NIKOLOVA, YANKA

English

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ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 1 (2012), no.2, 113119
ISSN 1857-8365, UDC: 517.956:517.44
SOLVING SOME FRACTIONAL ORDER DIFFERENTIAL
EQUATIONS BY MEANS OF INTEGRAL TRANSFORMS
AND SPECIAL FUNCTIONS
YANKA NIKOLOVA
Presented at the 8th International Symposium GEOMETRIC FUNCTION THEORY AND
APPLICATIONS, 27-31 August 2012, Ohrid, Republic of Macedonia.
Abstract. In this paper we propose some methods for solving fractional order dif-
ferential equations with variable coecients. To this aim, we consider suitable gen-
eralizations of the classical integral transforms of Fourier, Mellin and Laplace, and
study their basic properties. First we consider some ordinary dierential equations
of fractional order with variable coecients. The solutions obtained by means of in-
tegral transforms are expressed in terms of special functions, as the Wright function
and 1- and 2-parametric Mittag-Leer functions. The method of integral transforms
is used also to solve partial dierential equations of the same kind. Namely, a gen-
eralized Laplace transform is applied to solve the so-called Giona equation and the
fractional wave equation. Special attention is paid also to the generalized fractional
heat equation involving a generalization of the Riemann-Liouville fractional deriva-
tive. A combined application of the Laplace transform and of the generalized Fourier
transform leads to a solution of Cauchy problem for this equation in explicit integral
form, where the kernel is represented by the 1-parametric Mittag-Leer function.
1. Introduction
The theory of integrals and derivatives of arbitrary order, usually referred to as Frac-
tional Calculus (see for example [4], [15]), is one of the most intensively developing areas
of mathematical analysis as a result of its increasing range of applications in biology,
physics, electrochemistry, economics, probability theory and statistics. The operators
for fractional dierentiation and integration have been used in various elds such as hy-
draulics of dams, potential elds, diusion problems and waves in liquids and gases. The
main advantage of the fractional calculus is that the fractional order models provide an
instrument for the adequate description of the memory and hereditary properties of ma-
terials and processes. In 1980 Namias [7] introduced the Fractional Fourier transform
(FRFT) as a way to solve certain classes of ordinary and partial dierential equations
2010 Mathematics Subject Classication. 26A33, 34A08, 35R11, 33E12, 44A10, 44A15.
Key words and phrases. Fractional order dierential equation; integral transforms of Fourier, Mellin,
Laplace; Mittag-Leer functions.
113
114 Y.NIKOLOVA
appearing in quantum mechanics. Most probably, Namias was unaware of Wiener's paper
[16] published in 1929, where the FRFT in the form of fractional powers of Fourier oper-
ator has been already studied. The FRFT [6] became more popular after 1995 because of
its numerous applications in quantum mechanics, chemistry, optics, dynamical systems,
stochastic processes and signal processing.
The main goal of this paper is to apply the techniques of some generalized integral
transforms for solving some fractional dierential equations with variable coecients.
Since there are no common methods for solving fractional dierential equations, specially
of this kind, here we propose some four approaches based on integral transforms useful
for this purpose.
2. Preliminaries
For our purposes we use the notions of Riemann-Liouville (R L) fractional deriv-
ative and Riemann-Liouville fractional integral of a function f(x) of order , resp. of
order n  , dened as follows:
0D

x f(x) =
dn
dxn
h
0D
 (n )
x f(x)
i
; 0D
 (n )
x f(x) =
1
 (n  )
xZ
0
(x   )n  1f( )d;
where n is positive integer, and n  1   < n.
Let us mention that in a contrast to the classical calculus, if   0, x > 0 and  >  1,
then the fractional derivative of the power function x is given by
0D

xx
 =
 ( + 1)
 (   + 1)x
 ; that in the particular case  = 0 and 0   < 1 implies
0D

x 1 =
x 
 (1  ) : Thus, generally the R-L derivative of a constant is not zero (as in the
classical calculus). It is of primary importance for our later considerations to outline the
composition property valid for  > 0, x > 0; n 2 N , n  1   < n; and n  1   < n;
0D
 
x [0D

x f(t)] = f(t) 
nX
k=1

0D
 k
x f(t)

t=0
t k
 (  k + 1) :
Following an idea developed in [1], we consider the generalized Laplace-type integral
transform dened as LA[f(x); p] =
Z 1
0
A0(x)e A(x)(p)f(x)dx; where p is complex vari-
able and f(x) is piecewise continuous function such that jf(x)j  MeA(x)(c); for some
constants M and c. The functions A(x) and  are supposed to be increasing invertible
functions with A(0) = 0. Obviously, from here the classical Laplace transform follows as
a very special case (A(x) = x;(p) = p).
For a function u of the class S of a rapidly decreasing test functions on the real axis R,
the Fourier transformation is dened as u^(!) = F [u(t);!] =
1Z
 1
u(t)ei!tdt; ! 2 R; where
the inverse Fourier transformation has the form u(t) = F 1[u^(!); t] =
1
2
1Z
 1
u^(!)e i!td!
SOLVING SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS . . . 115
t 2 R. Denote further by V (R) the set of functions v(t) 2 S, satisfying the conditions
u^(!) = F
dnv
dtn

t=0
= 0; n = 0; 1; 2; :::. The Fourier pre-image of the space V (R), i.e.
(R) = f' 2 S : '^ 2 V (R)g, is called the Lizorkin space. As it is stated in [15], the
Lizorkin space is invariant with respect to the fractional integration and dierentiation
operators. The notion of Fractional Fourier transformation (FRFT) was introduced
also in Podlubny [15]. For a function u 2 (R) the FRFT u^ of the order  (0 <   1)
is dened as
u^(!) = F[u(t);!]=
1Z
 1
e(!; t)u(t)dt; ! 2 R; where e(!; t) :=
8<
:
e ij!j
1/t ; !  0
e ij!j
1/t ; ! > 0
:
If  = 1, the kernel e coincides with the kernel of the classical Fourier transforma-
tion: e1(!; t) :=
(
e ij!jt ; !  0
e ij!jt ; ! > 0
 ei!t; ! 2 R. Evidently, the fractional Fourier
transformation of the order 1 coincides with the Fourier transformation: F1  F:
The relation between the fractional and conventional Fourier transformation is given
by the following simple formula:
u^(!) = F[u(t);!]  F1[u(t);w] = u^(w); where w =
8<
:
 j!j1= ; !  0
j!j1= ; ! > 0
:
Thus, if F[u(t);!] = F1[u(t);w] = (w), then u(t) := F
 1[u^(!); t] = F
 1
1 [(w); t]:
We study the fractional diusion equation in terms of the Caputo fractional derivative
[2]:
aD


f(x) =
8>>>><
>>>>:
1
 (m  )
xZ
a
f (m)( )
(x   )+1 m d; m  1 <  < m
Dmx f(x) ;  = m
;
where m is a positive integer. The method we follow uses the rule for the Laplace
transform of Caputo derivative, see e.g. [15]:
L[0D

 f(x); s] = s
L[f(t); s] 
m 1X
k=0
s k 1f
(k)
(0) ; m  1 <  < m:
We use also a generalized fractional derivative operator of the form Du(x) =
(1   )D+u(x)   D u(x); 0 <   1;  2 R where D+ and D  are the Riemann-
Liouville fractional derivatives on the real axis, given as
D+u(x) =
1
 (1 )
d
dx
xZ
 1
(x  t) 1u(t)dt; D u(x) =  
1
 (1 )
d
dx
+1Z
x
(t x) 1u(t)dt:
A key role in our considerations is given to a relation established in Luchko, Martinez
and Trujillo [5], according to which for 0 <   1, any value of  and a function
u(x) = (R), F[D

u(x);!] = ( ic!)F[u(x);!], ! 2 R and c = sin (=2) +
isign!(1 2) cos (=2) : If 0 <   1 and the variable ð is a complex number, such that
116 Y.NIKOLOVA
p =

Re p  ij!j1=; !  0
Re p+ ij!j1=; ! > 0 , then the integral transform of the form Mff(t); pg =
M(psign!) =
1Z
0
tp sign! 1f(t)dt, is called -Mellin transform of the function f(t).
Using this denition, we prove some basic properties of the -Mellin transforms, see
for example Nikolova [8, 9]. Namely, in [8] we give a generalization of the classical Mellin
transformation, see also Debnath et al. [3].
3. Main results
3.1. Solving a fractional order dierential equation by the -Mellin transform.
We consider the Bessel dierential equation of fractional order, called also generalized
Bessel dierential equation:
t+10D
+1
t y(t) + t

0D

t y(t) = f(t); 0 <  < 1; t > 0;
and look for the solution of the boundary value problem for this equation with the con-
ditions y(0) = y0(0) = 0; y(1) = y0(1) = 0. The solution is found by applying the
-Mellin and the inverse -Mellin transforms consecutively, and it has the following form:
y(t)=
1Z
0
f(t )g( )d;
where g(t) = M 1 fG(psign!); tg, G(psign!) =
 (psign! )
(psign! ) (psign!) : The detailed re-
sults are given in the paper of Nikolova [9].
3.2. Solving a fractional order dierential equation by generalized Laplace-
type integral transform. We solve now the two-terms fractional dierential equation
of the form
0D

x [x
2f(x)]  a0Dx f(x) = 0; where n  1   < n; n 2 N; a > 0;
and  is real number such that n   1   <  < n and  =    : The particular case
 = n of this equation was studied by Nonnenmacher. First, we start by solving this
equation for the particular case as a =  and  = 12 . Then the solution has the form
f (x) = x 3=2e =x. In general, if f (x) is piecewise continuous on (0;+1) and integrable
on any nite subinterval of [0;+1), this two-terms fractional dierential equation with
variable coecients is solvable, and its solution is given by f (x) =
a
  ()
x  1e a=x.
This solution is obtained by using the Laplace transform, the inverse Laplace transform
(for details see e.g. [14]), the property of the   - function and the generalized Laplace-type
integral transform dened as LA [1].
Details on these results are can be found in the paper of Nikolova [12].
SOLVING SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS . . . 117
3.3. Using a generalization of the Frobenius method. We consider the fractional
dierential equation of the form
x20D
2
x y(x) + (x
2 + ax + b)y(x) = 0; where x > 0; 0 <   1; a; b 2 R;
and 0D
2
x is the Riemann-Liouville fractional derivative of order 2. Then we take  such
that     >  1 and  (+ + 1)
 ( + + 1) + b = 0. If this equation has a solution given by
convergent power series, then the solution has the form y(x) =
1X
k=1
ak(; )x
k+, where
the coecients k(; ) satisfy the recurrence relations a2(; )

 (2+ + 1)
 (+ 1)
+ b

+
a:a1(; ) = 0, and for k  3, ak(; )

 (k+ + 1)
 [(k   2)+ + 1] + b

+ a:ak 1(; ) +
ak 2(; ) = 0. By means of the Frobenious method, the fractional Coulomb equa-
tion is solved and the solution is obtained in power series form. Namely, we consider the
Coulomb wave equation
d2y
dx2
+

1  2
x
  m(m+ 1)
x2

y = 0, that evidently is a particu-
lar case of the above fractional equation when  = 1, a =  2,  = m, m = 1; 2; and
b =  m(m + 1). The solution of the fractional equation reduces in this particular case
to the regular solution of the Coulomb wave equation of the form
y(x) = Fm(; x) = Cm()x
m+1
1X
k=m+1
Ak;mx
k m 1 ;
where
Am+1;m = 1; Am+2;m =

m+ 1
; (k+m)(k m 1)Ak;m = 2 Ak 1;m Ak 2;m (k > m+2)
and
Cm() =
2me =2 j (m+ 1 + i)j
 (2m+ 2)
:
The detailed results are described in the paper of Nikolova et al. [13].
3.4. Solving a generalized fractional diusion equation by fractional Fourier
transform (FRFT). We consider the Cauchy-type problem for the fractional diusion
equation
D u(x; t) = 
@2u
@x2
; x 2 R; t > 0;
subject to the initial condition u(x; t)jt=0 = f(x), where D is the Caputo time-fractional
derivative of order , f(x) 2 (R) and  is a diusivity constant. The solution for this
fractional equation is given by the integral
u(x; t) =
1Z
 1
G(x  ; t)f()d ;
where G(x; t) =
1
2
1Z
 1
e iwxE( jwj
2
 t)dw. The solution is obtained by applying the
Laplace transform to the Caputo derivative (see [15]) and FRFT to the equation and the
118 Y.NIKOLOVA
initial condition. If  = 1, the solution of the Cauchy-type problem for above diusion
equation is given by the well-known integral formula
u(x; t) =
1p
4t
1Z
 1
e (x &)
2=4tf()d :
Consider the Cauchy-type problem for the generalization of the fractional diusion equa-
tion Du(x; t) = D
+1
 u(x; t); subject to the initial condition u(x; t)jt=0 = f(x) ; where
D

 is the Caputo time-fractional derivative of order , whereas D
+1
 is the space- frac-
tional derivative. If 0 <   1; 0 <   1 and for each value of  2 R, the Cauchy-type
problem for above fractional diusion equation is solvable and its solution u(x; t) is given
by the integral u(x; t) =
1R
 1
G(x  ; t)f()d; where
G(x; t) =
1
2
1Z
 1
e iwxE( ic+1wt)dw :
The solution is obtained by application of the Laplace transform to the Caputo derivative
and FRFT F+1 to the above equation and to the initial condition.
We consider also a generalization of the time-space diusion equation involving the
generalized fractional derivative
Du(x) = (1  )D+u(x)  D u(x); 0 <   1;  2 R ;
where D+ and D

  are the R-L fractional derivatives on the real axis. For  = 1 and
 = 1=2 the solution of the space-time diusion equation is studied by Luchko, Martinez,
Trujillo [5].
The details on our results are given in the papers [10, 11].
Acknowledgment
Partially supported under Project DID 02/25/2009: ITSFA, by National Science
Fund - Ministry of Education, Science and Youth, Bulgaria.
References
[1] A. Aghili, A. Ansari: Solving partial fractional dierential equations using the LA-transform,
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ical J. of Royal Astronomical Soc. 13, (1967), 529539.
[3] L. Debnath, D. Bhatta: Integral Transforms and Their Applications, 2006.
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SOLVING SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS . . . 119
[9] Y. Nikolova: - Mellin transform and one of its applications, Math. Balkanica, 26(1-2) (2012),
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[11] Y. Nikolova, L. Boyadjiev: Integral transforms method to solve a time-space fractional diusion
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[14] J. Paneva-Konovska: Laplace Transform in Examples and Problems, TU-Soa, Soa 2012.
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Faculty of Applied Mathematics and Informatics
Technical University of Sofia
8, Kl. Ohridski Str., 1000 - Sofia, Bulgaria
E-mail address: jvr@abv.bg


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