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An inverse problem for a generalized fractional diffusion
Applied Mathematics and Computation 249 (2014) 24–31
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An inverse problem for a generalized fractional diffusion Khaled M. Furati a,⇑, Olaniyi S. Iyiola a, Mokhtar Kirane b a b
King Fahd University of Petroleum & Minerals, Department of Mathematics & Statistics, Dhahran 31261, Saudi Arabia Laboratoire de Mathématiques Pôle Sciences et Technologie, Université de La Rochelle, Avenue M. Crépeau, 17042 La Rochelle Cedex, France
a r t i c l e
i n f o
a b s t r a c t We propose a method for determining the solution and source term of a generalized timefractional diffusion equation. The method is based on selecting a bi-orthogonal basis of L2 space corresponding to a nonself-adjoint boundary value problem. Uniqueness is proven and an existence result is obtained for smooth initial and final conditions. The asymptotic behavior of the generalized Mittag–Leffler function is used to relax the smoothness requirement on these conditions. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Inverse problem Fractional derivative Fractional diffusion equation Bi-orthogonal system Mittag–Leffler function Anomalous diffusion
1. Introduction and definitions We are concerned with the problem of determining the distribution uðx; tÞ, and the source term f ðxÞ, for the following problem
Da;c uðx; tÞ uxx ðx; tÞ ¼ f ðxÞ; I1c uðx; tÞjt¼0 ¼ gðxÞ; uð1; tÞ ¼ 0;
0 < x < 1; 0 < t < T; 0 < a 6 c 6 1;
uðx; TÞ ¼ hðxÞ; 0 < x < 1;
ð1Þ
ux ð0; tÞ ¼ ux ð1; tÞ; 0 < t 6 T;
where gðxÞ and hðxÞ are the initial and final conditions, respectively, and assumed to be in L2 ð0; 1Þ. The operators Ia and Da;c are defined by
Ia wðtÞ ¼
1 CðaÞ
Z
t
ðt sÞa1 wðsÞ ds;
t > 0;
a > 0;
0
" a;c
D wðtÞ ¼ D
a
# I1c wð0Þ c1 ; wðtÞ t CðcÞ
Da wðtÞ ¼ DI1a wðtÞ;
D¼
d ; dt
where C is the Gamma function. ⇑ Corresponding author. E-mail addresses: [email protected] (K.M. Furati), [email protected] (O.S. Iyiola), [email protected] (M. Kirane). http://dx.doi.org/10.1016/j.amc.2014.10.046 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð2Þ
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
25
The two parameter family of fractional derivatives Da;c of order a and type c allows for interpolation between the Riemann–Liouville and Caputo derivatives. The type-parameter c produces more types of stationary states and provide an extra degree of freedom for the initial condition. Models based on this derivative are considered in [6,10,12,14,31]. Although the initial condition is given as a fractional integral, it is possible to attribute a physical meaning to it and obtain it by appropriate measurements or observations. See the paper by Heymans and Podlubny [8]. Fractional calculus and fractional differential equations have become an essential tool in modeling many different phenomena in many fields such as control theory of dynamical systems [4,22], viscoelasticity [20], and nanotechnology [3]. Anomalous transport and anomalous diffusion [19], random walk [13,32], financial modeling [28] are also modeled using fractional models. Other physical and engineering processes are given in [23,25]. Also, more applications can be found in the surveys in [16,26] and the collection of applications in [9]. The main property of the fractional derivatives that enable them to play this role is the non-locality which is an intrinsic property of many complex systems. See for example [11]. In particular, fractional models are increasingly adopted for anomalous diffusion with slow diffusion [1,21,34]. Inverse problems for fractional diffusion equations that involve Caputo fractional derivative with respect to time are considered by a number of researchers. Zhang and Xu [33] showed that the source term can be uniquely identified given homogeneous boundary conditions. Kirane and Malik [18] considered a one dimensional fractional diffusion equation with a non-local non-self adjoint boundary condition and given solution at some later time. Using a bi-orthogonal system, they constructed a solution and source function and showed the uniqueness. Furthermore, they proved an existence result when the initial and final conditions are both smooth enough. Kirane et al. [17] extended the results to the problem in two-dimensions and showed the continuous dependence on the data. Recently, Aleroev et al. [2] considered an inverse source problem with an integral type over-determination condition. Özkum et al. [24] studied the inverse problem of determining the unknown source function of the linear fractional differential equation with variable coefficients using Adomian decomposition method. In this paper we generalize the problem in [18]. We use the same spatial bi-orthogonal system to obtain a formal separable solution and source function, and show their uniqueness. Then we utilize the asymptotic behavior of the generalized Mittag–Leffler function to obtain an existence result under some smoothness requirements of the initial and final conditions. The rest of the paper is organized as follows. In Section 2 we list some properties of the generalized fractional derivative operator and the generalized Mittag–Leffler functions. In Section 3 we construct the formal series solution and source term. In Sections 4 and 5 we obtain the uniqueness and existence results. In Section 6 we present some examples. 2. Generalized fractional derivative and Mittag–Leffler functions Notice that the derivative Da;c w in (2) reduces to the familiar Riemann–Liouville derivative and Caputo fractional derivative for c ¼ a and c ¼ 1, respectively,
Da;a wðtÞ ¼ Da wðtÞ;
Da;1 wðtÞ ¼ Da ½wðtÞ wð0Þ :¼ c Da wðtÞ:
Moreover, if I1c w 2 AC½0; T then
Da;c wðtÞ ¼ Ica Dc wðtÞ ¼ Ica DI1c wðtÞ:
ð3Þ
In particular, if c ¼ bð1 aÞ þ a for some 0 < b 6 1 then from (3) we have
Da;c ¼ Ibð1aÞ DIð1bÞð1aÞ ;
ð4Þ
which is the derivative operator introduced by Hilfer [12]. Srivastava and Tomovski [30] and Tomovski et al. [31] studied further the properties and applications of the derivative operator (4) and the associated generalizations of Mittag–Leffler function. In [30] the following generalization of Mittag– Leffler function,
Eqa;b;m ðzÞ ¼
1 X
ðqÞmk
zk
Cðak þ bÞ k! k¼0
;
z; b; q 2 C; Re a > maxf0; Re m 1g; Re m > 0;
ð5Þ
was introduced and investigated. This generalization is a special case of the Fox–Wright function p Wq :
Eqa;b;m ðzÞ ¼
ðq; mÞ 1 ;z : 1 W1 CðqÞ ðb; aÞ
More details on families of Fox–Wright function can be found in [29]. In this paper we use the following special case of (5), m ¼ 1, q
Ea;b ðzÞ ¼
1 X
ðqÞk zk ; Cðak þ bÞ k! k¼0
z; b; q 2 C; Re a > 0;
ð6Þ
26
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
where
Cðq þ kÞ : CðqÞ
ðqÞk ¼ qðq þ 1Þ ðq þ k 1Þ ¼
This function was introduced in [27] and further studied in [7]. Some special cases of this function are the Mittag–Leffler function in one and two parameters
Ea;b ðzÞ ¼ E1a;b ðzÞ;
Ea ðzÞ ¼ E1a;1 ðzÞ:
The function Eqa;b is an entire function [27] and thus bounded in any finite interval. Also, this function satisfies the recurrence relations [7,16]
kza Ea;aþb ðkza Þ ¼ Ea;b ðkza Þ
1
CðbÞ
;
z; b; k 2 C; Re a > 0;
aE2a;b ðkza Þ ¼ ð1 þ a bÞ Ea;b ðkza Þ þ Ea;b1 ðkza Þ; z; b; k 2 C; Re a > 0:
ð7Þ ð8Þ
The relations (7) and (8) yield the relation
akza E2a;b ðkza Þ ¼ ð1 þ a bÞ Ea;ba ðkza Þ þ Ea;ba1 ðkza Þ; z; b; k 2 C; Re a > 0:
ð9Þ
From Theorem 1.6 in [26] we have the bound
kt a jEa;b ðkt a Þj 6 M;
0 < a < 2; b 2 C; t P 0;
k P 0;
ð10Þ
for some M ¼ Mða; bÞ > 0. Thus, from (9) and (10) we obtain the bound,
k2 t2a E2a;b ðkt a Þ 6 M;
0 < a < 2; b 2 C; t P 0; k P 0;
ð11Þ
for some M > 0. Furthermore, from the positivity and monotonicity of Ea;b ðkta Þ we can show that
kEa;b ðkta Þ > k0 Ea;b ðk0 ta Þ > 0;
k > k0 > 0; t > 0; b P 2a; 0 < a < 1:
ð12Þ
From [7], we have the formula
h i I1c t b1 Eqa;b ðkt a Þ ¼ t bc Eqa;bcþ1 ðkt a Þ;
0 6 c 6 1; a; b > 0; k 2 R:
ð13Þ
For the Laplace transform we have the formula
n o q L t b1 Ea;b ðkt a Þ ¼
saqb ; ðsa kÞq
Re s > 0; jksa j < 1;
ð14Þ
where a; b; k; q 2 C; Re a > 0; Re b > 0, and Re q > 0. See for example (1.9.13) in [16] and (11.8) in [7]. 3. Formal solution and source term As in [18], the boundary conditions suggest the bi-orthogonal pair of dual Riesz bases for the space L2 ð0; 1Þ:
U ¼ f/0 ; /1n ; /2n g1 n¼1 ;
W ¼ fw0 ; w1n ; w2n g1 n¼1 ;
where,
/0 ðxÞ ¼ 2ð1 xÞ;
/1n ðxÞ ¼ 4ð1 xÞ cos kn x;
/2n ðxÞ ¼ 4 sin kn x;
ð15Þ
and
w0 ðxÞ ¼ 1;
w1n ðxÞ ¼ cos kn x;
w2n ðxÞ ¼ x sin kn x;
ð16Þ
where kn ¼ 2pn. In [15] it is proved that the sequences U and W are Riesz bases. Observe that neither sequence is orthogonal. We denote the inner product in L2 ð0; 1Þ by
hf ; gi ¼
Z
1
f ðxÞ gðxÞ dx:
0
We seek a solution and source function of (1) in the form
uðx; tÞ ¼ u0 ðtÞ /0 ðxÞ þ
1 X ukn ðtÞ /kn ðxÞ;
ð17Þ
n¼1 k¼1;2
f ðxÞ ¼ f 0 /0 ðxÞ þ
1 X f kn /kn ðxÞ: n¼1 k¼1;2
ð18Þ
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K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
Also we can write
g ¼ g 0 /0 þ
1 X
g kn /kn ;
n¼1 k¼1;2
where using the biorthogonal basis (16) we obtain
g 0 ¼ hg; w0 i;
g kn ¼ hg; wkn i; k ¼ 1; 2; n ¼ 1; 2; . . . :
ð19Þ
Substituting (17) and (18) into (1) yields the following system of ordinary fractional differential equations
Da;c u0 ðtÞ ¼ f 0 ;
ð20Þ
Da;c u1n ðtÞ þ k2n u1n ðtÞ ¼ f 1n ;
n ¼ 1; 2; . . . ;
ð21Þ
Da;c u2n ðtÞ þ k2n u2n ðtÞ 2kn u1n ðtÞ ¼ f 2n ;
n ¼ 1; 2; . . . :
ð22Þ
From the initial and final conditions in (1), we obtain
I1c u0 ð0Þ ¼ g 0 ; u0 ðTÞ ¼ h0 ;
I1c ukn ð0Þ ¼ g kn ; k ¼ 1; 2; n ¼ 1; 2; . . . ;
ukn ðTÞ ¼ hkn ; k ¼ 1; 2; n ¼ 1; 2; . . . ;
ð23Þ
where again fh0 ; h1n ; h2n g are the coefficients of the series expansion of h in the basis (15),
h0 ¼ hh; w0 i;
hkn ¼ hh; wkn i;
k ¼ 1; 2;
n ¼ 1; 2; . . . :
ð24Þ
For the Laplace transform, we have the formula [12]
L Da;c w ¼ sa Lfwg sac I1c wð0Þ:
ð25Þ
Accordingly, by applying the Laplace transform to (20) we obtain
U 0 ðsÞ ¼
f0 g þ 0; saþ1 sc
and hence
u0 ðtÞ ¼
f0
ta þ
Cð1 þ aÞ
g0
CðcÞ
t c1 :
ð26Þ
By applying Laplace transform to (21) we obtain
ðsa þ k2n ÞU 1n ðsÞ ¼
f 1n þ g 1n sac : s
So
U 1n ðsÞ ¼ f 1n
1 sðsa
þ
k2n Þ
þ g 1n
sac sa
þ k2n
ð27Þ
:
From formula (14) we obtain
u1n ðtÞ ¼ f 1n ta Ea;aþ1 ðk2n ta Þ þ g 1n tc1 Ea;c ðk2n t a Þ:
ð28Þ
Again, by applying Laplace transform to (22) and then substituting (27) we obtain
U 2n ðsÞ ¼
f 2n sðsa þ k2n Þ
þ
g 2n sac sa þ k2n
þ
2kn sa þ k2n
U 1n ðsÞ ¼
f 2n sðsa þ k2n Þ
þ
g 2n sac sa þ k2n
" þ 2kn f 1n
1 sðsa þ k2n Þ
þ g 1n 2
#
sac 2
ðsa þ k2n Þ
:
Thus formula (14) implies that
h i u2n ðtÞ ¼ f 2n ta Ea;aþ1 ðk2n t a Þ þ g 2n t c1 Ea;c ðk2n ta Þ þ 2kn f 1n t 2a E2a;2aþ1 ðk2n t a Þ þ g 1n t aþc1 E2a;aþc ðk2n t a Þ :
ð29Þ
As a result, we can write the coefficients u0 and ukn ; k ¼ 1; 2, in the form
u0 ðtÞ ¼
f0
Cð1 þ aÞ
ta þ
g0
CðcÞ
t c1 ;
ð30Þ
u1n ðtÞ ¼ f 1n E 1n ðtÞ þ g 1n E 2n ðtÞ;
ð31Þ
u2n ðtÞ ¼ f 2n E 1n ðtÞ þ g 2n E 2n ðtÞ þ 2kn ½f 1n E 3n ðtÞ þ g 1n E 4n ðtÞ;
ð32Þ
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K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
where
E 1n ðtÞ ¼ t a Ea;aþ1 ðk2n t a Þ; E 3n ðtÞ ¼ t
2a
E 2n ðtÞ ¼ t c1 Ea;c ðk2n ta Þ;
2
Ea;2aþ1 ðk2n t a Þ;
E 4n ðtÞ ¼ t aþc1 E2a;aþc ðk2n t a Þ:
By virtue of (30), (31), (32) and (23), we obtain the unknowns f 0 and f kn ; k ¼ 1; 2; n ¼ 1; 2; . . ., as
g 0 c1 ; T CðcÞ T h1n g 1n E 2n ðTÞ ¼ ; E 1n ðTÞ h2n g 2n E 2n ðTÞ 2kn ½f 1n E 3n ðTÞ þ g 1n E 4n ðTÞ ¼ : E 1n ðTÞ
f0 ¼ f 1n f 2n
Cð1 þ aÞ a
h0
ð33Þ ð34Þ ð35Þ
4. Uniqueness of solution In this section we present a uniqueness result for the solution and source term of the problem (1). Theorem 1. Let g; h 2 L2 ð0; 1Þ. Let u 2 ðLð0; TÞ \ Cð0; TÞÞ L2 ð0; 1Þ and f 2 L2 ð0; 1Þ. If fuðx; tÞ; f ðxÞg satisfies the inverse problem (1) then it is unique.
¼ u1 u2 and f ¼ f 1 f 2 . Then u satisfies the problem Proof. Suppose fu1 ðx; tÞ; f 1 ðxÞg and fu2 ðx; tÞ; f 2 ðxÞg satisfy (1). Let u
ðx; tÞ u xx ðx; tÞ ¼ f ðxÞ; Da;c u
0 < x < 1;
ðx; tÞjt¼0 ¼ 0; ðx; TÞ ¼ 0; I1c u u x ð0; tÞ ¼ u x ð1; tÞ; ð1; tÞ ¼ 0; u u
0 < t < T; ð36Þ
0 < x < 1; 0 < t 6 T:
Using the bases in (15) we can write
ðx; tÞ ¼ u 0 ðtÞ /0 ðxÞ þ u
1 X kn ðtÞ /kn ðxÞ; u
ð37Þ
n¼1 k¼1;2
f ðxÞ ¼ f 0 / ðxÞ þ 0
1 X f / ðxÞ: kn kn
ð38Þ
n¼1 k¼1;2
Then, the coefficients
0 ðtÞ; u
f 0 ;
kn ðtÞ; u
f ; k ¼ 1; 2; n ¼ 1; 2; . . . : kn
are given by (30)–(32) and (33)–(35), with
g 0 ¼ h0 ¼ 0;
g kn ¼ hkn ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . :
Thus clearly we have
f 0 ¼ f ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . : kn and
0 ðtÞ ¼ u kn ðtÞ 0; 0 < t 6 T; k ¼ 1; 2; n ¼ 1; 2; . . . : u ðx; tÞ ¼ 0 and f ðxÞ ¼ 0 and thus we have the uniqueness of the solution and the source term. Consequently, u
h
5. Existence of the solution In Section 3 we constructed a formal solution and source term for the problem (1). In this section we show that under some assumptions on the initial and final conditions we obtain the classical solution and continuous source term. Theorem 2. Let the following conditions hold g 2 C 2 ½0; 1; gð1Þ ¼ 0 00 0 0 h 2 C 4 ½0; 1; hð1Þ ¼ h ð1Þ ¼ 0; h ð1Þ ¼ h ð0Þ.
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
29
Then the formal solution constructed is a classical solution of problem (1). Proof. Let Q ¼ ð0; T ½0; 1 and Q ¼ ½; T ½0; 1 Q . We show that the series corresponding to u; ux ; uxx ; Da;c u are uniformly convergent and represent continuous functions on Q , for any > 0. Also we show that the series representation of f is uniformly convergent in ½0; 1. This is shown by bounding all these series by over-harmonic series then applying Weierstrass M-test. Since g 2 C 2 ½0; 1, by integration by parts we have
g 1n ¼ hg; w1n i ¼ g 2n ¼ hg; w2n i ¼
hg 00 ; w1n i k2n hg 00 ; w2n i k2n
þ
g 0 ð1Þ g 0 ð0Þ k2n 00 2hg ; w1n i k3n
þ
;
2½g 0 ð1Þ g 0 ð0Þ k3n
:
Also, since the sequence in (16) is a Riesz basis then it is a Bessel sequence [5] and thus there is C > 0 such that 1 X 2 hg 00 ; wkn i2 6 C kg 00 k :
ð39Þ
n¼1 k¼1;2
This implies that the set fhg 00 ; wkn ig is bounded. Accordingly there is G > 0 such that
jg kn j 6 G=k2n ; k ¼ 1; 2; n 2 N:
ð40Þ
ð4Þ
4
Similarly, since h 2 C ½0; 1; fhh ; wkn ig is bounded and through integration by parts we obtain ð4Þ
h1n ¼ hh; w1n i ¼
hh ; w1n i k4n
ð3Þ
ð4Þ
h2n ¼ hh; w2n i ¼
hh ; w2n i k4n
ð3Þ
h ð1Þ h ð0Þ k4n ð4Þ
þ4
;
000
000
hh ; w1n i þ h ð0Þ h ð1Þ : k5n
Thus, there is H > 0 such that
jhkn j 6 H=k4n ; k ¼ 1; 2; n 2 N:
ð41Þ
From the bounds (40), (41), (10), (11), and (12), the formulas (34) and (35) yield the bound
jf kn j 6 F=k2n ; k ¼ 1; 2; n ¼ 1; 2; . . . ;
ð42Þ
for some constant F > 0. Accordingly, by considering the bounds (40), (41), (42), (10) and (11), the formulas (31) and (32) imply that
t1cþa jukn ðtÞj 6 C=k4n ; k ¼ 1; 2; n ¼ 1; 2; . . . ; t 2 ð0; T;
ð43Þ
for some constant C > 0. Furthermore, the differential Eqs. (21), (22)the bounds (42), (43) imply that
t1cþa jDa;c ukn ðtÞj 6 K=k2n ; k ¼ 1; 2; n ¼ 1; 2; . . . ; t 2 ð0; T;
ð44Þ
for some constant K > 0. From (15) and (30) the first term of (17) is
u0 ðtÞ/0 ðxÞ ¼
f0 g ta þ 0 tc1 2ð1 xÞ: Cð1 þ aÞ CðcÞ
Clearly, this term is continuous on Q and has a continuous t fractional derivative (20) on Q. In addition, it has a continuous first and second derivative with respect to x on Q. Thus, the series in (17) and its term-by-term t fractional derivative and the first and second derivative with respect to x are all uniformly convergent in Q . Thus, being represented by uniformly convergent series of continuous functions on Q , u; Da;c u; ux , and uxx are all continuous on Q. Similarly, f is continuous on ½0; 1. Next we show that uðx; tÞ satisfies the initial and final conditions. By applying I1c to the formulas (30)–(32)), and using the formula (13) we obtain
I1c u0 ðtÞ ¼
f0
Cð2 c þ aÞ
t 1cþa þ g 0 ;
I1c u1n ðtÞ ¼ f 1n t 1þac Ea;2 ðk2n t a Þ þ g 1n Ea;1 ðk2n t a Þ;
h i I1c u2n ðtÞ ¼ f 2n t 1þac Ea;2 ðk2n t a Þ þ g 2n Ea;1 ðk2n t a Þ þ 2kn f 1n t2aþ1c E2a;2acþ2 ðk2n t a Þ þ g 1n t a E2a;aþ1 ðk2n t a Þ :
30
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
Thus from the bounds (40), (42) and (11) it is clear that the termwise fractional integral I1c of (17) converges to I1c uðx; tÞ and it is continuous on Q . When t ¼ 0 we have
I1c u0 ð0Þ ¼ g 0 ;
I1c u1n ð0Þ ¼ g 1n ;
I1c u2n ð0Þ ¼ g 2n :
ð45Þ
Therefore,
2 6 I1c uðx; tÞjt¼0 ¼ 4I1c u0 ðtÞ /0 ðxÞ þ
1 X n¼1 k¼1;2
3 7 I1c ukn ðtÞ /kn ðxÞ5 t¼0
¼ g 0 /0 ðxÞ þ
1 X g kn ukn ðtÞ /kn ðxÞ ¼ gðxÞ;
ð46Þ
n¼1 k¼1;2
as required. Finally, repeating the same argument as above and using formulas (33)–(35) we obtain uðx; TÞ ¼ hðxÞ; 0 6 x 6 1. The solution is therefore established. h
6. Examples
Example 1. Consider the problem (1) with
hðxÞ ¼ T a ð1 xÞ:
gðxÞ ¼ 0; Then clearly,
g 0 ¼ 0;
h0 ¼
Ta ; 2
and
g kn ¼ hkn ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . : So, from (33) and (30) we obtain
f0 ¼
Cð1 þ aÞ 2
;
u0 ðtÞ ¼
ta : 2
By substituting g kn and hkn into (34) and (35), we obtain f kn ¼ 0 and consequently, ukn ðtÞ ¼ 0. Thus, from (17) and (18) we obtain
f ðxÞ ¼ Cð1 þ aÞð1 xÞ;
uðx; tÞ ¼ t a ð1 xÞ:
Example 2. Consider the problem (1) with
gðxÞ ¼ CðcÞð1 xÞ;
hðxÞ ¼ T c1 ð1 xÞ:
Then clearly,
g0 ¼
CðcÞ 2
h0 ¼
;
T c1 ; 2
and
g kn ¼ hkn ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . : Accordingly, we have
f 0 ¼ 0;
u0 ðtÞ ¼
t c1 ; 2
and
f kn ¼ 0;
ukn ðtÞ ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . :
Hence the source and the solution functions are
f ðxÞ ¼ 0;
uðx; tÞ ¼ tc1 ð1 xÞ:
Acknowledgment The authors are grateful for the support provided by King Fahd University of Petroleum and Minerals through Project No. FT121003.
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
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Math., 2011, Article ID 298628, 51 pages, 2011. doi:10.1155/2011/298628. [8] N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives, Rheol. Acta 45 (2006) 765–771, http://dx.doi.org/10.1007/s00397-005-0043-5. [9] R. Hilfer, editor. Applications of Fractional Calculus in Physics, Singapore, 2000. World Scientific. [10] R. Hilfer, Fractional calculus and regular variation in thermodynamics, in: Applications of Fractional Calculus in Physics, pp. 429–463. [11] R. Hilfer, Fractional diffusion based on Riemann–Liouville fractional derivatives, J. Phys. Chem. B 104 (16) (2000) 3914–3917. [12] R. Hilfer, Fractional time evolution, in: Applications of Fractional Calculus in Physics, pp. 87–130. [13] R. Hilfer, L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995) R848–R851. [14] R. Hilfer, Y. Luchko, Zˇ. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann–Liouville fractional derivatives, Fract. Calculus Appl. Anal. 12 (2009) 299–318. [15] V.A. Il’in, L.V. Kritskov, Properties of spectral expansions corresponding to non-self-adjoint differential operators, J. Math. Sci. 116 (5) (2003) 3489– 3550. [16] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, volume 204 of Mathematics Studies, Elsevier, Amsterdam, 2006. [17] M. Kirane, S.A. Malik, M.A. Al-Gwaizb, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Methods Appl. Sci. 36 (9) (2012) 1056–1069. [18] M. Kirane, S.A. Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl. Math. Comput. 218 (2011) 163–170. [19] R. Klages, G. Radons, I. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley, 2008. [20] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, 2010. [21] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (1) (2000) 1–77. [22] C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional-order Systems and Controls, Advances in Industrial Control, Springer, 2010. [23] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lect. Notes Electr. Eng., volume 84, Springer, 2011. [24] G. Özkum, A. Demir, S. Erman, E. Korkmaz, B. Özgur. On the inverse problem of the fractional heat-like partial differential equations: determination of the source function, Adv. Math. Phys., (2013) 8 pages. doi:10.1155/2013/476154, Article ID 476154. [25] I. Petrás˘, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, 2011. [26] I. Podlubny, Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering, Acad. Press, 1999. [27] T.R. Prabhakar, A singular integral equation with a generalized Mittag–Leffler function in the kernel, Yokohama Math. J. 19 (1971) 7–15. [28] E. Scalas, R. Gorenflo, F. Mainardi, M. Meerschaert, Speculative option valuation and the fractional diffusion equation, in: J. Sabatier, J.T. Machado (Eds.), Proceedings of the IFAC Workshop on Fractional Differentiation and its Applications, (FDA 04), Bordeaux, 2004, 2004. [29] H.M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Limited, 1985. [30] H.M. Srivastava, Zˇ. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel, Appl. Math. Comput. 211 (2009) 198–210. [31] Zˇ. Tomovski, R. Hilfer, H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions, in: Integral Trans. Special Funct. 21 (11) (2010) 797–814. [32] Y. Zhang, D.A. Benson, M.M. Meerschaert, E.M. LaBolle, H.P. Scheffler, Random walk approximation of fractional-order multiscaling anomalous diffusion, Phys. Rev. E 74 (2006) 026706–026715. [33] Y. Zhang, X. Xu, Inverse source problem for a fractional diffusion equation, in: Inverse Prob. 27 (3) (2011), http://dx.doi.org/10.1088/0266-5611/27/3/ 035010. [34] L. Zhou, H.M. Selim, Application of the fractional advection-dispersion equation in porous media, Soil Sci. Soc. Am. J. 67 (4) (2003) 1079–1084.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An inverse problem for a generalized fractional diffusion Khaled M. Furati a,⇑, Olaniyi S. Iyiola a, Mokhtar Kirane b a b
King Fahd University of Petroleum & Minerals, Department of Mathematics & Statistics, Dhahran 31261, Saudi Arabia Laboratoire de Mathématiques Pôle Sciences et Technologie, Université de La Rochelle, Avenue M. Crépeau, 17042 La Rochelle Cedex, France
a r t i c l e
i n f o
a b s t r a c t We propose a method for determining the solution and source term of a generalized timefractional diffusion equation. The method is based on selecting a bi-orthogonal basis of L2 space corresponding to a nonself-adjoint boundary value problem. Uniqueness is proven and an existence result is obtained for smooth initial and final conditions. The asymptotic behavior of the generalized Mittag–Leffler function is used to relax the smoothness requirement on these conditions. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Inverse problem Fractional derivative Fractional diffusion equation Bi-orthogonal system Mittag–Leffler function Anomalous diffusion
1. Introduction and definitions We are concerned with the problem of determining the distribution uðx; tÞ, and the source term f ðxÞ, for the following problem
Da;c uðx; tÞ uxx ðx; tÞ ¼ f ðxÞ; I1c uðx; tÞjt¼0 ¼ gðxÞ; uð1; tÞ ¼ 0;
0 < x < 1; 0 < t < T; 0 < a 6 c 6 1;
uðx; TÞ ¼ hðxÞ; 0 < x < 1;
ð1Þ
ux ð0; tÞ ¼ ux ð1; tÞ; 0 < t 6 T;
where gðxÞ and hðxÞ are the initial and final conditions, respectively, and assumed to be in L2 ð0; 1Þ. The operators Ia and Da;c are defined by
Ia wðtÞ ¼
1 CðaÞ
Z
t
ðt sÞa1 wðsÞ ds;
t > 0;
a > 0;
0
" a;c
D wðtÞ ¼ D
a
# I1c wð0Þ c1 ; wðtÞ t CðcÞ
Da wðtÞ ¼ DI1a wðtÞ;
D¼
d ; dt
where C is the Gamma function. ⇑ Corresponding author. E-mail addresses: [email protected] (K.M. Furati), [email protected] (O.S. Iyiola), [email protected] (M. Kirane). http://dx.doi.org/10.1016/j.amc.2014.10.046 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð2Þ
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
25
The two parameter family of fractional derivatives Da;c of order a and type c allows for interpolation between the Riemann–Liouville and Caputo derivatives. The type-parameter c produces more types of stationary states and provide an extra degree of freedom for the initial condition. Models based on this derivative are considered in [6,10,12,14,31]. Although the initial condition is given as a fractional integral, it is possible to attribute a physical meaning to it and obtain it by appropriate measurements or observations. See the paper by Heymans and Podlubny [8]. Fractional calculus and fractional differential equations have become an essential tool in modeling many different phenomena in many fields such as control theory of dynamical systems [4,22], viscoelasticity [20], and nanotechnology [3]. Anomalous transport and anomalous diffusion [19], random walk [13,32], financial modeling [28] are also modeled using fractional models. Other physical and engineering processes are given in [23,25]. Also, more applications can be found in the surveys in [16,26] and the collection of applications in [9]. The main property of the fractional derivatives that enable them to play this role is the non-locality which is an intrinsic property of many complex systems. See for example [11]. In particular, fractional models are increasingly adopted for anomalous diffusion with slow diffusion [1,21,34]. Inverse problems for fractional diffusion equations that involve Caputo fractional derivative with respect to time are considered by a number of researchers. Zhang and Xu [33] showed that the source term can be uniquely identified given homogeneous boundary conditions. Kirane and Malik [18] considered a one dimensional fractional diffusion equation with a non-local non-self adjoint boundary condition and given solution at some later time. Using a bi-orthogonal system, they constructed a solution and source function and showed the uniqueness. Furthermore, they proved an existence result when the initial and final conditions are both smooth enough. Kirane et al. [17] extended the results to the problem in two-dimensions and showed the continuous dependence on the data. Recently, Aleroev et al. [2] considered an inverse source problem with an integral type over-determination condition. Özkum et al. [24] studied the inverse problem of determining the unknown source function of the linear fractional differential equation with variable coefficients using Adomian decomposition method. In this paper we generalize the problem in [18]. We use the same spatial bi-orthogonal system to obtain a formal separable solution and source function, and show their uniqueness. Then we utilize the asymptotic behavior of the generalized Mittag–Leffler function to obtain an existence result under some smoothness requirements of the initial and final conditions. The rest of the paper is organized as follows. In Section 2 we list some properties of the generalized fractional derivative operator and the generalized Mittag–Leffler functions. In Section 3 we construct the formal series solution and source term. In Sections 4 and 5 we obtain the uniqueness and existence results. In Section 6 we present some examples. 2. Generalized fractional derivative and Mittag–Leffler functions Notice that the derivative Da;c w in (2) reduces to the familiar Riemann–Liouville derivative and Caputo fractional derivative for c ¼ a and c ¼ 1, respectively,
Da;a wðtÞ ¼ Da wðtÞ;
Da;1 wðtÞ ¼ Da ½wðtÞ wð0Þ :¼ c Da wðtÞ:
Moreover, if I1c w 2 AC½0; T then
Da;c wðtÞ ¼ Ica Dc wðtÞ ¼ Ica DI1c wðtÞ:
ð3Þ
In particular, if c ¼ bð1 aÞ þ a for some 0 < b 6 1 then from (3) we have
Da;c ¼ Ibð1aÞ DIð1bÞð1aÞ ;
ð4Þ
which is the derivative operator introduced by Hilfer [12]. Srivastava and Tomovski [30] and Tomovski et al. [31] studied further the properties and applications of the derivative operator (4) and the associated generalizations of Mittag–Leffler function. In [30] the following generalization of Mittag– Leffler function,
Eqa;b;m ðzÞ ¼
1 X
ðqÞmk
zk
Cðak þ bÞ k! k¼0
;
z; b; q 2 C; Re a > maxf0; Re m 1g; Re m > 0;
ð5Þ
was introduced and investigated. This generalization is a special case of the Fox–Wright function p Wq :
Eqa;b;m ðzÞ ¼
ðq; mÞ 1 ;z : 1 W1 CðqÞ ðb; aÞ
More details on families of Fox–Wright function can be found in [29]. In this paper we use the following special case of (5), m ¼ 1, q
Ea;b ðzÞ ¼
1 X
ðqÞk zk ; Cðak þ bÞ k! k¼0
z; b; q 2 C; Re a > 0;
ð6Þ
26
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
where
Cðq þ kÞ : CðqÞ
ðqÞk ¼ qðq þ 1Þ ðq þ k 1Þ ¼
This function was introduced in [27] and further studied in [7]. Some special cases of this function are the Mittag–Leffler function in one and two parameters
Ea;b ðzÞ ¼ E1a;b ðzÞ;
Ea ðzÞ ¼ E1a;1 ðzÞ:
The function Eqa;b is an entire function [27] and thus bounded in any finite interval. Also, this function satisfies the recurrence relations [7,16]
kza Ea;aþb ðkza Þ ¼ Ea;b ðkza Þ
1
CðbÞ
;
z; b; k 2 C; Re a > 0;
aE2a;b ðkza Þ ¼ ð1 þ a bÞ Ea;b ðkza Þ þ Ea;b1 ðkza Þ; z; b; k 2 C; Re a > 0:
ð7Þ ð8Þ
The relations (7) and (8) yield the relation
akza E2a;b ðkza Þ ¼ ð1 þ a bÞ Ea;ba ðkza Þ þ Ea;ba1 ðkza Þ; z; b; k 2 C; Re a > 0:
ð9Þ
From Theorem 1.6 in [26] we have the bound
kt a jEa;b ðkt a Þj 6 M;
0 < a < 2; b 2 C; t P 0;
k P 0;
ð10Þ
for some M ¼ Mða; bÞ > 0. Thus, from (9) and (10) we obtain the bound,
k2 t2a E2a;b ðkt a Þ 6 M;
0 < a < 2; b 2 C; t P 0; k P 0;
ð11Þ
for some M > 0. Furthermore, from the positivity and monotonicity of Ea;b ðkta Þ we can show that
kEa;b ðkta Þ > k0 Ea;b ðk0 ta Þ > 0;
k > k0 > 0; t > 0; b P 2a; 0 < a < 1:
ð12Þ
From [7], we have the formula
h i I1c t b1 Eqa;b ðkt a Þ ¼ t bc Eqa;bcþ1 ðkt a Þ;
0 6 c 6 1; a; b > 0; k 2 R:
ð13Þ
For the Laplace transform we have the formula
n o q L t b1 Ea;b ðkt a Þ ¼
saqb ; ðsa kÞq
Re s > 0; jksa j < 1;
ð14Þ
where a; b; k; q 2 C; Re a > 0; Re b > 0, and Re q > 0. See for example (1.9.13) in [16] and (11.8) in [7]. 3. Formal solution and source term As in [18], the boundary conditions suggest the bi-orthogonal pair of dual Riesz bases for the space L2 ð0; 1Þ:
U ¼ f/0 ; /1n ; /2n g1 n¼1 ;
W ¼ fw0 ; w1n ; w2n g1 n¼1 ;
where,
/0 ðxÞ ¼ 2ð1 xÞ;
/1n ðxÞ ¼ 4ð1 xÞ cos kn x;
/2n ðxÞ ¼ 4 sin kn x;
ð15Þ
and
w0 ðxÞ ¼ 1;
w1n ðxÞ ¼ cos kn x;
w2n ðxÞ ¼ x sin kn x;
ð16Þ
where kn ¼ 2pn. In [15] it is proved that the sequences U and W are Riesz bases. Observe that neither sequence is orthogonal. We denote the inner product in L2 ð0; 1Þ by
hf ; gi ¼
Z
1
f ðxÞ gðxÞ dx:
0
We seek a solution and source function of (1) in the form
uðx; tÞ ¼ u0 ðtÞ /0 ðxÞ þ
1 X ukn ðtÞ /kn ðxÞ;
ð17Þ
n¼1 k¼1;2
f ðxÞ ¼ f 0 /0 ðxÞ þ
1 X f kn /kn ðxÞ: n¼1 k¼1;2
ð18Þ
27
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
Also we can write
g ¼ g 0 /0 þ
1 X
g kn /kn ;
n¼1 k¼1;2
where using the biorthogonal basis (16) we obtain
g 0 ¼ hg; w0 i;
g kn ¼ hg; wkn i; k ¼ 1; 2; n ¼ 1; 2; . . . :
ð19Þ
Substituting (17) and (18) into (1) yields the following system of ordinary fractional differential equations
Da;c u0 ðtÞ ¼ f 0 ;
ð20Þ
Da;c u1n ðtÞ þ k2n u1n ðtÞ ¼ f 1n ;
n ¼ 1; 2; . . . ;
ð21Þ
Da;c u2n ðtÞ þ k2n u2n ðtÞ 2kn u1n ðtÞ ¼ f 2n ;
n ¼ 1; 2; . . . :
ð22Þ
From the initial and final conditions in (1), we obtain
I1c u0 ð0Þ ¼ g 0 ; u0 ðTÞ ¼ h0 ;
I1c ukn ð0Þ ¼ g kn ; k ¼ 1; 2; n ¼ 1; 2; . . . ;
ukn ðTÞ ¼ hkn ; k ¼ 1; 2; n ¼ 1; 2; . . . ;
ð23Þ
where again fh0 ; h1n ; h2n g are the coefficients of the series expansion of h in the basis (15),
h0 ¼ hh; w0 i;
hkn ¼ hh; wkn i;
k ¼ 1; 2;
n ¼ 1; 2; . . . :
ð24Þ
For the Laplace transform, we have the formula [12]
L Da;c w ¼ sa Lfwg sac I1c wð0Þ:
ð25Þ
Accordingly, by applying the Laplace transform to (20) we obtain
U 0 ðsÞ ¼
f0 g þ 0; saþ1 sc
and hence
u0 ðtÞ ¼
f0
ta þ
Cð1 þ aÞ
g0
CðcÞ
t c1 :
ð26Þ
By applying Laplace transform to (21) we obtain
ðsa þ k2n ÞU 1n ðsÞ ¼
f 1n þ g 1n sac : s
So
U 1n ðsÞ ¼ f 1n
1 sðsa
þ
k2n Þ
þ g 1n
sac sa
þ k2n
ð27Þ
:
From formula (14) we obtain
u1n ðtÞ ¼ f 1n ta Ea;aþ1 ðk2n ta Þ þ g 1n tc1 Ea;c ðk2n t a Þ:
ð28Þ
Again, by applying Laplace transform to (22) and then substituting (27) we obtain
U 2n ðsÞ ¼
f 2n sðsa þ k2n Þ
þ
g 2n sac sa þ k2n
þ
2kn sa þ k2n
U 1n ðsÞ ¼
f 2n sðsa þ k2n Þ
þ
g 2n sac sa þ k2n
" þ 2kn f 1n
1 sðsa þ k2n Þ
þ g 1n 2
#
sac 2
ðsa þ k2n Þ
:
Thus formula (14) implies that
h i u2n ðtÞ ¼ f 2n ta Ea;aþ1 ðk2n t a Þ þ g 2n t c1 Ea;c ðk2n ta Þ þ 2kn f 1n t 2a E2a;2aþ1 ðk2n t a Þ þ g 1n t aþc1 E2a;aþc ðk2n t a Þ :
ð29Þ
As a result, we can write the coefficients u0 and ukn ; k ¼ 1; 2, in the form
u0 ðtÞ ¼
f0
Cð1 þ aÞ
ta þ
g0
CðcÞ
t c1 ;
ð30Þ
u1n ðtÞ ¼ f 1n E 1n ðtÞ þ g 1n E 2n ðtÞ;
ð31Þ
u2n ðtÞ ¼ f 2n E 1n ðtÞ þ g 2n E 2n ðtÞ þ 2kn ½f 1n E 3n ðtÞ þ g 1n E 4n ðtÞ;
ð32Þ
28
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
where
E 1n ðtÞ ¼ t a Ea;aþ1 ðk2n t a Þ; E 3n ðtÞ ¼ t
2a
E 2n ðtÞ ¼ t c1 Ea;c ðk2n ta Þ;
2
Ea;2aþ1 ðk2n t a Þ;
E 4n ðtÞ ¼ t aþc1 E2a;aþc ðk2n t a Þ:
By virtue of (30), (31), (32) and (23), we obtain the unknowns f 0 and f kn ; k ¼ 1; 2; n ¼ 1; 2; . . ., as
g 0 c1 ; T CðcÞ T h1n g 1n E 2n ðTÞ ¼ ; E 1n ðTÞ h2n g 2n E 2n ðTÞ 2kn ½f 1n E 3n ðTÞ þ g 1n E 4n ðTÞ ¼ : E 1n ðTÞ
f0 ¼ f 1n f 2n
Cð1 þ aÞ a
h0
ð33Þ ð34Þ ð35Þ
4. Uniqueness of solution In this section we present a uniqueness result for the solution and source term of the problem (1). Theorem 1. Let g; h 2 L2 ð0; 1Þ. Let u 2 ðLð0; TÞ \ Cð0; TÞÞ L2 ð0; 1Þ and f 2 L2 ð0; 1Þ. If fuðx; tÞ; f ðxÞg satisfies the inverse problem (1) then it is unique.
¼ u1 u2 and f ¼ f 1 f 2 . Then u satisfies the problem Proof. Suppose fu1 ðx; tÞ; f 1 ðxÞg and fu2 ðx; tÞ; f 2 ðxÞg satisfy (1). Let u
ðx; tÞ u xx ðx; tÞ ¼ f ðxÞ; Da;c u
0 < x < 1;
ðx; tÞjt¼0 ¼ 0; ðx; TÞ ¼ 0; I1c u u x ð0; tÞ ¼ u x ð1; tÞ; ð1; tÞ ¼ 0; u u
0 < t < T; ð36Þ
0 < x < 1; 0 < t 6 T:
Using the bases in (15) we can write
ðx; tÞ ¼ u 0 ðtÞ /0 ðxÞ þ u
1 X kn ðtÞ /kn ðxÞ; u
ð37Þ
n¼1 k¼1;2
f ðxÞ ¼ f 0 / ðxÞ þ 0
1 X f / ðxÞ: kn kn
ð38Þ
n¼1 k¼1;2
Then, the coefficients
0 ðtÞ; u
f 0 ;
kn ðtÞ; u
f ; k ¼ 1; 2; n ¼ 1; 2; . . . : kn
are given by (30)–(32) and (33)–(35), with
g 0 ¼ h0 ¼ 0;
g kn ¼ hkn ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . :
Thus clearly we have
f 0 ¼ f ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . : kn and
0 ðtÞ ¼ u kn ðtÞ 0; 0 < t 6 T; k ¼ 1; 2; n ¼ 1; 2; . . . : u ðx; tÞ ¼ 0 and f ðxÞ ¼ 0 and thus we have the uniqueness of the solution and the source term. Consequently, u
h
5. Existence of the solution In Section 3 we constructed a formal solution and source term for the problem (1). In this section we show that under some assumptions on the initial and final conditions we obtain the classical solution and continuous source term. Theorem 2. Let the following conditions hold g 2 C 2 ½0; 1; gð1Þ ¼ 0 00 0 0 h 2 C 4 ½0; 1; hð1Þ ¼ h ð1Þ ¼ 0; h ð1Þ ¼ h ð0Þ.
K.M. Furati et al. / Applied Mathematics and Computation 249 (2014) 24–31
29
Then the formal solution constructed is a classical solution of problem (1). Proof. Let Q ¼ ð0; T ½0; 1 and Q ¼ ½; T ½0; 1 Q . We show that the series corresponding to u; ux ; uxx ; Da;c u are uniformly convergent and represent continuous functions on Q , for any > 0. Also we show that the series representation of f is uniformly convergent in ½0; 1. This is shown by bounding all these series by over-harmonic series then applying Weierstrass M-test. Since g 2 C 2 ½0; 1, by integration by parts we have
g 1n ¼ hg; w1n i ¼ g 2n ¼ hg; w2n i ¼
hg 00 ; w1n i k2n hg 00 ; w2n i k2n
þ
g 0 ð1Þ g 0 ð0Þ k2n 00 2hg ; w1n i k3n
þ
;
2½g 0 ð1Þ g 0 ð0Þ k3n
:
Also, since the sequence in (16) is a Riesz basis then it is a Bessel sequence [5] and thus there is C > 0 such that 1 X 2 hg 00 ; wkn i2 6 C kg 00 k :
ð39Þ
n¼1 k¼1;2
This implies that the set fhg 00 ; wkn ig is bounded. Accordingly there is G > 0 such that
jg kn j 6 G=k2n ; k ¼ 1; 2; n 2 N:
ð40Þ
ð4Þ
4
Similarly, since h 2 C ½0; 1; fhh ; wkn ig is bounded and through integration by parts we obtain ð4Þ
h1n ¼ hh; w1n i ¼
hh ; w1n i k4n
ð3Þ
ð4Þ
h2n ¼ hh; w2n i ¼
hh ; w2n i k4n
ð3Þ
h ð1Þ h ð0Þ k4n ð4Þ
þ4
;
000
000
hh ; w1n i þ h ð0Þ h ð1Þ : k5n
Thus, there is H > 0 such that
jhkn j 6 H=k4n ; k ¼ 1; 2; n 2 N:
ð41Þ
From the bounds (40), (41), (10), (11), and (12), the formulas (34) and (35) yield the bound
jf kn j 6 F=k2n ; k ¼ 1; 2; n ¼ 1; 2; . . . ;
ð42Þ
for some constant F > 0. Accordingly, by considering the bounds (40), (41), (42), (10) and (11), the formulas (31) and (32) imply that
t1cþa jukn ðtÞj 6 C=k4n ; k ¼ 1; 2; n ¼ 1; 2; . . . ; t 2 ð0; T;
ð43Þ
for some constant C > 0. Furthermore, the differential Eqs. (21), (22)the bounds (42), (43) imply that
t1cþa jDa;c ukn ðtÞj 6 K=k2n ; k ¼ 1; 2; n ¼ 1; 2; . . . ; t 2 ð0; T;
ð44Þ
for some constant K > 0. From (15) and (30) the first term of (17) is
u0 ðtÞ/0 ðxÞ ¼
f0 g ta þ 0 tc1 2ð1 xÞ: Cð1 þ aÞ CðcÞ
Clearly, this term is continuous on Q and has a continuous t fractional derivative (20) on Q. In addition, it has a continuous first and second derivative with respect to x on Q. Thus, the series in (17) and its term-by-term t fractional derivative and the first and second derivative with respect to x are all uniformly convergent in Q . Thus, being represented by uniformly convergent series of continuous functions on Q , u; Da;c u; ux , and uxx are all continuous on Q. Similarly, f is continuous on ½0; 1. Next we show that uðx; tÞ satisfies the initial and final conditions. By applying I1c to the formulas (30)–(32)), and using the formula (13) we obtain
I1c u0 ðtÞ ¼
f0
Cð2 c þ aÞ
t 1cþa þ g 0 ;
I1c u1n ðtÞ ¼ f 1n t 1þac Ea;2 ðk2n t a Þ þ g 1n Ea;1 ðk2n t a Þ;
h i I1c u2n ðtÞ ¼ f 2n t 1þac Ea;2 ðk2n t a Þ þ g 2n Ea;1 ðk2n t a Þ þ 2kn f 1n t2aþ1c E2a;2acþ2 ðk2n t a Þ þ g 1n t a E2a;aþ1 ðk2n t a Þ :
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Thus from the bounds (40), (42) and (11) it is clear that the termwise fractional integral I1c of (17) converges to I1c uðx; tÞ and it is continuous on Q . When t ¼ 0 we have
I1c u0 ð0Þ ¼ g 0 ;
I1c u1n ð0Þ ¼ g 1n ;
I1c u2n ð0Þ ¼ g 2n :
ð45Þ
Therefore,
2 6 I1c uðx; tÞjt¼0 ¼ 4I1c u0 ðtÞ /0 ðxÞ þ
1 X n¼1 k¼1;2
3 7 I1c ukn ðtÞ /kn ðxÞ5 t¼0
¼ g 0 /0 ðxÞ þ
1 X g kn ukn ðtÞ /kn ðxÞ ¼ gðxÞ;
ð46Þ
n¼1 k¼1;2
as required. Finally, repeating the same argument as above and using formulas (33)–(35) we obtain uðx; TÞ ¼ hðxÞ; 0 6 x 6 1. The solution is therefore established. h
6. Examples
Example 1. Consider the problem (1) with
hðxÞ ¼ T a ð1 xÞ:
gðxÞ ¼ 0; Then clearly,
g 0 ¼ 0;
h0 ¼
Ta ; 2
and
g kn ¼ hkn ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . : So, from (33) and (30) we obtain
f0 ¼
Cð1 þ aÞ 2
;
u0 ðtÞ ¼
ta : 2
By substituting g kn and hkn into (34) and (35), we obtain f kn ¼ 0 and consequently, ukn ðtÞ ¼ 0. Thus, from (17) and (18) we obtain
f ðxÞ ¼ Cð1 þ aÞð1 xÞ;
uðx; tÞ ¼ t a ð1 xÞ:
Example 2. Consider the problem (1) with
gðxÞ ¼ CðcÞð1 xÞ;
hðxÞ ¼ T c1 ð1 xÞ:
Then clearly,
g0 ¼
CðcÞ 2
h0 ¼
;
T c1 ; 2
and
g kn ¼ hkn ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . : Accordingly, we have
f 0 ¼ 0;
u0 ðtÞ ¼
t c1 ; 2
and
f kn ¼ 0;
ukn ðtÞ ¼ 0; k ¼ 1; 2; n ¼ 1; 2; . . . :
Hence the source and the solution functions are
f ðxÞ ¼ 0;
uðx; tÞ ¼ tc1 ð1 xÞ:
Acknowledgment The authors are grateful for the support provided by King Fahd University of Petroleum and Minerals through Project No. FT121003.
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