Multi-dimensional fractional wave equation and some ...€¦ · 2. Fundamental solution of the...

$Page 1: Multi-dimensional fractional wave equation and some ...€¦ · 2. Fundamental solution of the fractional wave equation Fundamental solution G ;n In this talk, we are mostly interested$
Multi-dimensional fractional wave equation and some

properties of its fundamental solution

Yuri Luchko

Department of Mathematics, Physics, and Chemistry

Beuth Technical University of Applied Sciences Berlin

Berlin, Germany

Fractional Calculus, Probability and Non-local Operators:

Applications and Recent Developments

Basque Center for Applied Mathematics

November 6-8, 2013

Bilbao - Basque Country - Spain

Outline of the talk:

• Initial value problem for a multi-dimensional fractional wave equation

• Fundamental solution of the multi-dimensional fractional wave

equation and its properties

• Interpretation of the fundamental solution as a damped wave

• Plots and some open problems

Yu. Luchko (BHT Berlin) Fractional wave equation 2 / 32

Professor Mainardi's contribution to the topic

• R. Goren�o, Yu. Luchko, and F.Mainardi: Analytical properties and

applications of the Wright function. Fract. Calc. Appl. Anal. 2(1999),

383-415 (118 scholar.google citations).• R. Goren�o, Yu. Luchko, and F.Mainardi: Wright functions as

scale-invariant solutions of the di�usion-wave equation. Journal of

Computational and Applied Mathematics 11(2000), 175-191 (152

scholar.google citations).• F. Mainardi, Yu. Luchko, and G.Pagnini: The fundamental solution of

the space-time fractional di�usion equation. Fract. Calc. Appl. Anal. 4

(2001), 153-192 (450 scholar.google citations).• Yu. Luchko, F. Mainardi, and Yu. Povstenko: Propagation speed of the

maximum of the fundamental solution to the fractional di�usion-wave

equation. Computers and Mathematics with Applications 66(2013),

774-784.• Yu. Luchko, F. Mainardi: Some properties of the fundamental solution

to the signalling problem for the fractional di�usion-wave equation.

Central European Journal of Physics 11(2013), 666-675.Yu. Luchko (BHT Berlin) Fractional wave equation 3 / 32

1. Multi-dimensional fractional wave equation

Initial value problem for the fractional wave equation

We deal with the initial-value problem

u(x , 0) = ϕ(x) ,∂u

∂t(x , 0) = 0, x ∈ IRn

for the model multi-dimensional fractional wave equation

(neutral-fractional di�usion equation)

Dαt u(x , t) = −(−∆)

α2 u(x , t), x ∈ IRn , t ∈ IR+, 1 ≤ α ≤ 2.

In the equation, all quantities are supposed to be dimensionless, so that the

coe�cient by the Riesz space-fractional derivative can be taken to be equal

to one without loss of generality.



Caputo fractional derivative

In what follows, we employ the Caputo time-fractional derivative Dα of

order α:

(Dαf )(t) := (I n−αf (n))(t), n − 1 < α ≤ n, n ∈ IN

Iα, α ≥ 0 being the Riemann-Liouville fractional integral

(Iαf )(t) :=

{1

Γ(α)

∫ t

0(t − τ)α−1f (τ) dτ, α > 0,

f (t), α = 0.



Riesz fractional derivative

For a su�ciently well-behaved function f , the Riesz fractional derivative of

order α, 0 < α ≤ 2 is de�ned as a pseudo-di�erential operator with the

symbol −|κ|α:(F − (−∆)

α2 f )(κ) := −|κ|α(F f )(κ)

that can be represented as a hypersingular integral

−(−∆)α2 f (x) = − 1

dn,l (α)

∫IRn

(∆lhf )(x)

|h|n+αdh

with (∆lhf )(x) =

∑lk=0(−1)k

(lk

)f (x − kh).



Fractional wave equation

Mathematical theory:

R. Goren�o, A. Iskenderov, and Yu. Luchko: Mapping between solutions of

fractional di�usion-wave equations. Fract. Calc. Appl. Anal. 3(2000), 75-86.

F. Mainardi, Yu. Luchko, and G. Pagnini: The fundamental solution of the

space-time fractional di�usion equation. Fract. Calc. Appl. Anal. 4 (2001),

153-192.

R. Metzler, T.F. Nonnenmacher: Space- and time-fractional di�usion and

wave equations, fractional Fokker-Planck equations, and physical

motivation. Chemical Physics 284(2002), 67-90.

Yu. Luchko: Fractional wave equation and damped waves. J. Math. Phys.

54(2013), 031505.



Fractional wave equation

Applications:

G. Gudehus, A. Touplikiotis: Clasmatic seismodynamics - Oxymoron or

pleonasm? Soil Dyn. Earthq. Eng. 38 (2012), 1-14.

S.P. Näsholm, S. Holm: On a fractional Zener elastic wave equation. Fract.

Calc. Appl. Anal. 16 (2013), 26-50.

Yu. Luchko: Fractional wave equation and damped waves. J. Math. Phys.

54(2013), 031505.


2. Fundamental solution of the fractional wave equation

Fundamental solution Gα, nIn this talk, we are mostly interested in behavior and properties of the

fundamental solution (Green function) Gα, n of the equation, i.e. its solution

with the initial condition ϕ(x) = δ(x), δ being the Dirac delta function.

Application of the Fourier transform to the fractional wave equation leads

to the initial-value problem {Gα, n(κ, 0) = 1,∂Gα, n∂t (κ, 0) = 0

for the fractional di�erential equation

(DαGα, n)(t) + |κ|αGα, n(κ, t) = 0

with the unique solution

Gα, n(κ, t) = Eα(−|κ|αtα)

in terms of the Mittag-Le�er function Eα(z) =∑∞

k=0zk

Γ(1+αk) , α > 0.



Fundamental solution Gα, nInverse Fourier transform leads to the representation

Gα, n(x , t) =1

(2π)n

∫IRn

e−iκ·x Eα(−|κ|αtα) dκ, x ∈ IRn, t > 0.

But Eα(−|κ|αtα) is a radial function (spherically symmetric function) in κand thus

Gα, n(x , t) = (2π)−n/2|x |1−n/2∫ ∞0

τn/2 Eα(−ταtα) Jn/2−1(τ |x |) dτ, x 6= 0

with the Bessel function Jn/2−1 (see e.g. Samko, Kilbas, and Marichev:

Fractional Integrals and Derivatives, Gordon and Breach, 1993).

In the one-dimensional case (n = 1) we get:

Gα, 1(x , t) =1

π

∫ ∞0

Eα(−ταtα) cos(τ |x |) dτ, x ∈ IR , t > 0

because of the formula

J−1/2(x) =

√2

πzcos(z).



Fundamental solution Gα, nTaking into account the relation

Jν(z) =(z2

)νW1,ν+1

(−z

2

4

),

where Wα,β(z) is the Wright function

Wα,β(z) =∞∑

m=0

zm

m! Γ(αm + β), α < −1, β ∈ IR ,

we get the representation (x 6= 0)

Gα, n(x , t) =2

(4π)n/2

∫ ∞0

τn−1 Eα(−ταtα)W1,n/2−1

(−14τ2|x |2

)dτ

of Gα, n in terms of the two most important special functions of FC: the

Mittag-Le�er function Eα and the Wright function Wα,β .



Fundamental solution Gα, n

The representation

Gα, n(x , t) = (2π)−n/2|x |1−n/2∫ ∞0

τn/2 Eα(−ταtα) Jn/2−1(τ |x |) dτ

is valid under the conditions

0 < n < 2α + 1, x 6= 0 or 0 < n < α, x ∈ IRn

that follow from the asymptotical behavior of the Mittag-Le�er and the

Bessel functions at z = 0 and z = +∞ and guarantee the (conditional)

convergence of the integral at the RHS.

In our case 1 < α < 2, so that

n = 1, 2, 3

satisfy the conditions 0 < n < 2α + 1.



Fundamental solution Gα, n with n = 1, 21) n = 1:

Gα, 1(x , t) =1

π

∫ ∞0

Eα(−ταtα) cos(τ |x |) dτ

can be represented in the form:

Gα, 1(x , t) =1

π

|x |α−1tα sin(πα/2)

t2α + 2|x |αtα cos(πα/2) + |x |2α, x ∈ IR , t > 0.

(Derivation method: Mellin transform, Mellin-Barnes integral

representation, series representation of the Mellin-Barnes integral).

1) n = 2:

Gα, 2(x , t) =1

2π

∫ ∞0

τ Eα(−ταtα) J0(τ |x |) dτ, x 6= 0, t > 0,

where

J0(z) =

∫ ∞0

cos(z sin(φ)) dφ.



Fundamental solution Gα, n with n = 3

3) n = 3 (x 6= 0, t > 0):

Gα, 3(x , t) = (2π)−3/2|x |−1/2∫ ∞0

τ3/2 Eα(−ταtα) J1/2(τ |x |) dτ

can be represented in the form

Gα, 3(x , t) =1

2π2|x |

∫ ∞0

Eα(−ταtα) τ sin(τx) dτ.

because of the formula

J1/2(x) =

√2

πzsin(z).



Fundamental solution Gα, n with n = 3

Comparing the formulas

Gα, 1(x , t) =1

π

∫ ∞0

Eα(−ταtα) cos(τ |x |) dτ

and

Gα, 3(x , t) =1

2π2|x |

∫ ∞0

Eα(−ταtα) τ sin(τ |x |) dτ,

we get the relation (x 6= 0)

Gα, 3(x , t) = − 1

2π|x |∂

∂|x |Gα,1(x , t).

(Case of the fractional di�usion equation with 0 < α < 1, 0 < β < 2,

β 6= 1 was considered in A. Hanyga, Multi-dimensional solutions of

space-time-fractional di�usion equations, Proc. R. Soc. London. A 2002,

458, 429-450).



Properties of the fundamental solution Gα, 3

Surprisingly, there exists another formula that connects Gα, 3 and Gα, 1,

namely,

Gα, 3(x , t) =1

2π|x |2

(Gα,1(x , t) + t

∂

∂tGα,1(x , t)

), x 6= 0.

To prove the formula, let us �rst derive a Mellin-Barnes integral

representation for Gα,n(x , t).For x 6= 0, the RHS of the representation

Gα, n(x , t) = (2π)−n/2|x |1−n/2∫ ∞0

τn/2 Eα(−ταtα) Jn/2−1(τ |x |) dτ

can be interpreted as the Mellin convolution of the functions Eα(−tατα)and τ−n/2−1Jn/2−1(1/τ) at the point y = 1/x .



Mellin-Barnes representation of Gα,n

With the Mellin integral transform technique (Mellin transforms of the

Mittag-Le�er and the Bessel functions, some elementary properties of the

Mellin transform, and the Mellin convolution theorem) we get the

representation (x 6= 0)

Gα,n(x , t) =1

απn/2|x |n1

2πi

∫L

Γ(sα

)Γ(1− s

α

)Γ(n2− s

2

)Γ(1− s)2sΓ

(s2

) (t

|x |

)−sds

of the fundamental solution Gα,n in terms of the Mellin-Barnes integral

(Fox H-function).



Connection between Gα,3 and Gα,1Because of the factor

Γ(n2− s

2

),

the Mellin-Barnes representation of Gα,n has a di�erent structure for the

even (n = 2, 4, . . . ) and the odd (n = 1, 3, . . . ) dimensions.In particular, for n = 3 we have the relation (Γ(1 + z) = z Γ(z))

Γ

(3

2− s

2

)=

(1

2− s

2

)Γ

(1

2− s

2

)that connects the kernel of the Mellin-Barnes representation of Gα,3 with

the kernel of Gα,1:

Γ(sα

)Γ(1− s

α

)Γ(32− s

2

)Γ(1− s)2sΓ

(s2

) =

(1

2− s

2

)Γ(sα

)Γ(1− s

α

)Γ(12− s

2

)Γ(1− s)2sΓ

(s2

) .

Elementary properties of the Mellin integral transform lead then to the

relation

Gα, 3(x , t) =1

2π|x |2

(Gα,1(x , t) + t

∂

∂tGα,1(x , t)

), x 6= 0.



Closed form formula for Gα,3Employing the representation

Gα, 1(x , t) =1

π

|x |α−1tα sin(πα/2)

t2α + 2|x |αtα cos(πα/2) + |x |2α, x ∈ IR , t > 0

and the relation

Gα, 3(x , t) =1

2π|x |2

(Gα,1(x , t) + t

∂

∂tGα,1(x , t)

), x 6= 0

or the formula

Gα, 3(x , t) = − 1

2π|x |∂

∂|x |Gα,1(x , t), x 6= 0

we arrive at the nice closed form formula (x 6= 0, t > 0):

Gα, 3(x , t) =sin(πα/2)

2π2

(−(α− 1)t2α + 2|x |αtα cos(πα/2) + (1 + α)|x |2α

)|x |3−αt−α (t2α + 2|x |αtα cos(πα/2) + |x |2α)2

.


3. Fundamental solution as a damped wave

Physical interpretation of Gα,3Of course, Gα,3 is NOT a pdf because it is NOT everywhere

NONNEGATIVE (Gα,1 is a pdf). Let us denote |x | by r . Then we can

restrict our investigation to the function

Gα, 3(r , t) = Gα, 3(|x |, t) = Gα, 3(x , t), r > 0, t > 0

because of the spherical symmetry of the function Gα, 3(x , t).For the function Gα, 3(r , t), the following relations are valid:

Gα, 3(r , t) < 0 for r < r∗(α, t),

Gα, 3(r , t) = 0 for r = r∗(α, t),

Gα, 3(r , t) > 0 for r > r∗(α, t),

where

r∗(α, t) = zα t, zα =

− cos(πα/2) +√α2 − sin2(πα/2)

α + 1

1α

.



Connection between Gα,3 and Gα,1Of course, because of the relation

Gα, 3(r , t) = − 1

2πr

∂

∂rGα,1(r , t), r > 0, t > 0,

the point r∗(α, t) = zα t is the only maximum location of the fundamental

solution Gα,1(r , t) with r > 0, t > 0.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

x

G1.

5

t=0.1

t=0.2

t=0.3

Student Version of MATLAB

Plots of Gα,1(r , t) with α = 3/2.Yu. Luchko (BHT Berlin) Fractional wave equation 21 / 32


Properties of Gα,3From the formula (x 6= 0, t > 0)

Gα, 3(x , t) =sin(πα/2)

2π2

(−(α− 1)t2α + 2|x |αtα cos(πα/2) + (1 + α)|x |2α

)|x |3−αt−α (t2α + 2|x |αtα cos(πα/2) + |x |2α)2

the asymptotics of Gα, 3(r , t) for the �xed t, t > 0 and α, 1 < α < 2 easily

follows:

Gα,3(r , t) = O(rα−3), r → 0,

Gα,3(r , t) = O(r−α−3), r → +∞.Thus the integral (moment of Gα,3(r , t) of the order β)

Iα,β(t) =

∫ ∞0

rβGα,3(r , t) dr

exists for

2− α < β < 2 + α,

i.e., at least for

1 ≤ β ≤ 3 if 1 < α < 2.



Properties of Gα,3Employing the the Mellin-Barnes representation (x 6= 0)

Gα,3(x , t) =1

απ3/2|x |31

2πi

∫L

Γ(sα

)Γ(1− s

α

)Γ(32− s

2

)Γ(1− s)2sΓ

(s2

) (t

|x |

)−sds

that can be interpreted as the inverse Mellin integral transform of its kernel

we get the Mellin integral transform of Gα,3 and thus its �moments� of the

order β, 1 ≤ β ≤ 3

Iα,β(t) =(2t)β−2

απ3/2

Γ(2−βα

)Γ(1− 2−β

α

)Γ(β+12

)Γ(β − 1)Γ

(2−β2

) .

Using the known properties of the Gamma function, we get a simpler

representation

Iα,β(t) =tβ−2(β − 1)

2απ

sin(π β2

)sin(π (2−β)

α

) .Yu. Luchko (BHT Berlin) Fractional wave equation 23 / 32


Properties of Gα,3In particular, from the formula

Iα,β(t) =tβ−2(β − 1)

2απ

sin(π β2

)sin(π (2−β)

α

)we the following important particular cases:

1) β = 1 (mean value):

Iα,1(t) ≡ 0, for all 1 < α < 2, t > 0,

2) β = 2 (2nd �moment�):

Iα,2(t) ≡ 1

4π, for all 1 < α < 2, t > 0,

2) β = 3 (3rd �moment�):

Iα,3(t) ≡ t

απ sin(π/α), for all 1 < α < 2, t > 0.



Scaling properties of Gα,n

It follows from the Mellin-Barnes representation (x 6= 0)

Gα,n(x , t) =1

απn/2|x |n1

2πi

∫L

Γ(sα

)Γ(1− s

α

)Γ(n2− s

2

)Γ(1− s)2sΓ

(s2

) (t

|x |

)−sds

that Gα,n(x , t) can be represented via an auxiliary function that depends on

a single argument:

Gα,n(x , t) = Gα,n(|x |, t) = Gα,n(r , t) = r−n Lα,n

( rt

).

with

Lα,n (r) =1

απn/21

2πi

∫L

Γ(sα

)Γ(1− s

α

)Γ(n2− s

2

)Γ(1− s)2sΓ

(s2

) (r)s ds.




It is evident that if the function r−n Lα,n (r) has an extremum point, say,

r∗ = c(α, n)

then for a �xed t > 0 the fundamental solution can be represented as

Gα,n(r , t) = t−n( rt

)−nLα,n

( rt

)and therefore Gα,n(r , t) has an extremum point at the point

r∗(t, α, n) = r∗ t = c(α, n) t.

For n = 1, the function r−n Lα,n (r) is connected with the stable unimodal

distributions and is unimodal, too, i.e. it has only one maximum point.




The extremum value G ∗α,n(t) of Gα,n(r , t) at the extremum point

r∗(t, α, n) = c(α, n) t.

is equal to

G ∗α,n(t) = (c(α, n) t)−nLα,n (c(α, n)) =1

tn(c(α, n))−nLα,n (c(α, n))

and thus the product

(r∗(t, α, n))n G ∗α,n(t) = Lα,n (c(α, n))

is time independent. In particular, for n = 1 we get the relation

r∗(t, α, 1)G ∗α,1(t) = Lα,1 (c(α, 1)) .


4. Plots and open problems

Gα,3 as a damped wave

t = 0.2

t=0.3

t=0.4

0.2 0.4 0.6 0.8 1.0x

-2

-1

1

2

3

Abbildung: Green function Gα,3(r , t): Plots for α = 1.5 at the time instants

t = 0.2, 0.3, 0, 4

As we can see on the plots, for a �xed value of t, the Green function

Gα,3(r , t) has only one maximum point that depends on the time instant t

(and of course on the value of the parameter α).Yu. Luchko (BHT Berlin) Fractional wave equation 28 / 32


Phase velocity of the damped wavesThe velocity of the maximum location of Gα,3(r , t) (its phase velocity):

vp(α) :=dr∗(t, α, 3)

dt=

d(c(α, 3) t)

dt= c(α, 3),

where c(α, 3) is the location of the maximum point of the fundamental

solution Gα,3(r , t) at the time instant t = 1.

a=1.575, max =1.12

1.2 1.4 1.6 1.8 2.0

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Abbildung: Phase velocity vp(α) of Gα,3(r , t)Yu. Luchko (BHT Berlin) Fractional wave equation 29 / 32


Gα,3 as a damped wave

Finally, some plots of the fundamental solution Gα,3(x , t) with the �xed

α = 1.5 and |x | = r are presented.

r = 0.3

r = 0.5r = 0.7

0.2 0.4 0.6 0.8 1.0 1.2t

-1

1

2

3

4

5

Abbildung: Green function Gα,3(r , t): Plots for α = 1.5 and r = 0.3, 0.5, 0, 7



Open problems

• Determination of other velocities of the damped waves that are

described by the fractional wave equation like the velocity of the

mass-center, the pulse velocity, the group velocity, the

centrovelocities, etc.

• Analysis of the fractional wave equations with fractional derivatives of

other types

• Analysis of the fractional wave equations with the non-constant

coe�cients

• Qualitative behavior of solutions to the non-linear fractional wave

equations


5. Congratulations

Congratulations and best wishes, Francesco!

Abbildung: Berlin, Sept. 2007, visit to Rudolf and Else Goren�o


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