Historical origins of fractional calculus. # Fractional integral according to Riemann#Liouville. # Caputo fractional derivative. # Riesz#Feller fractional derivative.
Introduction to fractional calculus (Based on lectures by R. Goren‡o, F. Mainardi and I. Podlubny) R. Vilela Mendes
July 2008
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Contents
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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
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Fractional Calculus was born in 1695 What if the order will be
dn f dt n
n = ½? It will lead to a paradox, from which one day useful consequences will be drawn.
G.F.A. de L’Hôpital (1661–1704)
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G.W. Leibniz (1646–1716)
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G. W. Leibniz (1695–1697) In the letters to J. Wallis and J. Bernulli (in 1697) Leibniz mentioned the possible approach to fractional-order differentiation in that sense, that for non-integer values of n the definition could be the following: dnemx n mx = m e , dxn
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L. Euler (1730) dnxm = m(m − 1) . . . (m − n + 1)xm−n n dx Γ(m + 1) = m(m − 1) . . . (m − n + 1) Γ(m − n + 1) dnxm Γ(m + 1) m−n = x . dxn Γ(m − n + 1) Euler suggested to use this relationship also for negative or non-integer (rational) values of n. Taking m = 1 and n = 21 , Euler obtained: r 1/2 d x 4x 2 1/2 √ = x = dx1/2 π π
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S. F. Lacroix adopted Euler’s derivation for his successful textbook ( Trait´e du Calcul Diff´erentiel et du Calcul Int´egral, Courcier, Paris, t. 3, 1819; pp. 409–410).
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J. B. J. Fourier (1820–1822) The first step to generalization of the notion of differentiation for arbitrary functions was done by J. B. J. Fourier (Th´eorie Analytique de la Chaleur, Didot, Paris, 1822; pp. 499–508). After introducing his famous formula f (x) =
1 2π
Z∞
f (z)dz
−∞
Z∞
cos (px − pz)dp,
−∞
Fourier made a remark that Z∞ Z∞ n d f (x) 1 π = f (z)dz cos (px − pz + n )dp, dxn 2π 2 −∞
−∞
and this relationship could serve as a definition of the n-th order derivative for non-integer n.
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RiemannಥLiouville definition 1 §d· α = D f ( t ) ¨ ¸ a t Γ(n − α) © dt ¹
n t
³ a
f ( τ) dτ (t − τ) α −n +1
( n − 1 ≤ α < n)
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G.F.B. Riemann J. Liouville (1826–1866) (1809–1882)
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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 Z t 1 n (t τ )n 1 f ( τ ) d τ , n 2 N (1) Ja + f ( t ) : = (n 1) ! a vanishes at t = a with its derivatives of order 1, 2, . . . , n 1 . Require f (t ) and Jan+ f (t ) to be causal functions, that is, vanishing for t < 0 .
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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 Z t 1 n (t τ )n 1 f ( τ ) d τ , n 2 N (1) Ja + f ( t ) : = (n 1) ! a vanishes at t = a with its derivatives of order 1, 2, . . . , n 1 . Require f (t ) and Jan+ 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 )
()
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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 Z t 1 n (t τ )n 1 f ( τ ) d τ , n 2 N (1) Ja + f ( t ) : = (n 1) ! a vanishes at t = a with its derivatives of order 1, 2, . . . , n 1 . Require f (t ) and Jan+ 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) Jaα+ f (t ) :=
1 Γ (α)
Z t a
(t
τ )α
1
f (τ ) d τ ,
α2R
(2)
De…ne Ja0+ := I , Ja0+ f (t ) = f (t ) ()
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Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) Jbα f (t ) :=
1 Γ (α)
Z b t
(a = 0, b = +∞) Riemann
()
(τ
t )α
(a =
1
f (τ ) d τ ,
α2R
∞, b = +∞) Liouville
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Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) Jbα f (t ) :=
1 Γ (α)
Z b t
(a = 0, b = +∞) Riemann Let
(τ
t )α
(a =
1
f (τ ) d τ ,
α2R
∞, b = +∞) Liouville
J α := J0α+ Semigroup properties J α J β = J α+ β , α,β 0 Commutative property J βJ α = J αJ β E¤ect on power functions Γ ( γ +1 ) J α t γ = Γ ( γ +1 + α ) t γ + α , α > 0,γ > 1,t > 0 (Natural generalization of the positive integer properties).
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Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) Jbα f (t ) :=
1 Γ (α)
Z b t
(τ
(a = 0, b = +∞) Riemann Let
t )α
1
(a =
f (τ ) d τ ,
α2R
∞, b = +∞) Liouville
J α := J0α+ Semigroup properties J α J β = J α+ β , α,β 0 Commutative property J βJ α = J αJ β E¤ect on power functions Γ ( γ +1 ) J α t γ = Γ ( γ +1 + α ) t γ + α , α > 0,γ > 1,t > 0 (Natural generalization of the positive integer properties). Introduce the following causal function (vanishing for t < 0 ) Φ α (t ) : = ()
α 1 t+ , Γ(α)
α>0 July 2008
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Fractional integral according to Riemann-Liouville
Φ α (t )
Φ β (t ) = Φ α + β (t ) ,
J α f (t ) = Φ α (t )
()
f (t ) ,
α,β > 0 α>0
July 2008
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Fractional integral according to Riemann-Liouville
Φ α (t )
Φ β (t ) = Φ α + β (t ) ,
J α f (t ) = Φ α (t )
f (t ) ,
α,β > 0 α>0
Laplace transform
L ff (t )g :=
()
Z ∞
e
0
st
f (t ) dt = e f (s ) ,
s2C
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Fractional integral according to Riemann-Liouville
Φ α (t )
Φ β (t ) = Φ α + β (t ) ,
J α f (t ) = Φ α (t )
α,β > 0
f (t ) ,
α>0
Laplace transform
L ff (t )g :=
Z ∞
e
0
st
f (t ) dt = e f (s ) ,
De…ning the Laplace transform pairs by f (t ) J α f (t )
()
e f (s ) , sα
s2C
e f (s )
α>0
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Fractional derivative according to Riemann-Liouville Denote by D n with n 2 N , the derivative of order n . Note that Dn Jn = I ,
n2N
J n D n 6= I ,
D n is a left-inverse (not a right-inverse) to J n . In fact J n D n f (t ) = f (t )
n 1
tk
∑ f (k ) (0+ ) k! ,
t>0
k =0
()
July 2008
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Fractional derivative according to Riemann-Liouville Denote by D n with n 2 N , the derivative of order n . Note that Dn Jn = I ,
n2N
J n D n 6= I ,
D n is a left-inverse (not a right-inverse) to J n . In fact J n D n f (t ) = f (t )
n 1
tk
∑ f (k ) (0+ ) k! ,
t>0
k =0
Then, de…ne D α as a left-inverse to J α . With a positive integer m , m 1 < α m , de…ne: Fractional Derivative of order α : D α f (t ) : = D m J m α f (t ) i ( m h Rt f (τ ) d 1 d τ , m 1
1,t > 0
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Fractional derivative according to Riemann-Liouville
De…ne D 0 = J 0 = I . Then D α J α = I , α Dα tγ =
0
Γ ( γ + 1) tγ Γ(γ + 1 α)
α
,
α > 0,γ >
1,t > 0
The fractional derivative D α f is not zero for the constant function f (t ) 1 if α 62 N D α1 = Is
t Γ (1
α
α)
,
α
0,t > 0
0 for α 2 N, due to the poles of the gamma function
()
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Caputo fractional derivative D α f (t ) : = J m D α f (t ) : =
()
D m f (t ) with m 1 < α ( R t f (m ) ( τ ) 1 α+1 m d τ, Γ (m α ) 0
α
dm dt m f
(t τ )
(t ) ,
m , namely m
1 0
D α is not a right-inverse to J α J αD αt α
()
1
0,
but
D αJ αt α
1
= tα
1
,
α > 0,t > 0
July 2008
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Riemann versus Caputo D αt α
1
0,
α > 0,t > 0
D α is not a right-inverse to J α J αD αt α
1
0,
but
D αJ αt α
1
= tα
1
α > 0,t > 0
,
Functions which for t > 0 have the same fractional derivative of order α , with m 1 < α m . (the cj ’s are arbitrary constants) m
∑ cj t α
D α f (t ) = D α g (t ) () f (t ) = g (t ) +
j =1
D α f (t ) = D α g (t ) () f (t ) = g (t ) +
∑ cj t m
m
()
j
j
j =1
July 2008
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Riemann versus Caputo D αt α
1
0,
α > 0,t > 0
D α is not a right-inverse to J α J αD αt α
1
0,
D αJ αt α
but
1
= tα
1
α > 0,t > 0
,
Functions which for t > 0 have the same fractional derivative of order α , with m 1 < α m . (the cj ’s are arbitrary constants) m
∑ cj t α
D α f (t ) = D α g (t ) () f (t ) = g (t ) +
j =1
D α f (t ) = D α g (t ) () f (t ) = g (t ) +
∑ cj t m
m
Formal limit as α ! (m α ! (m
α ! (m ()
1)
+
1)
j
j
j =1
+
1)+ =) D α f (t ) ! D m J f (t ) = D m
=) D α f (t ) ! J D m f (t ) = D m
1
f (t )
1
f (t ) f (m July 2008
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Riemann versus Caputo The Laplace transform D α f (t )
sα e f (s )
m 1
∑
D k J (m
α)
f (0+ ) s m
1 k
,
m
1
