1 Jb f (t) := (τ t) f (τ) dτ , α R Γ (α) t Z 2 (a = 0, b = +∞) Riemann (a = ∞, b = +∞) Liouville Let α α J := J0+ Semigroup properties JαJ β = Jα+β , α , β 0 Commutative property J βJα = JαJ β E¤ect on power functions Jαtγ = Γ(γ+1) tγ+α , α > 0 , γ > 1 , t > 0 Γ(γ+1+α) (Natural generalization of the positive integer properties). Introduce the following causal function (vanishing for t < 0 ) α 1 t Φ (t) := + , α > 0 α Γ(α) () July2008 4/44 Laplace transform ∞ st f (t) := e f (t) dt = f (s) , s C L f g Z0 2 De…ning the Laplace transform pairs by f (et) f (s) f (s) e Jα f (t) , α > 0 sα e Fractional integral according to Riemann-Liouville Φ (t) Φ (t) = Φ + (t) , α , β > 0 α β α β Jα f (t) = Φ (t) f (t) , α > 0 α () July2008 5/44 De…ning the Laplace transform pairs by f (t) f (s) f (s) e Jα f (t) , α > 0 sα e Fractional integral according to Riemann-Liouville Φ (t) Φ (t) = Φ + (t) , α , β > 0 α β α β Jα f (t) = Φ (t) f (t) , α > 0 α Laplace transform ∞ st f (t) := e f (t) dt = f (s) , s C L f g Z0 2 e () July2008 5/44 Fractional integral according to Riemann-Liouville Φ (t) Φ (t) = Φ + (t) , α , β > 0 α β α β Jα f (t) = Φ (t) f (t) , α > 0 α Laplace transform ∞ st f (t) := e f (t) dt = f (s) , s C L f g Z0 2 De…ning the Laplace transform pairs by f (et) f (s) f (s) e Jα f (t) , α > 0 sα e () July2008 5/44 Then, de…ne Dα as a left-inverse to Jα. With a positive integer m , m 1 < α m , de…ne: α m m α Fractional Derivative of order α : D f (t) := D J f (t) d m 1 t f (τ) α dtm α+1 m dτ , m 1 < α < m D f (t) := Γ(m α) 0 (t τ) d m ( h dtm fR(t) , i α = m Fractional derivative according to Riemann-Liouville Denote by Dn with n N , the derivative of order n .