An interpolation between the Wave and Diffusion equa-
tions through the Fractional Evolution equations Dirac
like

T. Pierantozzi a and L. Vázqueza,b

a Departamento de Matemática Aplicada, Facultad de Informática,
Universidad Complutense de Madrid, Madrid, E-28040 Spain
Teresa Pierantozzi@mat.ucm.es

b Centro de Astrobioloǵıa (CSIC-INTA),
Torrejón de Ardoz, Madrid, 28850 Spain
lvazquez@fdi.ucm.es

(Received: )

Through the Fractional Calculus and following the method used by Dirac to obtain his
well-known equation from the Klein-Gordon equation, we analyze a possible interpola-
tion between the Dirac and the diffusion equations in one space dimension. We study
the transition between the hyperbolic and parabolic behaviors by means of the gener-
alization of the D’Alembert formula for the classical wave equation and the invariance
under space and time inversions of the interpolating Fractional Evolution equations Dirac
like. Such invariance depends on the values of the fractional index and it is related to
the non local property of the time fractional differential operator. For this system of
fractional evolution equations, we also find an associated conserved quantity analogous
of the Hamiltonian for the classical Dirac case.

PACS: 02.30.Gp, 02.50.-r, 03.65.Pm, 44.05.+e, 45.10.Hj.
Keywords: fractional differential equations, Riemann-Liouville fractional integrals

and derivatives, Caputo fractional derivative, Mittag-Leffler and Wright functions, Dirac-
type equations.

1 Introduction

Following the well known Dirac’s approach [19], the free Dirac equation can be considered
as the square root of the Klein-Gordon equation. In a more general context Morinaga
and Nono [11] analyzed the integer s-root of the partial differential equations of the form

∑

|I|=s

aI
∂|I|

∂xI
φ = φ, (1)

by defining them as the first order system

n∑

i=1

αi
∂Φ
∂xi

= Φ (2)

where α1, ...αn are matrices. From the physical point of view the αk describe internal
degrees of freedom of the associated system.

1



In the above context, Vázquez et al. recently considered in [21] and [22] the fractional
diffusion equations with internal degrees of freedom. They can be obtained as the s-roots
of the standard scalar linear diffusion equation. Thus, a possible definition of the square
root of the standard diffusion equation (SDE) in one space dimension, ut − uxx = 0, is
the following:

(A
∂1/2

∂t1/2
+ B

∂

∂x
)ψ(x, t) = 0 (3)

where A and B are 2× 2 matrices satisfying Pauli’s algebra

A2 = I, B2 = −I, AB + BA = 0, (4)

being I the identity operator.
It is worthy of mention that Oldham and Spanier [12] were the first authors to derive

a formulation involving the mathematical operation of semidifferentiation in replacement
of the Fick’s laws in a work of 1970.

Here ψ(x, t) is a multicomponent function with at least two scalar space-time compo-
nents. Also, each scalar component satisfies the SDE. Such solutions can be interpreted
as probability distributions with internal structure associated to internal degrees of free-
dom of the system. They are named diffunors in analogy with the spinors in Quantum
Mechanics.

In this paper we deal with a further generalization of Dirac’s method, considering the
system of fractional evolution equations

(A∂α
t + B∂x)ψ(t, x) = 0, ψ(t, x) =

(
u1(t, x)
u2(t, x)

)
, (5)

with 0 < α < 1, A and B satisfying (4), as the square root of the time Fractional
Diffusion-Wave Equation

∂2α
t u(t, x)− ∂xxu(t, x) = 0. (6)

Equation (6), associated to anomalous diffusion, has been widely studied in the lit-
erature by many authors, among which Schneider and Wyss [18], Metzler et al. ([7], [9],
[23]), Mainardi et al. [6], [5], [4], Sokolov et al. [20], Saichev et al. [16]. A more physical
discussion of this equation was given by Metzler and Klafter in [8] and it is worthwhile
mentioning that they proved that the resulting probability density is bimodal in charac-
ter. In this sense, Schneider and Wyss [18] showed that, for dimensions higher than 1,
the character of the solution of the fractional wave equation (when 1/2 < α ≤ 1) as a
proper probability density is lost.

Each component of ψ(t, x) satisfies (6) while the index property ∂t
α∂t

αu = ∂t
2αu

holds. Thus, in the interval 1/2 < α < 1, the decomposition (5) of (6), expressed in
terms of fractional evolution equations of Dirac-type, represents a fractional interpolation
between the diffusion (α = 1/2) and wave (α = 1) equations.

The applications of the fractional calculus range in a wide spectrum of areas like mate-
rial sciences (viscoelasticity, polymers,...), circuits, diffusion processes, Biology, Economy,
Geology, traffic problems, data analysis, and others, as it is illustrated, for example, in
the text books by Hilfer [2], Ross [15], Samko et al. [17]. Many of the associated models
amount to replace the time derivative in an evolution equation with a fractional derivative
of real order.

2



The fractional derivative operator ∂α
t appearing in (5) can be specified according to

several definitions available in the literature (see for example [17] and [14]); we will refer
to the two commonly used definitions of Riemann-Liouville and Caputo.

The Riemann-Liouville derivative of order α > 0 is defined as

(RL
aD

α
x f)(x) =

dn

dxn

1
Γ(n− α)

x∫

a

(x− t)n−α−1f(t)dt, (7)

with x > a, n = −[−α], whereas the Caputo fractional derivative, usually considered as
a regularised version of the Riemann-Liouville fractional differential operator, takes the
form

(C
aD α

x f)(x) =
1

Γ(n− α)

x∫

a

fn(τ)
(x− τ)α−n+1

dτ. (8)

There exists the following relation between the above definitions

( C
aD α

x f)(x) =RL
aDα

x


f(x)−

n−1∑

j=0

f (j)(a+)
(x− a)j

j!


 , (9)

and, a condition under which both derivatives hold is that f ∈ ACn−1(a,∞). Equivalence
(9) allows to use only initial conditions of the classical type when dealing with fractional
equations involving Riemann-Liouville or Caputo derivatives. The integral approach
pursued by Schneider and Wyss in [18] is probably the most physical way of incorporating
these initial values to the corresponding equation.

Definitions (7) and (8) reproduce the classical derivative dn

dxn when α = n (n ∈ N)
and the identity operator for α = 0, and they are non local operators being given by a
definite integral.

It is important to highlight that, in general (see [10] and [14], for example) the index
property aDα

x aDβ
xf =a Dα+β

x f does not hold using the definitions of fractional derivative
of Riemann-Liouville (7) or Caputo (8), unless the function f verifies:

f (j)(a+) = 0, j = 0, 1, 2, ...,m− 1, (10)

whenever f(x) ∈ ACm−1(a,∞) and fm(x) ∈ L1
loc(a,∞), m− 1 < β ≤ m.

In [10, Cap.4] it is analyzed a restricted class of functions C for which the Riemann-
Liouville fractional derivative satisfies the cited index property. Some examples of func-
tions in C are: xλ with λ > −1, the polynomials, the exponentials, the sine and cosine
functions and all the linear combinations of them.

The structure of the paper is as follows. In Sec. 2, a physical meaning to the solutions
of (5) is given, showing how they generalize the behavior of the solutions of the Dirac
system, recovered when α = 1, in relation with the D’Alembert solution. Following the
analogy with the classical Dirac case, the internal symmetries of the system (5) under
inversions and translations in time and/or space, and Galileo transform are considered
in Sec. 3, whereas in Sec. 4 a conserved quantity for (5), analogous of the Hamiltonian
for the classical Dirac equation, is found.

3



2 Physical meaning of the solutions of the system of

fractional evolution equations

It is well known that the general solution of the wave equation with zero initial velocity,




∂ttu(t, x)− c2∂xxu(t, x) = 0
u(0, x) = ϕ(x), ϕ ∈ C2

ut(0, x) = 0
, (11)

is given by the D’Alembert Formula

u(t, x) =
1
2

[ϕ(x− ct) + ϕ(x + ct)] . (12)

From a physical point of view, we can interpret this fact as that the amplitude at
time t of a perturbation created by a given starting deformation at rest, ϕ(x), is the
superposition of two waves, ϕ(x + ct) and ϕ(x − ct), whose shape is identical to the
starting one’s and travelling in opposite directions. The two waves are solutions of the
following first order problems

{
∂tu(t, x)− c ∂xu(t, x) = 0
u(0, x) = ϕ(x), ϕ ∈ C2 , (13)

and {
∂tu(t, x) + c ∂xu(t, x) = 0
u(0, x) = ϕ(x), ϕ ∈ C2 , (14)

and the D’Alembert solution is a linear combination of them with coefficients equals to
1/2.

Now, if we only shrink the study to the pure real matrices leading to a system (5) of
separated equations, then we have two possible choices:

A =
(

0 1
1 0

)
y B =

(
0 1
−1 0

)
, (15)

and the second pair of matrices given by the same A and −B.
Substituting (15) in (5), it reduces to the following system of equations





∂α
t u1(t, x)− ∂xu1(t, x) = 0

∂α
t u2(t, x) + ∂xu2(t, x) = 0

, (16)

where 0 < α < 1, and the equations appearing in (13) and (14) are devolved for the
limiting case of α = 1 and c = 1.

We want to show that the solution u(t, x) of the Time Fractional Diffusion Equation
(6) is still a linear combination of the solutions u1(t, x) and u2(t, x) of (16), for each
0 < α < 1 and, therefore, that the relation existing between the Dirac solutions and the
D’Alembert expression is extended to the fractional case.

In [3] the following initial value problem

(CDα
t u)(t, x) = λ∂xu(t, x) (t > 0, x ∈ R; 0 < α < 1),

lim
|x|→∞

u(t, x) = 0, u(0+, x) = g(x),
(17)

4



has been solved and the general solution, expressed in terms of the inverse of its Fourier
Transform, is given by

u(t, x) = ug,λ(t, x) =
1
2π

+∞∫

−∞
Eα,1 (−iλktα) G(k)e−ikxdk, (18)

where Eα,β(z) is the biparametric Mittag-Leffler special function [1]. This solution is
said to be localized due to the property lim

|x|→∞
u(t, x) = 0.

On the other hand, the general localized solution of the Cauchy problem for the time
fractional diffusion equation,

(CD2α
t f)(t, x) = λ2∂xxf(t, x)

lim
|x|→∞

f(t, x) = 0, f(0+, x) = g(x), [∂tf(t, x)]t=0 = 0, (t > 0, x ∈ R),
(19)

where 0 < α < 1, can be found in [4] and it reads

f(t, x) = fg(t, x) =
1
2π

+∞∫

−∞
E2α,1

(−(λk)2t2α
)
G(k)e−ikxdk. (20)

Now, if we apply the duplication formula [1] for the Mittag-Leffler function,

E2α,1(z) =
1
2

[
Eα,1(+z1/2) + Eα,1(−z1/2)

]
, (21)

then we can rewrite (20) as follows:

fg(t, x) =
1
2

[ug,−λ(t, x) + ug,λ(t, x)] , (22)

where ug,λ is given in (18).
So, we can conclude that the general solution of the Cauchy problem (19) for the time

fractional diffusion equation turns out to be a linear combination, with coefficients equals
to 1/2, of the two general solutions of the Cauchy problems for the fractional Dirac-type
equations represented by (17) and by the problem obtained when λ is replaced by −λ in
(17).

In particular, the fundamental solution of (19), when g(x) = δ(x), turns out to be

f(t, x) =
1

2λ tα
W

(
−| x |

tαλ
;−α, 1− α

)
, (0 < α < 1), (23)

where W (z; α, β) is the Wright special function ([1, 18.1(27)]).
Therefore, if we assume λ = 1 in (19), we obtain

f(t, x) =
1

2tα
W

(
−| x |

tα
;−α, 1− α

)
=

1
2

[u1(t, x) + u2(t, x)] , (24)

where

u1(t, x) =
{

1
tα W

(
x
tα ;−α, 1− α

)
x ≤ 0,

0 x > 0,
(25)

5



and

u2(t, x) =
{

0 x < 0,
1
tα W

(− x
tα ;−α, 1− α

)
x ≥ 0.

(26)

are the functions appearing in (16), fundamental solutions of (17) when λ = 1 and
λ = −1, respectively (see [3]).

3 Invariance with respect to Inversions and Transla-

tions in time and/or space

The aim of this section is to establish whether the system (5) turns out to be invariant
under inversions and translations in time and/or space, and Galileo transform, and how
the non-local property of the time fractional differential operator affects these results.

Let us firstly consider the spatial inversion P : x′ = −x of the system (5), and set

ψ′(t, x′) = Sψ(t, x(x′)). (27)

We look for a matrix S so that the transformed multicomponent function ψ′(t, x′) is
still a solution of the system (5), this is:

A∂α
t ψ′(t, x′) + B∂x′ψ

′(t, x′) = 0, ψ′(t, x′) = S

(
u1(t, x(x′))
u2(t, x(x′))

)
, (28)

where the fractional derivative ∂α
t could be specified in this context either through (7)

or (8).
We calculate

∂x′ψ
′(t, x′) = S∂x′ψ(t, x(x′)) = −S∂xψ(t, x) (29)

and this allows us to write:

A∂α
t ψ′(t, x′) + B∂x′ψ

′(t, x′) = AS∂α
t ψ(t, x)−BS∂xψ(t, x).

Therefore, in order to be (28) verified, it has to turn out

S−1AS∂α
t ψ(t, x)− S−1BS∂xψ(t, x) = 0,

and, according to (4), when S = A, this requirement is fulfilled, being ψ(t, x) solution
of (5). Thus, the system of fractional evolution equations (5) is invariant under spatial
inversion, as well as the Time Fractional Diffusion Equation (6) with respect to this
transformation.

Let us now consider the Time inversion T : t∗ = −t and set

ψ∗(t∗, x) = Tψ(t(t∗), x). (30)

Then, as for the spatial inversion, we have to find a matrix T such that ψ∗(t∗, x) is
solution of the system of equations (5), where now it is necessary to specify the used
definition of fractional derivative ∂α

t .

6



If we assume the Caputo fractional derivative (8) in (5), then we have

A C
aD α

t∗ ψ∗(t∗, x) + B∂xψ∗(t∗, x) = 0, ψ∗(t∗, x) = T

(
u1(t(t∗), x)
u2(t(t∗), x)

)
. (31)

It results
C
aD α

t∗ ψ∗(t∗, x) = T (−1)α C
−aD α

t ψ(t, x). (32)

and, in particular, when a = 0:

C
0D

α
t∗ ψ∗(t∗, x) = T (−1)α C

0D
α
t ψ(t, x). (33)

Therefore, in general, the left side of the equation in (31) takes the form

(−1)α AT C
−aD α

t ψ(t, x) + B T ∂xψ(t, x),

and we only can have invariance of (5), with respect to time inversion, if a = 0 and the
matrix T verifies {

(−1)αAT = TA

BT = TB
.

The matrix T = B fulfils the above conditions while (−1)α = −1, that means

(−1)α = ei(π+2nπ)α = −1 = ei(π+2kπ),

with n, k ∈ {N ∪ {0}}, and this implies

α =
1 + 2k

1 + 2n
, k = 0, 1, 2, ..., , n = k + 1, k + 2, k + 3...,

as the hypothesis 0 < α < 1 has to be preserved.
Some particular values of α, for which the invariance of the system (5) under time

inversion holds, are the followings

α =
1
3
,
1
5
,
1
7
, ...,

3
5
,
3
7
,
3
9
, ...,

5
7
,
5
9
,

5
11

, ....

Note that the Time Fractional Diffusion Equation (6) is invariant under time inversion
while (−1)2α = 1, that holds and it is well defined for

α =
k

1 + 2n
, n = 1, 2, 3, ..., , k = 1, 2, ..., 2n,

this is,

α =
1
3
,
2
3
,
1
5
,
2
5
,
3
5
,
4
5
,
1
7
,
2
7
, ....,

6
7
,
1
9
, ...,

in agreement with known results for the classical one-dimensional Time Diffusion Equa-
tion (α = 1/2).

The invariance properties under Time Inversion of the systems (5) and (6) are a partial
characterization of the transition between the parabolic (α = 1/2) and hyperbolic (α = 1)
behaviors.

If we repeat the computations when ∂tα =RL
a D α

t∗ in (5), we obtain the same results
so that we can deal in this context with any of the two definitions of fractional derivative
without losing generality.

7



Now, if we consider the space-time inversion PT : t∗ = −t, x′ = −x and we assume

ψ̄(t∗, x′) = Rψ(t(t∗), x(x′)), (34)

over the system of equations (5), repeating exactly the same calculations as above the
system turns out to be invariant under space-time inversion if we still restrict our study
to the case of a = 0 in the lower extreme of integration of the fractional derivative and
when the matrix R fulfils {

(−1)αAR = RA

BR + RB = 0
.

Therefore it must be distinguished between two cases:

• if (−1)α = −1, that means

α =
1 + 2k

1 + 2n
, k = 0, 1, 2, ..., , n = k + 1, k + 2, k + 3...,

then for R = AB = −BA the system (5) is invariant under space-time inversion.

• if (−1)α = 1, that means

α =
2k

1 + 2n
, k = 1, 2, 3, ..., , n = k, k + 1, k + 2, ...,

then the space-time invariance only holds when R = A.

The invariance property of system (5) fails, in general, with respect to Space-Time
Translation and Galileo Transform. In fact, if we operate with a space-time translation
on the system (5), assuming the changes of variable t∗ = t + t0, x′ = x + x0, where t0
and x0 are constants, and setting

ψ̃(t∗, x′) = V ψ(t(t∗), x(x′)), (35)

the system turns out to be invariant under this space-time transformation just for the
trivial case of t0 = 0 and V = I. Actually, for the Caputo derivative, for example, it
results

C
aD α

t∗ ψ̃(t∗, x′) = V a−t0
CD

α

t ψ(t, x).

In the case of the Galileo Transform: x′ = x + vt, t′ = t, with v > 0 constant, we set

ψ̂(t, x′) = Wψ(t, x(x′, t)) (36)

in (5). In general, we do not have invariance of the system (5) with respect to Galileo
Transform, as well as it occurs with the classical Dirac equation. Indeed, it is

C
aD α

t ψ̂(t, x′) = W C
aD α

t ψ(t, x)− vW aI1−α
t ψx(t, x),

where Iα
t is the Riemann-Liouville fractional integral defined as

(aIα
x f)(x) =

1
Γ(α)

x∫

a

(x− t)α−1f(t)dt,

8



with x > a, α > 0, f ∈ L1
loc(a,∞) and (aI0

t f)(x) = f(x).
Then, the transformed multicomponent function ψ̂(t, x′) is still a solution of (5) if the

system
A C

aD α
t ψ̂(t, x′) + B∂x′ ψ̂(t, x′) =

= AWC
aD α

t ψ(t, x)− vAW aI1−α
t ψx(t, x) + BW∂xψ(t, x), (37)

is equal to the zero vector, that implies

AW aI1−α
t ψx(t, x) = 0. (38)

Therefore, in general, we do not have invariance of the system (5) with respect to
Galileo Transform because, in view of the evaluation of the beta integral [1], valid for all
p > 0 and q > 0 (or Re(p) > 0 and Re(q) > 0), namely

B(p, q) :=

1∫

0

up−1(1− u)q−1du =
Γ(p)Γ(q)
Γ(p + q)

,

it can be proved the following property for the Riemann-Liouville fractional integral

aIα
t (t− a)β =

Γ(β + 1)
Γ(β + α + 1)

(t− a)β+α (39)

when β > −1. As a consequence, it results aI1−α
t ψx(t, x) = 0 just in case ψx(t, x) = 0

for any 0 < α < 1, that means ψ(t, x) = ψ(t), constant function in x. If this condition

is not fulfilled, as it occurs in general, then, being A =
(

0 1
1 0

)
in our case, we can

obtain a restricted class of functions u1 and u2 for which (5) is invariant under Galileo
transform.

Given W =
(

w11 w12

w21 w22

)
, the system (38) can be reduced to

{
w21I

1−α
t

∂
∂xu1(t, x) + w22I

1−α
t

∂
∂xu2(t, x) = 0

w11I
1−α
t

∂
∂xu1(t, x) + w12I

1−α
t

∂
∂xu1(t, x) = 0

and, if det(W ) = 0, that means w12w21−w11w22 = 0, then it has the solution ∂
∂xu1(t, x) =

−w22
w21

∂
∂xu2(t, x) that implies u1(t, x) = −w22

w21
u2(t, x) + c(t), where c(t) is a constant

function in x.
The general non invariance of the fractional evolution equations under Time Trans-

lation and Galileo transform is due to the non-local property of the time fractional
derivative and it is in contrast with the invariance results holding for the Dirac equation,
with respect to the Time Translation, as a fundamental requirement to be the relativity
principle verified [19].

4 A Conserved Quantity: the Fractional Hamiltonian

The invalidity of the invariance of the fractional Dirac-type system (5) under Time Trans-
lation does not prevent it from possessing a fractional conserved quantity, analogous of
the Hamiltonian for the classical Dirac system.

9



It is well known (see, for example, [19]) that the Lagrangian density for the classical
Dirac equation obtained from (5) when α = 1, is given by

L(t, x) = ψ̄A∂tψ + ψ̄B∂xψ,

with ψ = ψ(t, x) = (u1(t, x), u2(t, x))T , ψ̄ = ψ̄(t, x) = ψ+A, and ψ+ = ψ+(t, x) =
(u∗1(t, x), u∗2(t, x)), complex conjugated of ψ that verifies the conjugated equation of (5)
with α = 1:

∂tψ
+A+ + ∂xψ+B+ = 0.

Therefore, the Hamiltonian density will be

H(t, x) =
∂L(t, x)
∂(∂tψ)

∂tψ − L(t, x) = ψ̄A∂tψ − L(t, x) = −ψ+C∂xψ, (40)

with C = AB = −BA, the Hamiltonian

H(t, x) =

+∞∫

−∞
H(t, x)dx = −

+∞∫

−∞
ψ+C∂xψdx, (41)

and, its time derivative,

d

dt
H(t, x) =

+∞∫

−∞
∂x

[
ψ+∂xψ

]
dx, (42)

provided that we restrict ourselves to the pure real matrices A and B so that the equiv-
alence ∂tψ

+ = −∂xψ+C is verified. In this case, if we assume, for example, the initial
condition ψ+∂xψ → 0 when | x |→ ∞, we can conclude that there exists a conserved
quantity associated to equation (5) with α = 1, given by the Hamiltonian (41).

In what follows, we want to show that, as well as we did above, it is possible to find
a conserved quantity associated to the system of fractional Dirac-type equations (5) for
general 0 < α < 1, even if it does not present invariance with respect to time translation.

We start defining, by analogy with the classical Dirac case, a formal “fractional
Lagrangian density” related to (5)

Lα(t, x) = ψ̄A∂α
t ψ + ψ̄B∂xψ, (43)

and a formal “fractional Hamiltonian density”

Hα(t, x) =
∂Lα(t, x)
∂(∂α

t ψ)
∂α

t ψ − Lα(t, x) = ψ̄A∂α
t ψ − Lα(t, x) = −ψ+C∂xψ. (44)

The final expression in (44) is equivalent to (40) and can be simplified observing
that, being A and B pure real matrices, if ψ is a solution of (5), then also ψ+ has to
solve it, reason why a pure real solution of (5) can always be found. Therefore, we will
assume that ψ it is a pure real solution of (5) and this allows us to write the “fractional
Hamiltonian” as

Hα(t, x) =

+∞∫

−∞
Hα(t, x)dx = −

+∞∫

−∞
ψT C∂xψdx (45)

10



and, consequently,

d

dt
Hα(t, x) = −

+∞∫

−∞

[
∂tψ

T C∂xψ + ψT C∂x∂tψ
]
dx = −

+∞∫

−∞
∂x

[
ψT C∂tψ

]
dx. (46)

It has been used the equivalence ∂tψ
T C∂xψ = ∂xψT C∂tψ, due to the fact that the

matrix C = AB, with pure real A and B, only can be of two types:

C1 =
(

c11 0
0 −c11

)
o C2 =

(
0 c12

c12 0

)
,

where c11 and c12 take the values ±1.
At this point it is necessary to specify the definition of the fractional derivative in

use.
When 0 < α < 1 the Riemann-Liouville fractional derivative ∂t

α =RLDα
t , according

to (9), fulfils the following identity

(RLDα
t ψ)(t, x) = (I1−α

t ∂tψ)(t, x) +
t−α

Γ(1− α)
ψ(0, x), (47)

being (RLDα
t ψ(0, x))(t, x) = t−α

Γ(1−α)ψ(0, x) and (CDα
t ψ)(t, x) = (I1−α

t ∂tψ)(t, x) by defini-
tion (8).

Now, if we introduce the Riemann-Liouville fractional derivative in (5), using the
fact that this derivative is the left inverse operator of the Riemann-Liouville fractional
integral, we can write

(∂tψ)(t, x) = −C∂x(RLD1−α
t ψ)(t, x)−A(RLD1−α

t

t−α

Γ(1− α)
ψ(0, x))(t, x) =

= −C∂x(RLD1−α
t ψ)(t, x)−A∂tψ(0, x) = −C∂x(RLD1−α

t ψ)(t, x). (48)

In a similar straightforward way it can be proved that result (48) holds exactly the
same when the Caputo derivative appears in (5).

Therefore, when α = 1/2 both derivatives verify

(∂tψ)(t, x) = C2(∂2
xψ)(t, x) = (∂2

xψ)(t, x). (49)

In agreement with (48), the expression for the Hamiltonian time derivative (46) takes
the form:

d

dt
Hα(t, x) = −

+∞∫

−∞
∂x

[
ψT C∂tψ

]
dx =

+∞∫

−∞
∂x

[
ψT ∂x

RLD1−α
t ψ

]
, (50)

when 0 < α < 1 and, in particular,

d

dt
H1/2(t, x) =

+∞∫

−∞
−∂x

[
ψT C ∂2

x ψ
]
dx, (51)

11



for α = 1/2, when the fractional derivative is either of the Riemann-Liouville or of the
Caputo type.

Therefore, we can conclude that, when 0 < α < 1, if the condition

d

dt
Hα(t, x) =

+∞∫

−∞
∂x

[
ψT ∂x

RLD1−α
t ψ

]
dx =

=
[
u1 ∂x

RLD1−α
t u1 + u2 ∂x

RLD1−α
t u2

] |x=+∞
x=−∞= 0, (52)

is fulfilled, then a conserved quantity exists associated to equation (5) and it is given by
the fractional Hamiltonian (45).

For the particular case of α = 1/2, we deduce from (51) an alternative condition,
equivalent to (52), to provide the existence of the conserved quantity H1/2(t, x):

d

dt
H1/2 =

+∞∫

−∞
−∂x

[
ψT C ∂2

x ψ
]
dx = −ψT C ∂2

xψ|x=+∞
x=−∞ = 0,

that means (
u1∂

2
xu1 − u2∂

2
xu2

) |x=+∞
x=−∞ = 0, (53)

when C = C1, and (
u1∂

2
xu2 + u2∂

2
xu1

) |x=+∞
x=−∞ = 0, (54)

when C = C2. Both conditions (53) and (54) come true when, for example, | uh |→ 0
and ∂2

xuh is bounded when | x |→ ∞, for h = 1, 2.
We can conclude this section recalling some results obtained in [3].
If we consider the system of fractional equations of Dirac-type (5) when ∂t

α =C Dα
t ,

when A and B are given by (15) so that it turns out to be

C = C1 =
( −1 0

0 1

)
,

and we complete (5) with the initial conditions

lim
|x|→∞

ψ(t, x) = 0, ψ(0+, x) = δ(x),

then the localized fundamental solutions are given by (25) y (26).
In [5] the asymptotic behavior of the Wright function W (z; α, β), for the case of

α = −ν and β = 1 − ν, has been studied. In particular, if the function M(z; ν) =
W (−z;−ν, 1 − ν) is introduced, when the argument z = r > 0 is real and positive and
r → +∞, it holds

M
( r

ν
; ν

)
∼ a(ν)r(ν−1/2)/(1−ν) exp

[
−b(ν)r1/(1−ν)

]
, (55)

where a(ν) = 1/
√

2π(1− ν), b(ν) = (1− ν)/ν.
This implies an asymptotic exponential decay of our solutions u1 y u2, and, conse-

quently, that | uh |→ 0 when | x |→ ∞ for h = 1, 2.

12



In order to ensure the existence of the conserved quantity (45) when 0 < α < 1, we
also have to analyze the asymptotic behavior of ∂x

RLD1−α
t uh for h = 1, 2.

It turns out to be:

RLD1−α
t

(
1
tα

W

(
−|x|

tα
;−α, 1− α

))
= ∂tI

α
t

[
+∞∑

k=0

−(|x|)k t−αk−α

k!Γ(−αk + 1− α)

]
=

= ∂t

[
+∞∑

k=0

−(|x|)k t−αk

k!Γ(1− αk)

]
=

[
+∞∑

k=0

−(|x|)k t−αk−1

k!Γ(−αk)

]
=

=
1
t
W

(
−|x|

tα
;−α, 0

)
=

α|x|
tα+1

W

(
−|x|

tα
;−α, 1− α

)

if we use the property W (−z;−ν, 1− ν) = 1
νz W (−z;−ν, 0); therefore, (55) implies that

RLD1−α
t uh, as well as ∂x

RLD1−α
t uh, will also decay exponentially when | x |→ ∞ for

h = 1, 2.

5 Conclusions

We have treated a generalization of the classical free Dirac equations, namely the frac-
tional evolution equations of Dirac-type. For the localized solutions of these equations
we have derived their relation with the corresponding solution of the fractional diffusion
equation, showing how the latter turns out to be a linear combination of the formers, as
well as the D’Alembert solution of the classical wave equation is a linear combination of
the solutions of the first order equations derived from the decomposition of the second
order wave operator into its corresponding square-root operators.

Following the analogy existing between the fractional evolution equations and the
classical Dirac equation, we have studied their internal symmetries with respect to certain
transformations in space and/or time. The system of fractional Dirac-type equations is
invariant under Spatial Inversion for each 0 < α < 1, whereas it possesses invariance
under Time Inversion only for certain values of the fractional index α, so inclosing the
hyperbolic behavior of the classical Dirac equation or of the Time Fractional Diffusion
equation (6) when 1

2 < α ≤ 1 (including the classical wave equation corresponding
to α = 1), and the parabolic one of the Time Fractional Diffusion equation (6) when
0 < α ≤ 1

2 (including the classical diffusion equation corresponding to α = 1/2). In
keeping with the joint Space-Time Inversion, a range of validity for the invariance is
located, still depending on the index α, but wilder then the which one corresponding to
the only Time Inversion.

The system proves to be never invariant under Time Translation and Galileo trans-
form, due to the non-local property of the time fractional derivative. This lack of in-
variance of the fractional evolution equations under time translation, is in contrast with
the invariance results holding for the classical Dirac equation, with respect to the same
transformation, as a fundamental requirement to be the relativity principle verified, but
does not prevent the fractional Dirac-type system from possessing a fractional conserved
quantity, analogous of the Hamiltonian for the classical Dirac system.

13



6 Acknowledgements

T.P. and L.V. thank the partial support of the Ministerio de Ciencia y Tecnologa of Spain

under grant BFM 2002-02359. T.P. acknowledges the predoctoral fellowship of the Programa

de Formación de Personal Investigador de la Comunidad de Madrid (5793/2002)

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