Interdisciplinary Information Sciences Vol. 16, No. 1 (2010) 127–137 #Graduate School of Information Sciences, Tohoku University ISSN 1340-9050 print/1347-6157 online DOI 10.4036/iis.2010.127
Neumann-Series Solution of Fractional Differential Equation Tohru MORITA 1; and Ken-ichi SATO2 1
2
3-4-34 Nakayama-Yoshinari, Aoba-ku, Sendai 989-3203, Japan College of Engineering, Nihon University, Koriyama 963-8642, Japan Received February 6, 2009; final version accepted October 1, 2009
For the initial-value problem of a fractional differential equation with constant coefficients, the solution is given by the Neumann series for the corresponding Volterra integral equation, and is expressed by a generalized MittagLeffler function. The derivation is presented in the style of operational calculus, which is formulated by distribution theory. KEYWORDS: fractional differential equation, Neumann series, generalized Mittag-Leffler function, operational calculus, distribution theory
1. Introduction The fractional derivative and fractional differential equation have long been discussed; see e.g. [4, 8, 12]. Recently its applicability is discussed, e.g. in the problem of viscoelastic behaviors of polymeric substances [3, 12, 15]. In Chapter 3 in the book [12] and in [9–11], the existence and uniqueness of a fractional differential equation (fDE) were proved with the aid of the corresponding facts on a Volterra integral equation. It is known that the unique solution of a Volterra integral equation is given by the Neumann series [13, p. 146]. In the present paper, we study the fDE with constant coefficients which are simply called fDE below in this paper. The solution of simple fDE was given in [2, 6], with the aid of Mikusin´ski’s operational calculus [7], where the operators are differentiation and convolution with a function. We can regard the calculation as an application of the Neumann series to the Volterra integral equations corresponding to the simple fDE. In a recent paper by Morita and Sato [11], the initial-value problem of an fDE was studied in terms of distribution theory, where the description was given in the style of classical operational calculus. In that theory, distributions are right-sided distributions in the space D0R , which consists of regular distributions and their derivatives, where regular distributions in D0R are those locally integrable functions in R that have a support bounded on the left. The operators which appear in solving an fDE in D0R are the fractional integrals, fractional differentiations, their linear combinations, their inverses, which always exist, and their products. These operators in D0R commute with each other, and hence algebraic manipulation in the style of classical operational calculus is applicable. The primary purpose of the present paper is to show how the Neumann-series solution of the initial-value problem of the general fDE with constant coefficients is obtained in this way. In particular, it is shown that the Green’s function of general fDE obtained in this way takes the form of the generalized Mittag-Leffler function. In this place, the definition of the Riemann and Liouville (RL) fractional integral and derivative is given and then the calculation is presented to show how the Green’s function of the simplest fDE, which is given by (8) below, is obtained in the form of the two-parameter Mittag-Leffler function, as an illustrative example. In the explanation, we use the notations a Rb :¼ ða; bÞ, a R :¼ ða; 1Þ, Rb :¼ ð1; bÞ and R :¼ ð1; 1Þ for real numbers a and b satisfying a < b, to represent the sets of real numbers. We use the notations Z and N to denote the sets of all integers and all positive integers, respectively. For a 2 Z and b 2 Z satisfying a b, a Zb and a Z are used for the sets of all integers i satisfying a i b and i a, respectively. Hence N ¼ 1 Z. We use the Heaviside step function Hðt bÞ for b 2 R, such that the product f ðtÞHðt bÞ with a function f ðtÞ is assumed to be equal to f ðtÞ for t > b and to 0 for t 5 b. We use dxe for real number x to denote the least integer not less than x. Let f ðtÞ be integrable in the interval ðb; cÞ for a fixed b 2 R and all c 2 b R, so that the product f ðtÞHðt bÞ is locally integrable in R. Then the RL fractional integral b D R f ðtÞ for 2 0 R, is defined by Z t 1 1 D f ðtÞ ¼ ðt xÞ f ðxÞ dx Hðt bÞ; ð1Þ b R ðÞ b and we set b D0R f ðtÞ ¼ f ðtÞHðt bÞ. For 2 0 R, we put n ¼ de, and then the RL fractional derivative b DR f ðtÞ is defined by
Corresponding author. E-mail:
[email protected]
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MORITA and SATO
b DR f ðtÞ
¼
dn n D f ðtÞ Hðt bÞ; b dtn R
ð2Þ
when b DRn f ðtÞ exists and is n times differentiable for t 2 b R. We use 0 DR f ðtÞ in place of 0 D1R f ðtÞ. Here b DR f ðtÞ for < 0 and > 0 is defined for t 2 R, but they are usually defined only for t 2 b R, without the factor Hðt bÞ on the righthand sides of (1) and (2) [4, 8, 12]. For the operator b DR , there exists theR problem that the operators do not commute with each other. For instance, t 1 0 when b ¼ 0 and f ðtÞ ¼ 1, 0 D1 R f ðtÞ ¼ 0 f ðxÞdx HðtÞ ¼ tHðtÞ and 0 DR 0 DR f ðtÞ ¼ HðtÞ ¼ 0 DR f ðtÞ ¼ f ðtÞHðtÞ, while 1 0 0 DR f ðtÞ ¼ 0 and 0 DR 0 DR f ðtÞ ¼ 0 6¼ 0 DR f ðtÞ ¼ f ðtÞHðtÞ. Let f ðtÞHðt bÞ be locally integrable in R. Then it is a regular distribution in D0R , and its fractional integral D ½ f ðtÞHðt bÞ for 2 0 R is given by D ½ f ðtÞHðt bÞ ¼ b D R f ðtÞ;
ð3Þ
and we set D0 ½ f ðtÞHðt bÞ ¼ f ðtÞHðt bÞ. If 2 0 R, n ¼ de, and b Dn R f ðtÞ is n times differentiable in R, then the fractional derivative D ½ f ðtÞHðt bÞ is given by D ½ f ðtÞHðt bÞ ¼ b DR f ðtÞ:
ð4Þ
For a right-sided distribution hðtÞ 2 D0R , we use DhðtÞ in place of D1 hðtÞ, and the index law D D hðtÞ ¼ Dþ hðtÞ;
ð5Þ
is always valid for 2 R and 2 R. For instance, D1 HðtÞ ¼ tHðtÞ, DD1 HðtÞ ¼ D0 HðtÞ ¼ HðtÞ, DHðtÞ ¼ ðtÞ is Dirac’s delta function, and D1 DHðtÞ ¼ HðtÞ ¼ D0 HðtÞ. In Appendix A, the definitions of right-sided distributions in D0R and of their fractional integral and derivative are given, and then equalities (3) and (4) and the index law (5) are proved. 1 Let f ðtÞ :¼ ðþ1Þ t for 2 R excluding negative integers, and f ðtÞ :¼ 0 for 2 N. Then, for 2 ð1Þ R and 2 R, we have 8 < þ 1, > 0 DR f ðtÞ ¼ f ðtÞHðtÞ; > > > < D f ðtÞ ¼ HðtÞ; ¼ , 0 R ð6Þ D ½ f ðtÞHðtÞ ¼ > ðtÞ; 0 D f ðtÞ ¼ 0; ¼ þ 1, > R > > : 1 ðtÞ; 0 DR f ðtÞ ¼ f ðtÞHðtÞ; = þ 1. D By putting ¼ and ¼ 0 in (6) and then using the fact that D HðtÞ ¼ D1 ðtÞ, we obtain 1 t HðtÞ; ð þ 1Þ
ð7Þ
þ cuðtÞ ¼ f ðtÞ; t > 0;
ð8Þ
D1 ðtÞ ¼ for 2 ð1Þ R. The simplest fDE is given by 0 DR uðtÞ
where 2 0 R and c 2 R satisfy c 6¼ 0. In solving the initial-value problem of this equation, we use the Green’s function GðtÞ, which satisfies ðD þ cÞGðtÞ ¼ ðtÞ;
ð9Þ
1 1 and GðtÞ ¼ 0 for t < 0. This (9) is converted to ð1 þ cD ÞGðtÞ ¼ D ðtÞ ¼ ðÞ t HðtÞ. By (3) and (1), this is regarded as a Volterra integral equation and its Neumann-series solution is given by
GðtÞ ¼
1 1 X X 1 ðcÞn tnþ1 n n D ðtÞ ¼ ðcÞ D ðtÞ ¼ HðtÞ 1 þ cD ðn þ Þ n¼0 n¼0
¼ E0 ð; ; ct Þt1 HðtÞ; by using (7). Here E0 ð; ; xÞ for 2 0 R and 2 0 R, is given by 1 X xn E0 ð; ; xÞ ¼ : ðn þ Þ n¼0
ð10Þ
ð11Þ
This is identical with the two-parameter Mittag-Leffler function E; ðxÞ [1, 4, 12]. As stated above, the primary purpose of the present paper is to show how the Neumann-series solution of the initialvalue problem of the general fDE with constant coefficients is obtained in this way. In section 2, we state the assumption imposed on the RL fractional derivative of a function when it appears in an initial-value problem of a fDE, and its relation with the fractional derivative of a right-sided distribution. Based on it, we convert the general fDE for a function
Neumann-Series Solution of Fractional Differential Equation
129
uðtÞ of t 2 0 R, to the corresponding fDE for a regular distribution uðtÞHðtÞ in D0R , and give the solution, in section 3. The Green’s function, which appears in the solution, is obtained by generalizing the derivation in (10) for the simplest case. As the result, the Green’s function is expressed by a generalized Mittag-Leffler function, in section 4. The solution is compared with the one given in a recent book by Kilbas et al. [4], where the solution is confirmed to satisfy the given equation with the aid of Laplace transform. In section 5, we present the Green’s function for some simple fDE. A brief discussion is given on an fDE in which the solution may take nonzero values in all range of R, in section 6. A concluding remark is presented in section 7. Another typical fractional differentiation is the one due to Caputo. The solution of the general fDE, where the fractional derivatives are Caputo’s, was given by Luchko [5], with the aid of Mikusin´ski’s operational calculus [7]. In Appendix B, it is shown that the same problem is solved by simple alterations of preceding calculations, in the present formalism. The result obtained is confirmed to be the same as that given by Luchko [5]. In Appendix B, a remark is given on the review of Luchko’s work in [4]. Comparisons of the present results with related ones in preceding publications are stated in Remarks 1–4, in sections 4 and 5, and in Appendices A and B. It is well-known that the order of the fractional integral and differentiation may take a complex value [4]. In Appendix C, comment is given what corrections are needed in the arguments in the text, when the order is assumed to take a complex value. In the rest of this section, we give some explanations of terminologies. A function f ðtÞ of real variable t in R is said to be continuous and differentiable, when it is absolutely continuous on R [13, p. 50], so that f ðtÞ is continuous on R, Rc 0 d 0 f ðtÞ ¼ f ðtÞ is locally integrable in R, and f ðcÞ f ðbÞ ¼ f ðtÞdt if 1 < b < c < 1. A function f ðtÞ is said to be b dt n times differentiable in a Rb , when f ðtÞ itself and its derivatives up to ðn 1Þth order are continuous and differentiable in a Rb . When f ðtÞ is continuous on ðb; cÞ for b 2 R and c 2 b R, f ðbþÞ is used to denote limt!bþ f ðtÞ. A function f ðtÞ is said to have a support bounded on the left, when there exists such a finite b 2 R that f ðtÞ ¼ 0 for t b. A function gðtÞ is said to have a support bounded on the right, when there exists such a finite c 2 R that gðtÞ ¼ 0 for t c. In the present paper as well as in [11], a function is simply said to be locally integrable, when it is defined in R and is integrable in the Lebesgue sense over every finite interval in R [13, p. 7].
2. Fractional derivatives in the intial-value problem of fDE 2.1
Riemann-Liouville fractional integral and derivative
Let b 2 R and f ðtÞ be such a function that f ðtÞHðt bÞ is locally integrable. We then define the Riemann-Liouville (RL) fractional integral b D R f ðtÞ of f ðtÞ, for 2 0 R, by (1). Now b DR f ðtÞ ¼ b DR ½ f ðtÞHðt bÞ, b DR f ðtÞ is locally integrable, and b DR f ðbþÞ ¼ 0 for 1. For 2 0 R, the RL fractional derivative b DR f ðtÞ is defined by (2). Let b DR uðtÞ for b 2 R and 2 0 R be the RL fractional derivative of a function uðtÞ of t 2 R, let u for 2 ð1Þ R be given by u :¼ b DR uðbþÞ
ð12Þ
when it exists, and let N :¼ de. When b DR uðtÞ appears in an initial-value problem of fDE, we assume for every n 2 0 ZN1 that b DnþN uðtÞ is continuous and differentiable at t > b, and there exists a finite value unþN . R Because of this assumption, b DR uðtÞ is locally integrable, and the Taylor series expansion of b DN uðtÞ around t ¼ b R is expressed as N1 X ðt bÞn N uðtÞ ¼ uNþn Hðt bÞ þ wðtÞ; ð13Þ b DR n! n¼0 where wðtÞ is a function of oððt bÞN1 Þ as t ! bþ and satisfies wðtÞ ¼ wðtÞHðt bÞ and b DR uðtÞ ¼ In the following, when b DR uðtÞ appears for 2 0 R, we adopt this assumption. 2.2
dN dtN
wðtÞ.
Fractional integral and derivative of a regular distribution
In Introduction, the fractional integral and derivative of a regular distribution are given by (3) and (4), but the latter is useful only for a special function. We now give the corresponding formula for the function uðtÞ studied in the preceding subsection. Lemma 1. Let 2 0 R and b DR uðtÞ exist. Then the product uðtÞHðt bÞ and b DR uðtÞ are regular distributions belonging to D0R , and they are related by de1 X D ½uðtÞHðt bÞ ¼ b DR uðtÞ þ uk1 Dk ðt bÞ: ð14Þ k¼0
Proof. The fact that b DR uðtÞ 2 D0R is due to the assumption stated in the preceding subsection. We put N ¼ de, note that DN ½uðtÞHðt bÞ ¼ b DN uðtÞ due to (3), and apply DN to (13). Then we obtain (14), by using (5) and (7), the R N N N dN equalities D wðtÞ ¼ b DR wðtÞ ¼ dtN wðtÞ due to (4) and b DR uðtÞ ¼ dtd N wðtÞ as noted after (13).
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In solving the initial-value problem of fDE, we stressed repeatedly the following lemma [10, 11]. Lemma 2. Let 1 2 R and 2 2 R be two numbers satisfying the conditions that 2 < 1 , the difference 1 2 is not an integer, and b DR1 uðtÞ and b DR2 uðtÞ exist. Then u ¼ 0 for 2 R satisfying 2 1. Proof. We write the expression which is obtained by using (14) for ¼ 2 and ¼ 1 , in D2 ½uðtÞHðt bÞ D2 1 D1 ½uðtÞHðt bÞ ¼ 0. Then the condition that the expression is locally integrable results in the conclusion that u2 k1 ¼ 0 for k = 0 and hence for 2 k 1 5 2 1, and u1 k1 ¼ 0 for k = 1 2 and hence for 1 k 1 5 2 1.
3. Solution of fDE 3.1
Initial-value problem of fDE
We now consider the fDE m 0 DR uðtÞ
þ
m1 X
cl 0 DRl uðtÞ ¼ f ðtÞHðtÞ;
ð15Þ
l¼0
where m 2 N, cl 2 R for l 2 0 Zm1 are constants, l 2 R for l 2 0 Zm and m > 0. Here we assume that f ðtÞHðtÞ is locally integrable, and l < m and cl 6¼ 0 for l 2 0 Zm1 . Usually l > 0 for l 2 1 Zm1 and 0 ¼ 0, but it may happen that l < 0 for some l 2 0 Zm1 . Using Lemma 1 in (15), we obtain ! m1 X m l D þ cl D ½uðtÞHðtÞ ¼ f ðtÞHðtÞ þ vðtÞ; ð16Þ l¼0
vðtÞ :¼
m X
"
dX l e1
cl
# k
ul k1 D ðtÞ ;
ð17Þ
k¼0
l¼0 l >0
where cm ¼ 1. For the initial values ful k1 g, the following lemma follows from Lemma 2. Lemma 3. When the fDE (15) is considered, ul k1 ¼ 0 must hold for every l 2 0 Zm and k 2 0 Z satisfying l k 1 1, where is the greater of 0 and the greatest number l which belongs to the set fl gl20 Zm1 and for which m l is not an integer. Now vðtÞ may be expressed as vðtÞ ¼
2
m X l¼0 l >
Applying Dm to (16), we obtain 1þ
m1 X
6 4 cl
dX l e1 k¼0 k
ð22Þ
k¼0 k 0, and on m complex variables fzl gl20 Zm1 , by
¼
n!
Qm1
E ðfl gl20 Zm1 ; m ; fzl gl20 Zm1 Þ Qm1 kl 1 n X X ðn þ Þ! l¼0 zl ¼ : Qm1 Pm1 n¼0 fk0 ;k1 ;;km1 g ! l¼0 kl ! ð l¼0 kl l þ m Þ
ð26Þ
Then GðtÞ given by (25) is expressed as GðtÞ ¼ E0 ðfm l gl20 Zm1 ; m ; fcl tm l gl20 Zm1 Þ tm 1 HðtÞ: Here we define G ðtÞ for 2 0 Z, by D
and hence G0 ðtÞ ¼ GðtÞ. If we express (28) using the index law (5). It follows that Appendix A.2.
m
m1 X
ð27Þ
!þ1 l
cl D G ðtÞ ¼ ðtÞ; ð28Þ l¼0 by F þ1 G ðtÞ ¼ ðtÞ, we confirm F þ1 D G ðtÞ ¼ D F þ1 G ðtÞ ¼ D ðtÞ by 1 1 D G ðtÞ ¼ D F þ1 ðtÞ ¼ Fþ1 D ðtÞ. The last equality is proved also in þ
Lemma 4. Let 2 0 Z and 2 R satisfying < ð þ 1Þm , and let G ðtÞ be defined by (28). Then D G ðtÞ is given by
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D G ðtÞ ¼
ðDm
þ
1 Pm1 l¼0
cl Dl Þþ1
D ðtÞ
¼ E ðfm l gl20 Zm1 ; ð þ 1Þm ; fcl tm l gl20 Zm1 Þ tðþ1Þm 1 HðtÞ:
ð29Þ
Proof. When K is a Volterra integral operator, (24) is valid, which corresponds to the algebraic equality 1 2 expansion ð1 zÞ1 ¼ 1z ¼ 1 þ z þ z þ for jzj < 1. From this equality, we have the Taylor series P P1 1 ðnþÞ! n 1 n n ¼ n¼0 ðnþÞ! n¼0 n!! z for jzj < 1 and 2 0 Z. Corresponding to this, we have ð1 KÞ n!! K from (24). Now the second member of (29) is expressed as !n 1 m1 X X 1 ðn þ Þ! m l D ðtÞ ¼ cl D Dm ðtÞ; P l Þþ1 n!! ð1 þ m1 c D l n¼0 l¼0 l¼0 where l ¼ m l for l 2 0 Zm1 , and m ¼ ð þ 1Þm . By applying the calculation as in (25) to this expression and comparing with (26), we obtain (29). (29) for ¼ 0 is required when we calculate the solution of (16), with the aid of the formula (22). If the righthand side of (21) is expressed by a sum of partial fractions of the form of the second member in (29), GðtÞ given by (21) is expressed by a sum of generalized Mittag-Leffler functions with less parameters and less variables. Remark 1. In section 5.2.4 of a recent book by Kilbas et al. [4, p. 49], the solution of (15) is given. In that solution, the leading two terms 0 DRm uðtÞ þ cm1 0 DRm1 uðtÞ in (15) are treated separately from the other terms and the Green’s function G0 ðtÞ and the solution of the homogeneous equation Dk G0 ðtÞ in the solution take different forms from those given by (29).
5. Solution of simple fDE 5.1
Two-term fDE
The simplest fDE, which has two terms on the lefthand side, is given by (8), that is 0 DR uðtÞ
þ cuðtÞ ¼ f ðtÞ; t > 0;
ð30Þ
where 2 0 R and c 2 R satisty c 6¼ 0. By (15) and (16), the corresponding equation for uðtÞHðtÞ 2 D0R is as follows: de1 X uk1 Dk ðtÞ: ð31Þ ðD þ cÞ½uðtÞHðtÞ ¼ f ðtÞHðtÞ þ k¼0
We now introduce the Green’s function G ðtÞ for 2 0 Z, by ðD þ cÞþ1 G ðtÞ ¼ ðtÞ:
ð32Þ
The Green’s function GðtÞ for (30) and (31) is defined by (9), that is (32) for ¼ 0, and G0 ðtÞ is obtained as (10). For 2 R satisfying < þ , as a simple case of Lemma 4, we have D G ðtÞ ¼
1 D ðtÞ ¼ E ð; þ ; ct Þtþ1 HðtÞ: ðD þ cÞþ1
Here E ð; ; xÞ for 2 0 Z, 2 0 Z and 2 0 Z, is defined by (26), as follows: 1 X ðn þ Þ! xn E ð; ; xÞ ¼ : !n! ðn þ Þ n¼0
ð33Þ
ð34Þ
E ð; ; xÞ for ¼ 0 is given by (11). The solution of (30) is obtained by using (22) with the aid of this formula (33) for ¼ 0. The same result is derived in section 4.1.1 in the book [4], by solving the Volterra equation corresponding to (30) by the basic method of iterations. The formula (33) is useful when the partial fractions of the Green’s function is used as stated at the end of section 4. Remark 2. If we take the partial differentiations D G ðtÞ ¼ where E0ðÞ ð; ; xÞ ¼ transform.
d dx
ðD
@ @ðcÞ
D G0 ðtÞ of D G0 ðtÞ given by (33), we obtain
1 1 D ðtÞ ¼ E0ðÞ ð; ; ct Þtþ1 HðtÞ; þ1 ! þ cÞ
ð35Þ
E0 ð; ; xÞ. In [12, p. 155], this formula for ¼ 0 is derived with the aid of Laplace
Neumann-Series Solution of Fractional Differential Equation
5.2
133
Three-term fDE
The simple fDE, which has three terms on the lefthand side, is given by 0 DR uðtÞ
þ b 0 DR uðtÞ þ cuðtÞ ¼ f ðtÞ; t > 0;
ð36Þ
where > > 0, b 6¼ 0 and c 6¼ 0. The corresponding equation for uðtÞHðtÞ is as follows: ðD þ bD þ cÞ½uðtÞHðtÞ ¼ f ðtÞHðtÞ þ vðtÞ;
ð37Þ
where vðtÞ is given by vðtÞ ¼
de1 X
ðuk1 þ buk1 ÞDk ðtÞ þ
de1 X
uk1 Dk ðtÞ;
ð38Þ
k¼de
k¼0
if is an integer, and by de1 X
uk1 Dk ðtÞ;
ð39Þ
ðD þ bD þ cÞGðtÞ ¼ ðtÞ:
ð40Þ
vðtÞ ¼
k¼0
if is not an integer. The Green’s function GðtÞ is now defined by
This is converted to ð1 þ bD þ cD ÞGðtÞ ¼ D ðtÞ ¼ GðtÞ ¼
1 1 þ bD þ cD
1 1 HðtÞ. ðÞ t
D ðtÞ ¼
¼ E0 ð; ; ; ct ; bt Þ t
1 X
The solution of this equation is given by
ðbD cD Þn D ðtÞ
n¼0 1
HðtÞ:
ð41Þ
Here the three-parameter, two-variable function E ð; ; ; x; yÞ for 2 0 Z, > 0, > 0 and > 0 is defined by 1 X n X ðn þ Þ! xnk yk E ð; ; ; x; yÞ ¼ : ð42Þ !ðn kÞ!k! ððn kÞ þ k þ Þ n¼0 k¼0 We now define G ðtÞ for 2 0 Z, by ðD þ bD þ cÞþ1 G ðtÞ ¼ ðtÞ: For 2 R satisfying < þ , as a simple case of Lemma 4, we have 1 D ðtÞ D G ðtÞ ¼ ðD þ bD þ cÞþ1 ¼ E ð; ; þ ; ct ; bt Þ tþ1 HðtÞ:
ð43Þ
ð44Þ
This formula for ¼ 0 is required when we calculate the solution of (36), with the aid of the formula (22). The formula (44) is useful when the partial fractions of the Green’s function is used as stated at the end of section 4.
6. Solution of fDE in R In the preceding sections, we consider functions belonging to D0R , which is the space of right-sided distributions, where a regular function belonging to D0R and a testing function in DR have a support bounded on the left and on the right, respectively. We now consider the space DexiL;bR of testing functions, which are infinitely differentiable, may increase at most exponentially as t tends to 1 and have a support bounded on the right. est HðtÞ for s 2 R is such a function. Then a 2 function, which is locally integrable and decay to 0 faster than any exponential function as t tends to 1, e.g. et , is a regular distribution belonging to the space D0exiL;bR . In the present section, we consider the fDE given by m1 X m cl 1 DRl uðtÞ ¼ f ðtÞ: ð45Þ 1 DR uðtÞ þ l¼0
Here 1 DR uðtÞ for f ðtÞ 2 D0exiL;bR and the
2 R is defined by (1) and (2) with f ðtÞ and b replaced by uðtÞ and 1, respectively. If solution uðtÞ is assumed to belong to D0exiL;bR , the fDE becomes m1 X Dm uðtÞ þ cl Dl uðtÞ ¼ f ðtÞ; ð46Þ l¼0
in place of (16). Then the solution of (46) is given by
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MORITA and SATO
uðtÞ ¼
Dm þ
1 Pm1 l¼0
Z cl Dl
t
Gðt t0 Þ f ðt0 Þ dt0 ;
f ðtÞ ¼
ð47Þ
1
where GðtÞ is the Green’s function defined by (20) or (21) and its explicit expressions are given for various situations in sections 4 and 5. The fact that there exists 2 R satisfying GðtÞet ! 0 as t ! 1, is guaranteed by a theorem given in the Appendix of [9]. In (47), we have no terms due to the initial values. When GðtÞet ! 0 as t ! 1, we consider the space DexiL;bR of testing functions, in which every function gðtÞ is infinitely differentiable, satisfies gðtÞejtj ! 0 as t ! 1, and has a support bounded on the right. Then a locally integrable function f ðtÞ, which satisfies f ðtÞejtj ! 0 as t ! 1, belongs to D0exiL;bR . Now if f ðtÞ on the righthand side of (46) belongs to D0exiL;bR , the solution of (46) is given by (47).
7. Concluding Remark In the preceding sections, we solve three types of fDE (15), (15C) and (45), which take the form FR uðtÞ ¼ fR ðtÞ, P l where FR ¼ b DRm þ m1 c l b DR for b ¼ 0 or b ¼ 1 and fR ðtÞ is a locally integrable function. The discussions are l¼0 summarizedPas follows. The fDE FR uðtÞ ¼ fR ðtÞ is first transformed to an equivalent equation FvðtÞ ¼ hðtÞ where l F ¼ Dm þ m1 or vðtÞ ¼ uðtÞ, and hðtÞ is a distribution. When hðtÞ is a regular distribution, l¼0 cl D , vðtÞ ¼ uðtÞHðtÞ Rt vðtÞ is given by vðtÞ ¼ F 1 hðtÞ ¼ 1 Gðt t0 Þhðt0 Þdt0 . When hðtÞ is expressed as hðtÞ ¼ DN ðtÞ R t in terms of N 2 N and a locally integrable function ðtÞ, vðtÞ is given by vðtÞ ¼ F 1 hðtÞ ¼ DN F 1 ðtÞ ¼ DN 1 Gðt t0 Þ ðt0 Þdt0 . The solution uðtÞ is obtained by this. The expressions of GðtÞ ¼ F 1 ðtÞ are obtained in sections 4 and 5.
Appendix A:
The space of right-sided distributions
We consider distributions in the space D0R . A function f ðtÞ which is locally integrable and has a support bounded on the left, belongs to D0R and is called a regular distribution. The distributions in D0R are called right-sided distributions. The testing functions in the space DR which is dual to D0R , are infinitely differentiable and have a support bounded on the right. For hðtÞ 2 D0R , a number hh; gi is associated with every function gðtÞ in DR . For a regular distribution f ðtÞ 2 D0R , the number h f ; gi is calculated by Z1 f ðtÞgðtÞ dt: ðA:1Þ h f ; gi ¼ h f ðtÞ; gðtÞi :¼ 1
An operator FW in the space DR is such that, to every gðtÞ 2 DR , FW gðtÞ 2 DR is associated. Then the operator F in the space D0R , that is conjugate to FW , is defined so that hFh; gi ¼ hh; FW gi
ðA:2Þ
is valid for every hðtÞ 2 D0R and gðtÞ 2 DR . A.1
Fractional integral and derivative of a right-sided distribution
For gðtÞ 2 DR , the fractional integral D W gðtÞ 2 DR is defined by Z1 1 DW gðtÞ :¼ ðx tÞ1 gðxÞ dx; ðÞ t
ðA:3Þ
for positive 2 R, and we set D0W gðtÞ ¼ gðtÞ. For positive 2 R, the fractional derivative DW gðtÞ 2 DR is given by DW gðtÞ ¼ ð1Þn Dn W
dn gðtÞ; dtn
ðA:4Þ
where n ¼ de. We set DW gðtÞ ¼ D1W gðtÞ ¼ dtd gðtÞ. A proof to show that D W gðtÞ 2 DR is given at the end of this appendix. Lemma 5. Let gðtÞ 2 DR . Then the index law: DW DW gðtÞ ¼ Dþ W gðtÞ;
ðA:5Þ
is valid for every 2 R and every 2 R. R1 Proof. We first note that (A·5) is valid when < 0 and < 0, by using (A·3). By (A·3), D1 W gðtÞ ¼ t gðxÞdx, 1 1 and hence DW D1 W gðtÞ ¼ DW DW gðtÞ ¼ gðtÞ. It follows that, for > 0, DW DW gðtÞ ¼ DW DW DW DW gðtÞ ¼ 1 1 DW DW DW gðtÞ ¼ DW DW DW DW gðtÞ ¼ DW DW gðtÞ. We can now easily complete the proof. Definition 1. For hðtÞ 2 D0R , the fractional integral and derivative D hðtÞ 2 D0R , for negative and positive 2 R, respectively, are defined by (A·2) with F and FW replaced by D and DW , respectively, where DW gðtÞ for ¼ < 0 and for > 0, are given by (A·3) and (A·4).
Neumann-Series Solution of Fractional Differential Equation
135
Definition 1 implies that, when f ðtÞ is continuous and differentiable over R, D f ðtÞ ¼ dtd f ðtÞ. When > 0 and n ¼ de, D hðtÞ ¼ Dn Dn hðtÞ by (A·4) and Definition 1. The statements in the paragraph including (3) in section 1 are confirmed with the aid of (A·1)–(A·4), Definition 1 and (1). The index law given by (5) follows from Definition 1 and Lemma 5. Remark 3. In the book of Miller and Ross [8, Chapter VII], one chapter is used to discuss the Weyl fractional integral and differentiation, where notation W gðtÞ is used in place of the present DW gðtÞ. They are defined by (A·3) and (A·4) for function gðtÞ belonging to the class DrdR of functions, which are infinitely differentiable in R and decay more rapidly than power tN for all N 2 N as t tends to 1. Lemma 5 is proved with DR replaced by DrdR . The space DR is a subclass of DrdR . A.2
Inverse operator expressed by the Green’s function
Lemma 6. Let GðtÞ be the Green’s function of an operator F in the space D0R , so that it satisfies FGðtÞ ¼ ðtÞ, GðtÞ ¼ 0 for t < 0, and it is locally integrable. Let FW be the operator in the space DR , that is conjugate to F. Then the 1 operators FW and F 1 , which are inverse to FW and F, respetively, are given by Z1 Z1 1 1 Gðt xÞgðtÞdt; F f ðtÞ ¼ Gðt xÞ f ðxÞdx; ðA:6Þ FW gðxÞ ¼ 1
1
D0R ,
and the following relations are valid for 2 R, gðtÞ 2 DR and for gðtÞ 2 DR and a regular distribution f ðtÞ in hðtÞ 2 D0R : 1 1 DW FW gðxÞ ¼ FW DW gðxÞ; F 1 D hðxÞ ¼ D F 1 hðxÞ: ðA:7Þ 1 1 Proof of (A·6). When FW gðxÞ 2 DR and FW FW gðxÞ ¼ gðtÞ are assumed, the equations in (A·6) are consequences of the following equations: 1 1 1 gðxÞ ¼ hðt xÞ; FW gðtÞi ¼ hFGðt xÞ; FW gðtÞi ¼ hGðt xÞ; gðtÞi FW Z1 ¼ Gðt xÞgðtÞdt;
ðA:8Þ
1
Z 1 Z1 1 hF 1 f ðtÞ; gðtÞi ¼ h f ðxÞ; FW gðxÞi ¼ f ðxÞ Gðt xÞgðtÞdt dx 1 Z 1 1 Gðt xÞ f ðxÞdx; gðtÞ : ¼
ðA:9Þ
1
In the last equality, Fubini’s theorem is used.
1 Proof of (A·7), and proof to show that FW gðtÞ 2 DR . When gðtÞ 2 DR , gðtÞ has a support bounded on the right, so that 1 there exists c 2 R such that gðtÞ ¼ 0 for all t > c. Then we confirm that FW gðtÞ ¼ 0 for all t > c by using (A·6) and the fact that GðtÞ ¼ 0 for t < 0. By using the fact that two convolutions of fuctions HðtÞ and GðtÞ, which have a support bounded on the left, commute with each other [14, p. 125], we conclude (A·7) for < 0. We now confirm (A·7) 1 1 1 1 1 for ¼ 1 as DW FW gðtÞ ¼ DW FW DW DW gðtÞ ¼ DW D1 W FW DW gðtÞ ¼ FW DW gðtÞ. We can now easily complete the 1 1 1 proofs, since DW FW gðtÞ ¼ FW DW gðtÞ implies that FW gðtÞ is infinitely differentiable. 1 For > 0, D W is a special one of FW , and hence we obtain DW gðtÞ 2 DR .
Appendix B:
Use of the Caputo fractional derivative
We now consider the Caputo fractional derivative b DC f ðtÞ for b 2 R and positive . It is defined by n n d f ðtÞ ; b DC f ðtÞ ¼ b DR dtn
ðB:1Þ
when f ðtÞ is n times differentiable for t > b, where n ¼ de. We express the equation in which 0 DCl uðtÞ appears in place of 0 DRl uðtÞ for l 2 0 Zm in (15), by (15C). In place of (14), we have the relation de1 X D ½uðtÞHðtÞ ¼ 0 DC uðtÞ þ uk Dk1 ðtÞ: ðB:2Þ k¼0
Using this in (15C), we obtain (16) with vðtÞ given by
136
MORITA and SATO
vðtÞ :¼
m X
"
l¼0 l >0
cl
dX l e1
# uk Dl k1 ðtÞ ;
ðB:3Þ
k¼0
where cm ¼ 1. Now the solution of (16) is given by " # Zt dX m l e1 X Gðt t0 Þ f ðt0 Þ dt0 þ cl uk Dl k1 GðtÞ : uðtÞHðtÞ ¼ 0
ðB:4Þ
k¼0
l¼0 l >0
GðtÞ and Dl k1 GðtÞ are expressed in terms of the generalized Mittag-Leffler function, by using the formula (29) for ¼ 0 and 2 R: D GðtÞ ¼ D G0 ðtÞ ¼ tm 1 E0 ðfl g; m ; fzl gÞ; where fl g and fzl g represent fm l gl20 Zm1 and fcl tm l gl20 Zm1 , respectively. We now rewrite (B·4) as Zt dX m e1 uðtÞHðtÞ ¼ Gðt t0 Þ f ðt0 Þ dt0 þ uk yk ðtÞHðtÞ; 0
ðB:5Þ
ðB:6Þ
k¼0
where yk ðtÞ ¼ tk E0 ðfl g; k þ 1; fzl gÞ þ
m1 X
cl0 tm l
0
þk
E0 ðfl g; l0 þ k þ 1; fzl gÞ:
ðB:7Þ
l0 ¼0 dl0 ekþ1
We can show that yk ðtÞ is expressed also as yk ðtÞ ¼
m1 X tk ðcl0 Þtm l þk E0 ðfl g; k þ 1 þ l0 ; fzl gÞ: þ k! l0 ¼0 0
ðB:8Þ
dl0 ek
Derivation of (B·8). We note here the formula: E0 ðfl g; ; fzl gÞ ¼
m1 X 1 þ zl E0 ðfl g; þ l0 ; fzl gÞ; ðÞ l0 ¼0 0
ðB:9Þ
which is derived by observing the following formula: 1 X m1 m1 X X 1 þ f ðl1 ; ; ln Þ; ðB:10Þ E0 ðfl g; ; fzl gÞ ¼ ðÞ n¼1 l1 ¼0 ln ¼0 Q P where f ðl1 ; ; ln Þ ¼ nk¼1 zlk =ð nk¼1 lk þ Þ. By using (B·9) in the first term on the righthand side of (B·7), we obtain (B·8). The second term on the righthand side of (B·8) consists of terms of the form constant t with > m , and hence k we can easily confirm the initial condition that limt!0þ dtd k uðtÞ ¼ uk for k 2 0 Zdm e1 , for the solution given by (B·6) combined with (B·8). Here we consider the two-term fDE. We express the equation in which 0 DP C uðtÞ appears in place of 0 DR uðtÞ in (30), de1 k1 by (30C). By using (B·2) in (30C), we obtain (16) with vðtÞ given by vðtÞ :¼ k¼0 uk D ðtÞ. Now the solution of (30C) is given by Zt de1 X uðtÞHðtÞ ¼ Gðt t0 Þ f ðt0 Þ dt0 þ uk Dk1 GðtÞ; ðB:11Þ 0 1
GðtÞ ¼ E0 ð; ; ct Þt
k¼0
HðtÞ; Dk1 GðtÞ ¼ E0 ð; k þ 1; ct Þtk HðtÞ:
This is the special case of (B·4) or (B·6), for m ¼ 1, 1 ¼ , 0 ¼ 0, and c0 ¼ c. The functions yk ðtÞ in (B·6) are given by (B·7) without the second term on the righthand side, and by (B·8) with the second term on the righthand side for l0 ¼ 0. Remark 4. Luchko [5] derived the solutions of (15C) and (30C), with the aid of Mikusin´ski’s operational calculus. The solution (B·11) of (30C) agrees with his equation given on the top of p. 481 in [5]. The solution of (15C) given in Theorem 3.1 in [5] agrees with (B·6) combined with (B·8). Theorem 3.1 in [5] is reviewed in section 4.3.4 in the book [4], but we note some confusion in the indices of fractional differentiations in the review.
Neumann-Series Solution of Fractional Differential Equation
Appendix C:
137
fDE of complex order
In the text, we assume that the orders and of fractional integral and differentiation D hðtÞ and D hðtÞ are real numbers. In the book [4], complex values are assumed for these values. If we assumed that the orders , , , l and m , and the coefficients cl , c and b in fDE in the text take complex values, the statements there would be valid, providing that the following corrections were introduced. 1. When the indices , , , l and m appear in an inequality, they must be replaced by their real values; e.g., > 0 by Re > 0. 2. When z is a complex number, dze must be replaced by dRe ze, except when Re z is an integer and z 6¼ Re z. In the latter case, dze must be replaced by Re z þ 1 ¼ dRe z þ e, where is an infinitesimal positive number. 3. In Lemma 4, the set fl gl20 Zm1 must be replaced by fRe l gl20 Zm1 .
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