research article analytic solution of a class of...
TRANSCRIPT
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 109582, 5 pageshttp://dx.doi.org/10.1155/2013/109582
Research ArticleAnalytic Solution of a Class of Fractional Differential Equations
Yue Hu1 and Zuodong Yang2
1 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China2 Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China
Correspondence should be addressed to Yue Hu; [email protected]
Received 8 July 2013; Revised 23 October 2013; Accepted 13 November 2013
Academic Editor: Shawn X. Wang
Copyright © 2013 Y. Hu and Z. Yang. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods.Weobtain three theorems which extend the Garraâs results to the general case.
1. Introduction
Recently, Garra [1] studied the analytic solution of a class offractional differential equations with variable coefficients byusing operatorial methods.
The proof of his main results is strongly based onoperatorial properties of Caputo fractional derivative and thefollowing theorem by Miller and Ross [2].
TheoremM.-R. (Theorem 1.1 in [1]). Let ð(ð¡) be any functionof the form ð¡ðð(ð¡) or ð¡ð(ln ð¡)ð(ð¡), where ð > â1, and
ð (ð¡) =â
âð=0
ððð¡ð (1)
has a radius of convergence ð > 0 and 0 < ð < ð . Then
ð·ðŒð¡ð·ðœ
ð¡ð (ð¡) = ð·
ðŒ+ðœ
ð¡ð (ð¡) (2)
holds for all ð¡ â (0, ð), provided(1) ðœ < ð + 1 and ðŒ are arbitrary, or(2) ðœ ⥠ð + 1 and ðŒ are arbitrary and ðð = 0 for ð =0, 1, . . . , ð â 1, whereð = âðœâ (ceiling of the number).
Garraâs main results are as follows.
Theorem Garra (Theorem 1.2 in [1]). Consider the followingboundary value problem (BVP):
ð·ð¿ð¥
ð (ð¥, ð¡) = ð·ðŒ
ð¡ð (ð¥, ð¡) ,
ð (0, ð¡) = ð (ð¡) ,(3)
in the half plane ð¡ > 0, with analytic boundary condition ð(ð¡)such that the conditions of Theorem 1.1 (Theorem M.-R.) aresatisfied. The operatorial solution of BVP (3) is given by
ð (ð¥, ð¡) =â
âð=0
ð¥ðð·ðŒðð¡
(ð!)2ð (ð¡) , (4)
where ð·ð¿ð¥
is the Laguerre derivative and ð·ðŒð¡denotes Caputo
fractional derivative. On the basis of the previous result, Garraproved (Example 1 in [1]) that if ð(ð¡) = ð¡ð, ð â N, then theanalytic solution of the BVP (3) is given by
ð (ð¥, ð¡) =â
âð=0
ð¥ðð·ðŒðð¡ð¡ð
(ð!)2=[ð/ðŒ]
âð=0
Î (ð + 1)
Î (ð + 1 â ðŒð)â ð¥ðð¡ðâðŒð
(ð!)2,
(5)
where Î(ðŒ) = â«â0ðâð¡ð¡ðŒâ1ðð¡.
Motivated by the results, in the present note, we extendGarraâs results to the general case.
2. Preliminaries
Definition 1 (see [3]). For every positive integer ð, theoperatorð·ðð¿
ð¥
:= ð·ð¥, . . . , ð·ð¥ð·ð¥ð· (containing ð+1 ordinaryderivatives) is called the ð-order Laguerre derivatives and theðð¿-exponential function is defined by
ðð (ð¥) :=â
âð=0
ð¥ð
(ð!)ð+1. (6)
2 Abstract and Applied Analysis
In [4], the following result is proved.
Lemma 2. Let ð be an arbitrary real or complex number. Thefunction ðð(ðð¥) is an eigenfunction of the operator ð·ðð¿
ð¥
; thatis,
ð·ðð¿ð¥
ðð = ððð (ðð¥) . (7)
For ð = 0, we have ð·0ð¿ := ð·. Thus, (7) leads to theclassical property of the exponential functionð·ððð¥ = ðððð¥.
Similarly, the spectral properties can be obtained [3] byusing the general Laguerre derivatives
ð·1ð¿ + ðð· = ð· (ð¥ð· + ð) (8)
(here ð is a real or complex constant) or more generally theoperator
ð·(ð¥ð· + ð)ð =ð
âð=0
(ðð)ð·ðð¿ð
ðâð (9)
(ð â N) and the corresponding eigenfunctions
â
âð=0
ð¥ð
ð! (ð + ð)!(10)
orâ
âð=0
ð¥ð
ð!((ð + ð)!)ð. (11)
Throughout this paper, we use the Caputo fractionalderivatives as in [1].
Definition 3 (see [5]). The Caputo derivative of fractionalorder ðŒ of function ð(ð¡) is defined as
ð·ðŒð (ð¡) :=
{{{{{{{{{
1
Î (ð â ðŒ)â«ð¡
0
ð(ð) (ð)
(ð¡ â ð)ðŒ+1âððð, ð â 1 < ðŒ < ð
ððð
ðð¡ð, ðŒ = ð,
(12)
whereð := âðŒâ.
Lemma 4 (see [5]). Let ð(ð¡) = ð¡ð for some ð ⥠0. Then
ð·ðŒð (ð¡)
=
{{{{{{{
0, ðð ð â 0, 1, . . . , ð â 1,Î (ð + 1)
Î (ð + 1 â ðŒ)ð¡ðâðŒ, ðð ð â N, ð ⥠ð ðð
ð â N, ð > ð â 1,
(13)
whereð =: âðŒâ.
Lemma 5. Let ð > 0. Let ðŒ > 0, ðœ > 0, and ðŒ + 𜠆ð. Then
ð·ðŒ+ðœð¡ð = ð·ðŒð·ðœð¡ð (ð¡ > 0) . (14)
Proof. This follows immediately from Lemma 4.
Remark 6. In general, for ðŒ, ðœ > 0, ð·ðŒð·ðœ = ð·ðŒ+ðœ is not true.For example,
ð·0.6ð·0.6ð¡ =Î (2)
Î (0.8)ð¡â0.2, (15)
but
ð·0.6+0.6ð¡ = ð·1.2ð¡ = 0. (16)
Lemma 7. Suppose ðŒ > 0, and let ð(ð¡) = ââð=0ððð¡ð have a
radius of convergence ð > 0 and 0 < ð < ð . If [ðŒð] = âðŒ(ð +1)â â 1 for some ð â N, then
ð·(ð+1)ðŒð (ð¡) = ð·ðŒð·ððŒð (ð¡) . (17)
Proof. Let âðŒ(ð + 1)â = ð. Thenð â 1 < ðŒ(ð + 1) †0.By lemmas 4 and 5, we have
ð·(ð+1)ðŒð (ð¡) = ð·(ð+1)ðŒ(
â
âð=0
ððð¡ð)
= ð·(ð+1)ðŒ(â
âð=ð
ððð¡ð) .
(18)
Since [ðŒð] = âðŒ(ð + 1)â â 1 = ð â 1, together withDefinition 3, we have
ð·ðŒð·ðŒð(ðâ1
âð=0
ððð¡ð) = 0, (19)
and therefore (18) implies that
ð·(ð+1)ðŒð (ð¡) = ð·ðŒð·ðŒð(
â
âð=0
ððð¡ð) = ð·ðŒð·ðŒðð (ð¡) . (20)
This completes the proof.
3. Main Results
We first study the following BVP in the plane ð¡ > 0:
ð·ðð¿ð¥
ð (ð¥, ð¡) = ð·ðŒ
ð¡ð (ð¥, ð¡) ,
ð (0, ð¡) = ð (ð¡) .(21)
Theorem 8. Let ð(ð¡) = ð¡ð with ð > 0 and assume that ð/ðŒ is apositive integer. Then the operatorial form solution of BVP (21)is
ð (ð¥, ð¡) =ð/ðŒ
âð=0
ð¥ðð·ðŒðð¡ð
(ð!)ð+1
=ð/ðŒ
âð=0
Î (ð + 1)
Î (ð + 1 â ðŒð)â ð¥ð
(ð!)ð+1ð¡ðâðŒð.
(22)
Abstract and Applied Analysis 3
Proof. Let ð/ðŒ = ð. By Lemmas 4 and 5, we have
ð·ðð¿ð¥
ð (ð¥, ð¡) =ð
âð=1
ðð+1ð·ðŒðð¡ð¡ð
(ð!)ð+1=ð
âð=1
ð¥ðâ1ð·ðŒðð¡ð¡ð
[(ð â 1)!]ð+1
=ðâ1
âð=0
ð·ðŒ(ð+1)ð¡
ð¡ð
(ð!)ð+1= ð·ðŒð¡(ðâ1
âð=0
ð¥ðð·ðŒðð¡ð
(ð!)ð+1)
= ð·ðŒð¡(ð
âð=0
ð¥ðð·ðŒðð¡ð
(ð!)ð+1) = ð·ðŒ
ð¡ð (ð¥, ð¡) .
(23)
In the fifth previous equality, we use the fact that
ð·ðŒð¡(ð¥ðð·ðŒðð¡ð
(ð!)ð+1) = ð·ðŒ
ð¡(ð¥ðð·ðð¡ð
(ð!)ð+1) = 0. (24)
Remark 9. We point out that the result of Example 1 in [1] isincorrect. A counterexample is as follows. Let ð(ð¡) = ð¡ andðŒ = 0.6. By Lemma 5, we have
ð·ð¿ð¥
([1/0.6]
âð=0
ð¥ðð·ðŒðð¡
(ð!)2ð¡)
= ð·ð¥ð¥ð·ð¥ (ð¡ +ð¥ð·0.6ð¡
(1!)2ð¡) = ð·0.6
ð¡(ð¡) .
(25)
On the other hand, we have
ð·0.6ð¡([1/0.6]
âð=0
ð¥ðð·ðŒðð¡
(ð!)2ð¡)
= ð·0.6ð¡(ð¡ +
ð¥ð·0.6ð¡
(1!)2ð¡) = ð·0.6
ð¡(ð¡) + ð¥ð·
0.6
ð¡(ð·0.6ð¡ð¡) .
(26)
Note that by Remark 6, we have
ð·0.6+0.6ð¡
ð¡ = 0,
ð·0.6ð¡ð·0.6ð¡ð¡ =
1
Î (0.8)ð¡â0.2 Ìž= 0.
(27)
Hence,
ð·ð¿ð¥
([1/0.6]
âð=0
ð¥ðð·ðŒðð¡
(ð!)2ð¡) Ìž=ð·0.6
ð¡([1/0.6]
âð=0
ð¥ðð·ðŒðð¡
(ð!)2ð¡) . (28)
Theorem 10. Let ð(ð¡) satisfy the assumptions of Lemma 7. If[ðŒð] = âðŒ(ð + 1)â â 1, (ð = 1, 2, . . .), then the operatorial formsolution of BVP (21) is given by
ð (ð¥, ð¡) = ðð (ð¥ð·ðŒ
ð¡) ð (ð¡) =
â
âð=0
ð¥ðð·ðŒðð¡
(ð!)ð+1ð (ð¡) . (29)
Proof. By Lemmas 2 and 7, we have
ð·ðð¿ð¥
ð (ð¥, ð¡) = ð·ðð¿ð¥
(ðð (ð¥ð·ðŒ
ð¡) ð (ð¡))
= ð·ðŒð¡ðð (ð¥ð·
ðŒ
ð¡) ð (ð¡) = ð·
ðŒ
ð¡ð (ð¥, ð¡) .
(30)
This completes the proof.
The following generalization of the Theorem 10 can beproved similarly.
Theorem 11. Let ð be a real or complex constant and ð â N.Consider the following BVP:
ð·ð¥(ð¥ð·ð¥ + ð)ðð (ð¥, ð¡) = ð·
ðŒ
ð¡ð (ð¥, ð¡) ,
ð (0, ð¡) = ð (ð¡) ,(31)
in the half plane ð¡ > 0, with analytic boundary condition ð(ð¡)such that the conditions of Lemma 7 are satisfied. If [ðŒð] =âðŒ(ð + 1)â â 1, (ð = 1, 2, . . .), then the operatorial solution ofBVP (31) is given by
ð (ð¥, ð¡) =â
âð=0
ð¥ðð·ðŒðð¡
ð!((ð + ð)!)ðð (ð¡) . (32)
Proof. Using spectral properties of Laguerre derivative,together with Lemma 7, we have
ð·ð¥(ð¥ð·ð¥ + ð)ðð (ð¥, ð¡) = ð·ð¥(ð¥ð·ð¥ + ð)
ð
à (â
âð=0
ð¥ðð·ðŒðð¡
ð!((ð + ð)!)ðð (ð¡))
=â
âð=1
ð(ð + ð)ðð¥ðâ1ð·ðŒðð¡
ð!((ð + ð)!)ðð (ð¡)
=â
âð=0
ð¥ðð·ðŒ(ð+1)
ð!((ð + ð)!)ð+1ð (ð¡)
= ð·ðŒ(â
âð=0
ð¥ðð·ðŒð
ð!((ð + ð)!)ðð (ð¡))
= ð·ðŒð (ð¥, ð¡) .
(33)
This completes the proof.
Remark 12. Conditions [ðŒð] = âðŒ(ð + 1)â â 1 (ð = 1, 2, . . .)are very harsh. However, if we remove them,Theorems 10 and11 no longer remain valid. It should be noted that the mainresult in [1] (Theorem 1.2 in [1]) is incorrect in general case. Acounterexample is as follows. Let ð(ð¡) = ðð¡ and ðŒ = 0.6. Onehas
ð·ð¿ð¥
(â
âð=0
ð¥ðð·ðŒðð¡ðð¡
(ð!)2) = ð·ð¥ð¥ð·ð¥ (
ð¥
(1!)2ð·0.6ð¡ðð¡ +
ð¥2
(2!)2ð·1.2ð¡ðð¡
+ð¥3
(3!)2ð·1.8ð¡ðð¡ + â â â )
= ð·0.6ð¡ðð¡ +
ð¥
(1!)2ð·1.2ð¡ðð¡
+ð¥2
(2!)2ð·1.8ð¡ðð¡ + â â â .
(34)
4 Abstract and Applied Analysis
On the other hand, we have
ð·ðŒð¡(â
âð=0
ð¥ðð·ðŒðð¡ðð¡
(ð!)2)
= ð·0.6ð¡(ðð¡ +
ð¥
(1!)2ð·0.6ð¡ðð¡ +
ð¥2
(2!)2ð·1.2ð¡ðð¡ + â â â ) .
(35)
Set ð¥ = 1 in (34) and (35). By Definition 3 and Lemma 4,we deduce that all terms in (34) are positive and cannotcontain negative exponent of variable ð¡. For example,
ð·0.6ð¡ðð¡ = ð·0.6
ð¡(1 + ð¡ +
ð¡2
2!+ â â â ) = ð·0.6
ð¡(ð¡ +
ð¡2
2!+ â â â )
=1
Î (1.4)ð¡0.4 +
1
Î (2.4)ð¡1.4 + â â â ,
ð·1.8ð¡ðð¡ = ð·1.8
ð¡(1 + ð¡ +
ð¡2
2!+ â â â ) = ð·1.8
ð¡(ð¡2
2!+ð¡3
3!+ â â â )
=1
Î (1.2)ð¡0.2 +
1
Î (2.2)ð¡1.2 + â â â ,
and so forth.(36)
Similarly, all terms in (35) are also positive, except thatsome terms contain negative exponent of variable ð¡. Forexample,
ð·0.6ð¡ð·0.6ð¡ðð¡ = ð·0.6
ð¡ð·0.6ð¡(1 + ð¡ +
ð¡2
2!+ â â â )
= ð·0.6ð¡ð·0.6ð¡(ð¡ +
ð¡2
2!+ â â â )
=1
Î (0.8)ð¡â0.2 + â â â ,
ð·0.6ð¡ð·6.6ð¡ðð¡ = ð·0.6
ð¡ð·6.6ð¡(1 + ð¡ +
ð¡2
2!+ â â â )
= ð·0.6ð¡ð·6.6ð¡(ð¡ +
ð¡2
2!+ â â â )
=1
Î (0.8)ð¡â0.2 + â â â ,
and so forth.
(37)
Thus, we conclude that
ð·ð¿ð¥
(â
âð=0
ð¥ðð·0.6ðð¡ðð¡
(ð!)2) Ìž=ð·0.6
ð¡(â
âð=0
ð¥ðð·0.6ðð¡ðð¡
(ð!)2) . (38)
4. Conclusion and Discussion
In this paper, we point out that Garraâs results are incorrectand give some necessary counterexamples. In addition, we
established three theorems (Theorems 8, 10, and 11) whichcorrect and extend the corresponding results of [1].
Different from integer-order derivative, there are manykinds of definitions for fractional derivatives, includingRiemann-Liouville, Caputo, Grunwald-Letnikov, Weyl,Jumarie, Hadamard, Davison and Essex, Riesz, Erdelyi-Kober, and Coimbra (see [1, 6â8]). These definitions aregenerally not equivalent to each other. Every derivative hasits own serviceable range. In other words, all these fractionalderivatives definitions have their own advantages anddisadvantages. For example, the Caputo derivative is veryuseful when dealing with real-world problem, since it allowstraditional initial and boundary conditions to be included inthe formulation of the problem and the Laplace transformof Caputo fractional derivative is a natural generalizationof the corresponding well-known Laplace transform ofinteger-order derivative. So, the Caputo fractional-ordersystem is often used in modelling and analysis. However, thefunctions that are not differentiable do not have fractionalderivative, which reduces the field of application of Caputoderivative (see [1, 8, 9]).
When solving fractional-order systems, the law of expo-nents (semigroup property) is the most important. Unlikeinteger-order derivative, for ðŒ > 0 and ðœ > 0, ðŒ derivative oftheðœ derivative of a function is, in general, not equal to theðŒ+ðœ derivative of such function. About the semigroup propertyof the fractional derivatives, under suitable assumptions offractional order, there have existed some studies, but only afew studies provide valuable judgment methods (see [1, 9]).
Fortunately, if we define Ω as the class of all functionsð which are infinitely differentiable everywhere and aresuch that ð and all its derivatives are of order ð¡âð for allð, ð = 1, 2, . . ., then, for all functions of class Ω, Weylfractional derivatives possess the semigroup property [1].This has brought us great convenience for studying Weylfractional differential equations.Wewill considered this topicin a forthcoming paper.
Acknowledgments
The authors are grateful to the anonymous reviewers forseveral comments and suggestions which contributed tothe improvement of this paper. This work is supportedby the National Natural Science Foundation of China (no.11171092) and the Natural Science Foundation of JiangsuHigher Education Institutions of China (no. 08KJB110005).
References
[1] R. Garra, âAnalytic solution of a class of fractional differentialequations with variable coefficients by operatorial methods,âCommunications in Nonlinear Science and Numerical Simula-tion, vol. 17, no. 4, pp. 1549â1554, 2012.
[2] K. S. Miller and B. Ross, An Introduction to the FractionalCalculus and Fractional Differential Equations, John Wiley &Sons, New York, NY, USA, 1993.
[3] G. Bretti and P. E. Ricci, âLaguerre-type special functions andpopulation dynamics,â Applied Mathematics and Computation,vol. 187, no. 1, pp. 89â100, 2007.
Abstract and Applied Analysis 5
[4] G. Dattoli and P. E. Ricci, âLaguerre-type exponentials, andthe relevant ð¿-circular and ð¿-hyperbolic functions,â GeorgianMathematical Journal, vol. 10, no. 3, pp. 481â494, 2003.
[5] K. Diethelm, The Analysis of Fractional Differential Equations,Springer, Berlin, Germany, 2010.
[6] I. Podlubny, Fractional Differential Equations, vol. 198, Aca-demic Press, New York, NY, USA, 1999.
[7] P. L. Butzer and U. Westphal, An Introduction to FractionalCalculus, World Scientific, Singapore, 2000.
[8] A. Atangana and A. Secer, âA note on fractional order deriva-tives and table of fractional derivatives of some special func-tions,â Abstract and Applied Analysis, vol. 2013, Article ID279681, 8 pages, 2013.
[9] C. Li andW. Deng, âRemarks on fractional derivatives,âAppliedMathematics andComputation, vol. 187, no. 2, pp. 777â784, 2007.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of