Fractional Differential Equations 2012downloads.hindawi.com/journals/specialissues/714979.pdf ·...

Fractional Differential Equations 2012Guest Editors: Fawang Liu, Om P. Agrawal, Shaher Momani, Nikolai N. Leonenko, and Wen Chen

International Journal of Differential Equations

Fractional Differential Equations 2012

International Journal of Differential Equations

Fractional Differential Equations 2012

Guest Editors: Fawang Liu, Om P. Agrawal,Shaher Momani, Nikolai N. Leonenko,and Wen Chen

Copyright q 2013 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “International Journal of Differential Equations.” All articles are open access articles dis-tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

Editorial BoardOm P. Agrawal, USABashir Ahmad, Saudi ArabiaChérif Amrouche, FranceSabri Arik, TurkeyDumitru Baleanu, TurkeyVieri Benci, ItalyElena Braverman, CanadaAlberto Cabada, SpainJinde Cao, ChinaDer-Chen Chang, USAYang Chen, UKGui Qiang Chen, USAFengde Chen, ChinaCharles E. Chidume, ItalyI. D. Chueshov, UkraineShangbin Cui, ChinaToka Diagana, USAJinqiao Duan, USAM. A. El-Gebeily, Saudi ArabiaAhmed El-Sayed, EgyptKhalil Ezzinbi, MoroccoZhaosheng Feng, USAGiovanni P. Galdi, USAD. D. Ganji, USAWeigao Ge, ChinaYoshikazu Giga, JapanJaume Giné, SpainJerome A. Goldstein, USAS. R. Grace, EgyptMaurizio Grasselli, ItalyEmmanuel Hebey, FranceHelge Holden, NorwayMayer Humi, USAElena I. Kaikina, MexicoQingkai Kong, USA

Dexing Kong, ChinaM. O. Korpusov, RussiaA. M. Krasnosel’skii, RussiaMiroslav Krstic, USAP. A. Krutitskii, RussiaMin Ku, PortugalMustafa Kulenovic, USAKarl Kunisch, AustriaAlexander Kurganov, USAJose A. Langa, SpainPhilippe G. Lefloch, FranceDaniel Franco Leis, SpainNikolai N. Leonenko, UKYuji Liu, ChinaF. Liu, ChinaWen Xiu Ma, USARuyun Ma, ChinaT. R. Marchant, AustraliaMarco Marletta, UKRoderick Melnik, CanadaS. A. Messaoudi, Saudi ArabiaStanisław Migórski, PolandA. Mikelic, FranceShaher Momani, JordanGaston M. N’Guerekata, USAJuan J. Nieto, SpainSotiris K. Ntouyas, GreeceDonal O’Regan, IrelandJong Yeoul Park, KoreaKanishka Perera, USARodrigo Lopez Pouso, SpainRamón Quintanilla, SpainYoussef Raffoul, USAT. M. Rassias, GreeceYuriy Rogovchenko, Norway

Julio D. Rossi, ArgentinaSamir H. Saker, EgyptMartin Schechter, USAWilliam E. Schiesser, USALeonid Shaikhet, UkraineZhi Qiang Shao, ChinaQin Sheng, USAJunping Shi, USAStevo Stević, SerbiaIoannis G. Stratis, GreeceJian-Ping Sun, ChinaGuido Sweers, GermanyNasser-eddine Tatar, Saudi ArabiaRoger Temam, USAGunther Uhlmann, USAJ. van Neerven, The NetherlandsA. Vatsala, USAPeiguang Wang, ChinaZhi-Qiang Wang, USALihe Wang, USAMingxin Wang, ChinaGershon Wolansky, IsraelPatricia J. Y. Wong, SingaporeJen-Chih Yao, TaiwanJingxue Yin, ChinaJianshe Yu, ChinaVjacheslav Yurko, RussiaQi S. Zhang, USASining Zheng, ChinaSongmu Zheng, ChinaYong Zhou, ChinaFeng Zhou, ChinaWenming Zou, ChinaXingfu Zou, Canada

Contents

Fractional Differential Equations 2012, Fawang Liu, Om P. Agrawal, Shaher Momani,Nikolai N. Leonenko, and Wen ChenVolume 2013, Article ID 802324, 2 pages

Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications,Lakshman Mahto, Syed Abbas, and Angelo FaviniVolume 2013, Article ID 704547, 11 pages

Solving the Fractional Rosenau-Hyman Equation via Variational Iteration Method andHomotopy Perturbation Method, R. Yulita Molliq and M. S. M. NooraniVolume 2012, Article ID 472030, 14 pages

Fractional Order Difference Equations, J. Jagan Mohan and G. V. S. R. DeekshituluVolume 2012, Article ID 780619, 11 pages

Analytical Study of Nonlinear Fractional-Order Integrodifferential Equation: Revisit Volterra’sPopulation Model, Najeeb Alam Khan, Amir Mahmood, Nadeem Alam Khan, and Asmat AraVolume 2012, Article ID 845945, 8 pages

Solving Fractional-Order Logistic Equation Using a New Iterative Method,Sachin Bhalekar and Varsha Daftardar-GejjiVolume 2012, Article ID 975829, 12 pages

Generalized Multiparameters Fractional Variational Calculus, Om Prakash AgrawalVolume 2012, Article ID 521750, 38 pages

Generalized Monotone Iterative Technique for Caputo Fractional Differential Equation withPeriodic Boundary Condition via Initial Value Problem, J. D. Ramı́rez and A. S. VatsalaVolume 2012, Article ID 842813, 17 pages

A Time-Space Collocation Spectral Approximation for a Class of Time Fractional DifferentialEquations, Fenghui HuangVolume 2012, Article ID 495202, 19 pages

Axisymmetric Solutions to Time-Fractional Heat Conduction Equation in a Half-Space underRobin Boundary Conditions, Y. Z. PovstenkoVolume 2012, Article ID 154085, 13 pages

Chaos Control and Synchronization in Fractional-Order Lorenz-Like System, Sachin BhalekarVolume 2012, Article ID 623234, 16 pages

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013, Article ID 802324, 2 pageshttp://dx.doi.org/10.1155/2013/802324

EditorialFractional Differential Equations 2012

Fawang Liu,1 Om P. Agrawal,2 Shaher Momani,3 Nikolai N. Leonenko,4 and Wen Chen5

1 School of Mathematical Sciences, Queensland University of Technology, P.O. Box 2434, Brisbane, QLD 4001, Australia2 Department of Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, USA3Department of Mathematics, The University of Jordan, Amman 11942, Jordan4 School of Mathematics, Cardiff University, Cardiff CF2 4YH, UK5Department of Engineering Mechanics, Hohai University, Xikang Road No. 1, Nanjing, Jiangsu 210098, China

Correspondence should be addressed to Fawang Liu; [email protected]

Received 8 January 2013; Accepted 8 January 2013

Copyright © 2013 Fawang Liu et al. 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.

It is our pleasure to bring this third special issue of theInternational Journal of Differential Equations dedicated toFractional Differential Equations (FDEs).

In recent years, a growing number of papers by manyauthors from various fields of science and engineering dealwith dynamical systems described by fractional partial differ-ential equations. Due to the extensive applications of FDEsin engineering and science, research in this area has grownsignificantly all around the world.

This third special issue on fractional differential equationsconsists of one review article and 9 original articles coveringvarious aspects of FDEs and their applications written byprominent researchers in the field.

In the paper titled as “Generalized multiparameters frac-tional variational calculus” by O. P. Agrawal, the authorintroduces some new one-parameter GFDs, investigates theirproperties, and uses them to develop several parts of FVC.The author also shows that many of the fractional derivativesand fractional variational formulations proposed recentlyin the literature can be obtained from the GFDs and thegeneralized FVC.

The papers titled as “Solving the fractional Rosenau-Hyman equation via variational iteration method and homo-topy perturbation method,” by R. Y. Molliq and M. S. M.Noorani, titled as “Generalized monotone iterative techniquefor Caputo fractional differential equation with periodicboundary condition via initial value problem” by J. D. Ramı́rezandA. S. Vatsala, and titled as “Solving fractional-order logisticequation using a new iterative method” by S. Bhalekar and

V. Daftardar-Gejji introduce variational iteration and homo-topy perturbation methods for solving fractional Rosenau-Hyman, fractional differential (with periodic boundary con-ditions), and fractional-order logistic equations, respectively.

The paper titled as “Axisymmetric solutions to time-fractional heat conduction equation in a half-space underRobin boundary conditions,” by Y. Z. Povstenko derivesanalytical solutions to time-fractional heat equation in a half-space under Robin boundary conditions using an integraltransform technique. The paper titled as “Analytical study ofnonlinear fractional-order integrodifferential equation: revisitVolterra’s population model” by N. A. Khan et al. proposes atwo-component homotopy method to solve Volterra’s popu-lation model.

The paper titled as “A time-space collocation spectralapproximation for a class of time fractional differential equa-tions” by F. Huang develops a time-space collocation spectralmethod for a class of time fractional differential equations.

The paper titled as “Analysis of Caputo impulsive fractionalorder differential equations with applications” by L. Mahtoet al. studies the existence and uniqueness of the theoremof Caputo impulsive fractional order differential equationsusing Sadavoskii’s fixed point method. The paper titled as“Fractional order difference equations” by J. J.Mohan andG. V.S. R. Deekshitulu establishes a theorem on the existence anduniqueness of solutions for various classes of fractional orderdifference equations.

Finally, The paper titled as “Chaos control and synchro-nization in fractional-order Lorenz-like system” by S. Bhalekar

2 International Journal of Differential Equations

investigates Chaos control and synchronization in fractional-order Lorenz-like system.

Thus, this special issue provides a wide spectrum ofcurrent research in the area of FDEs, andwe hope that expertsin this and related fields find it useful.

Fawang LiuOm P. AgrawalShaher Momani

Nikolai N. LeonenkoWen Chen

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013, Article ID 704547, 11 pageshttp://dx.doi.org/10.1155/2013/704547

Research ArticleAnalysis of Caputo Impulsive Fractional Order DifferentialEquations with Applications

Lakshman Mahto,1 Syed Abbas,1 and Angelo Favini2

1 School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, India2Dipartimento di Matematica, Universitá di Bologna, Pizza di Porta S. Donato 5, 40126 Bologna, Italy

Correspondence should be addressed to Syed Abbas; [email protected]

Received 6 May 2012; Accepted 21 November 2012

Academic Editor: Nikolai Leonenko

Copyright © 2013 Lakshman Mahto et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We use Sadovskii’s fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractionaldifferential equations of order 𝛼 ∈ (0, 1) with one example of impulsive logistic model and few other examples as well. We alsodiscuss Caputo impulsive fractional differential equations with finite delay.The results proven are new and compliment the existingone.

1. Introduction

Dynamics ofmany evolutionary processes from various fieldssuch as population dynamics, control theory, physics, biology,andmedicine. undergo abrupt changes at certainmoments oftime like earthquake, harvesting, shock, and so forth. Theseperturbations can be well approximated as instantaneouschange of states or impulses. These processes are modeled byimpulsive differential equations. In 1960,Milman andMyšhkisintroduced impulsive differential equations in their paper[1]. Based on their work, several monographs have beenpublished by many authors like Samoilenko and Perestyuk[2], Lakshmikantham et al. [3], Bainov and Simeonov [4, 5],Bainov and Covachev [6], and Benchohra et al. [7]. All theauthorsmentioned perviously have considered impulsive dif-ferential equations as ordinary differential equations coupledwith impulsive effects. They considered the impulsive effectsas difference equations being satisfied at impulses time. So,the solutions are piecewise continuous with discontinuities atimpulses time. In the fields like biology, population dynamics,and so forth, problems with hereditary are best modeledby delay differential equations [8]. Problems associated withimpulsive effects and hereditary property are modeled byimpulsive delay differential equations.

The origin of fractional calculus (derivatives (𝑑𝛼/𝑑𝑡𝛼)𝑓and integrals 𝐼𝛼𝑓 of arbitrary order 𝛼 > 0) goes backto Newton and Leibniz in the 17th century. In a lettercorrespondence with Leibniz, L’Hospital asked “What if theorder of the derivative is 1/2”? Leibniz replied, “Thus itfollows that will be equal to 𝑥√𝑑𝑥 : 𝑥, an apparent paradox,from which one day useful consequences will be drawn.”This letter of Leibniz was in September 1695. So, 1695 isconsidered as the birthday of fractional calculus. Fractionalorder differential equations are generalizations of classicalinteger order differential equations and are increasingly usedto model problems in fluid dynamics, finance, and otherareas of application. Recent investigations have shown thatsometimes physical systems can be modeled more accuratelyusing fractional derivative formulations [9].There are severalexcellent monographs available on this field [10–15]. In [11],the authors give a recent and up-to-date description of thedevelopments of fractional differential and fractional integro-differential equations including applications. The existenceand uniqueness of solutions to fractional differential equa-tions has been considered bymany authors [16–21]. Impulsivefractional differential equations represent a real frameworkfor mathematical modeling to real world problems. Signif-icant progress has been made in the theory of impulsive


fractional differential equations [7, 22–24]. Xu et al. in theirpaper [25] have described an impulsive delay fishing model.

Fractional derivatives arise naturally in mathematicalproblems, for 𝛼 > 0 and a function 𝑓 : [0, 𝑇] → R, recall[10, Definitions 3.1, 2.2]

(a) the Caputo fractional derivative

𝐶

𝐷𝛼

𝑓 (𝑡) =

1

Γ (1 − 𝛼)

∫

𝑡

0

(𝑡 − 𝑠)−𝛼

𝑓

(𝑠) 𝑑𝑠, (1)

(b) the Riemann-Liouville fractional derivative

𝐷𝛼

𝑓 (𝑡) =

𝑑𝛼

𝑑𝑡𝛼𝑓 (𝑡) =

1

Γ (1 − 𝛼)

𝑑

𝑑𝑡

∫

𝑡

0

(𝑡 − 𝑠)−𝛼

𝑓 (𝑠) 𝑑𝑠, (2)

provided that the right-hand sides exist pointwise on [0, 𝑇] (Γdenotes the gamma function). Using the Riemann-Liouvillefractional integral [10, Definition 2.1] 𝐼𝛼

0𝑓(𝑡) = (1/Γ(𝛼)) ∫

𝑡

0

(𝑡−

𝑠)𝛼−1

𝑓(𝑠)𝑑𝑠, we have 𝐶𝐷𝛼𝑓(𝑡) = 𝐼1−𝛼0

(𝑑/𝑑𝑡)𝑓(𝑡) and𝐷𝛼

𝑓(𝑡) = (𝑑/𝑑𝑡)𝐼1−𝛼

0𝑓(𝑡). 𝐼𝛼

0𝑓 exists, for instance, for all

𝛼 > 0, if 𝑓 ∈ 𝐶0([0, 𝑇]) ∩ 𝐿1loc([0, 𝑇]); moreover, 𝐼𝛼

0𝑓(0) = 0.

Throughout the paper, we assume that 𝐼 = [0, 𝑇].One can see that both of the fractional derivatives are

actually nonlocal operator because integral is a nonlocaloperator.Moreover, calculating time fractional derivative of afunction at some time requires all the past history, and hencefractional derivatives can be used for modeling systems withmemory. Fractional differential equations can be formulatedusing both Caputo and Riemann-Liouville fractional deriva-tives. A Riemann-Liouville initial value problem can be statedas follows:

𝐷𝛼

𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡)) , 𝑡 ∈ 𝐼 = [0, 𝑇] ,

𝐷𝛼−1

𝑥 (0) = 𝑥0,

(3)

or equivalently,𝑥(𝑡) = 𝑥0𝑡𝛼−1

/Γ(𝛼)+∫

𝑡

0

(𝑡−𝑠)𝛼−1

𝑓(𝑠, 𝑥(𝑠))𝑑𝑠 inits integral representation [11, Theorem 3.24]. For a physicalinterpretation of the initial conditions in (3), see [26–28].If derivatives of Caputo type are used instead of Riemann-Liouville type, then initial conditions for the correspondingCaputo fractional differential equations can be formulated asfor classical ordinary differential equations, namely, 𝑥(0) =𝑥0.Ourmain objective is to discuss existence and uniqueness

of solutions of the following impulsive fractional differentialequation of Caputo type in a Banach space𝑋with norm ‖ ⋅ ‖

𝑋:

𝐶

𝐷𝛼

𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡)) , 𝑡 ∈ 𝐼 = [0, 𝑇] , 𝑡 ̸= 𝑡𝑘,

Δ𝑥 (𝑡) |𝑡=𝑡𝑘

= 𝐼𝑘(𝑥 (𝑡−

𝑘)) , 𝑘 = 1, 2, . . . , 𝑚,

𝑥 (0) = 𝑥0,

(4)

where 𝑓 ∈ 𝐶(𝐼 × 𝑋,𝑋), 𝐼𝑘

: 𝑋 → 𝑋 and 𝑥0∈ 𝑋. 0 = 𝑡

0<

𝑡1

< 𝑡2

< ⋅ ⋅ ⋅ < 𝑡𝑚

< 𝑡𝑚+1

= 𝑇, Δ𝑥(𝑡)|𝑡=𝑡𝑘

= 𝑥(𝑡+

𝑘) − 𝑥(𝑡

−

𝐾),

𝑥(𝑡+

𝑘) = lim

ℎ→0𝑥(𝑡+ℎ), and𝑥(𝑡−

𝑘) = lim

ℎ→0𝑥(𝑡−ℎ).We break

our function 𝑓 into two components which satisfy different

conditions. We observed that this kind of functions occurs inecological modeling. We have given the example of logisticequation in the last section. Our main tool is Sadovskii’s fixedpoint theorem.

Now, we define some important spaces and norm whichwill encounter frequently:

PC (𝐼, 𝑋) = {𝑥 : [0, 𝑇] → 𝑋 | 𝑥 ∈ 𝐶 ([𝑡0, 𝑡1] , 𝑋)

∪ 𝐶 ((𝑡𝑘, 𝑡𝑘+1

] , 𝑋) 𝑘 = 1, 2, . . . , 𝑚,

𝑥 (𝑡+

𝑘) and 𝑥 (𝑡−

𝑘) exist, 𝑥 (𝑡

𝑘) = 𝑥 (𝑡

−

𝑘) } ,

(5)

with sup-norm ‖ ⋅ ‖, defined by ‖𝑥‖ = sup{‖𝑥(𝑡)‖𝑋

: 𝑡 ∈ 𝐼}.

Definition 1. A solution of fractional differential equation (4)is a piecewise continuous function 𝑥 ∈ PC([0, 𝑇], 𝑋) whichsatisfied (4).

Definition 2 ([29], Definition 11.1). Kuratowskii noncom-pactness measure: let 𝑀 be a bounded set in metric space(𝑋, 𝑑), then Kuratowskii noncompactness measure, 𝜇(𝑀)is defined as inf{𝜖 : 𝑀 covered by a finitemany sets suchthat the diameter of each set ≤ 𝜖}.

Definition 3 ([29], Definition 11.6). Condensing map: Let Φ :𝑋 → 𝑋 be a bounded and continuous operator on Banachspace 𝑋 such that 𝜇(Φ(𝐵)) < 𝜇(𝐵) for all bounded set 𝐵 ⊂𝐷(Φ), where 𝜇 is the Kuratowskii noncompactness measure,then Φ is called condensing map.

Definition 4. Compact map: a map 𝑓 : 𝑋 → 𝑋 is said to becompact if the image of every bounded subset of 𝑋 under 𝑓is precompact (closure is compact).

Theorem 5 (see [30]). Let B be a convex, bounded, and closedsubset of a Banach space𝑋 and letΦ : 𝐵 → 𝐵 be a condensingmap. Then, Φ has a fixed point in 𝐵.

Lemma6 ([29], Example 11.7). AmapΦ = Φ1+Φ2: 𝑋 → 𝑋

is 𝑘-contraction with 0 ≤ 𝑘 < 1 if

(a) Φ1is 𝑘-contraction, that is, ‖Φ

1(𝑥) − Φ

1(𝑦)‖𝑋

≤

𝑘‖𝑥 − 𝑦‖𝑋;

(b) Φ2is compact,

and hence Φ is a condensing map.

The structure of the paper is as follows. In Section 2, weprove the existence (Theorem 8) and uniqueness of solutionsto (4). We show in Section 3 the existence and uniquenessof solutions for a general class of impulsive functionaldifferential equations of fractional order 𝛼 ∈ (0, 1). InSection 4, we give some examples in favor of our sufficientconditions.

International Journal of Differential Equations 3

2. Impulsive Fractional Differential Equation

Consider the initial value problem (4) on the cylinder 𝑅 ={(𝑡, 𝑥) ∈ R×𝑋 : 𝑡 ∈ [0, 𝑇], 𝑥 ∈ 𝐵(0, 𝑟)} for some fixed 𝑇 > 0,𝑟 > 0, and assume that there exist 𝑝 ∈ (0, 𝛼), 𝛼 ∈ (0, 1),𝑀1,𝑀2, 𝐿1

∈ 𝐿1/𝑝

([0, 𝑇],R+) and functions 𝑓1, 𝑓2

∈ 𝐶(𝑅 ×

𝑋,𝑋) such that 𝑓 = 𝑓1+ 𝑓2, and the following assumptions

are satisfied:

(A.1) 𝑓1

is bounded and Lipschitz, in particular,‖𝑓1(𝑡, 𝑥)‖

𝑋≤ 𝑀

1(𝑡) and ‖𝑓

1(𝑡, 𝑥) − 𝑓

1(𝑡, 𝑦)‖

𝑋≤

𝐿1(𝑡)‖𝑥 − 𝑦‖

𝑋for all (𝑡, 𝑥), (𝑡, 𝑦) ∈ 𝑅,

(A.2) 𝑓2

is compact and bounded, in particular,‖𝑓2(𝑡, 𝑥)‖

𝑋≤ 𝑀2(𝑡) for all (𝑡, 𝑥) ∈ 𝑅,

(A.3) 𝐼𝑘

∈ 𝐶(𝑋,𝑋) such that ‖𝐼𝑘(𝑥)‖𝑋

≤ 𝑙1and ‖𝐼

𝑘(𝑥) −

𝐼𝑘(𝑦)‖𝑋

≤ 𝑙2‖𝑥 − 𝑦‖

𝑋,

where 𝐶(𝑅 × 𝑋,𝑋) denotes set of continuous functions from𝑅 × 𝑋 to 𝑋 and 𝐿

1/𝑝([0, 𝑇],R+) denotes space 1/𝑝-Lesbegue

measurable functions from [0, 𝑇] to R+ with norm ‖ ⋅ ‖1/𝑝

.

Lemma 7 (Fečken et al. [24], Lemma 2). The initial valueproblem (4) is equivalent to the nonlinear integral equation

𝑥 (𝑡) = 𝑥0+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓1(𝑠, 𝑥 (𝑠)) 𝑑𝑠

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓2(𝑠, 𝑥 (𝑠)) 𝑑𝑠, 𝑡 ∈ [0, 𝑡

1]

= 𝑥0+ 𝐼1(𝑥 (𝑡−

1)) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓2(𝑠, 𝑥 (𝑠)) 𝑑𝑠, 𝑡 ∈ (𝑡

1, 𝑡2]

= 𝑥0+

2

∑

𝑘=1

𝐼𝑘(𝑥 (𝑡−

𝑘)) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓2(𝑠, 𝑥 (𝑠)) 𝑑𝑠, 𝑡 ∈ (𝑡

2, 𝑡3]

= 𝑥0+

𝑚

∑

𝑘=1


𝑘)) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓2(𝑠, 𝑥 (𝑠)) 𝑑𝑠, 𝑡 ∈ (𝑡

𝑚, 𝑇] .

(6)

In other words, every solution of the integral equation (6) is alsosolution of our original initial value problem (4) and conversely.

Theorem 8 (existence of solutions). Under the assumptions(A.1)–(A.3), the problem (4) has at least one solution in [0, 𝑇],provided that

𝛾1= 𝑚𝑙2+

𝑐𝐿1

1/𝑝

𝑇𝛼−𝑝

Γ (𝛼 + 1)

< 1, where 𝑐 = (1 − 𝑝

𝛼 − 𝑝

)

1−𝑝

.

(7)

Proof. Let 𝐵𝜆be the closed bounded and convex subset of

PC([0, 𝑇], 𝑋), where 𝐵𝜆is defined as 𝐵

𝜆= {𝑥 : ‖𝑥‖ ≤ 𝜆}, 𝜆 =

max{𝜆0, 𝜆1, . . . , 𝜆

𝑚}, and

𝜆𝑘=

𝑥0

𝑋

+ 𝑘𝑙1+

𝑐 (𝑀1

1/𝑝

+𝑀2

1/𝑝

)

Γ (𝛼)

𝑇𝛼−𝑝

,

𝑘 = 0, 1, 2, . . . , 𝑚.

(8)

Define a map 𝐹 : 𝐵𝜆

→ 𝑋 such that

𝐹𝑥 (𝑡) = 𝑥0+ ∑

0


For 𝑡 ∈ (𝑡1, 𝑡2],

‖𝐹𝑥 (𝑡)‖𝑋

≤𝑥0

𝑋

+𝐼1(𝑥 (𝑡−

1))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠 ⋅ 𝑥 (𝑠))

𝑋

𝑑𝑠

≤𝑥0

𝑋

+ 𝑙1+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓1(𝑠, 𝑥 (𝑠))

𝑋

𝑑𝑠

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓2(𝑠, 𝑥 (𝑠))

𝑋

𝑑𝑠

≤𝑥0

𝑋

+ 𝑙1+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑀1(𝑠) 𝑑𝑠

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


≤𝑥0

𝑋

+ 𝑙1+

1

Γ (𝛼)

× (∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

(∫

𝑡

0

𝑀1/𝑝

1(𝑠) 𝑑𝑠)

𝑝

+

1

Γ (𝛼)

(∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

× (∫

𝑡

0

𝑀1/𝑝

2(𝑠) 𝑑𝑠)

𝑝

≤𝑥0

𝑋

+ 𝑙1+

𝑐 (𝑀1

1/𝑝

+𝑀2

1/𝑝

)

Γ (𝛼)

𝑇𝛼−𝑝

= 𝜆1.

(12)For 𝑡 ∈ (𝑡

2, 𝑡3],

‖𝐹𝑥 (𝑡)‖ ≤𝑥0

𝑋

+

2

∑

𝑘=0


𝑘))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠 ⋅ 𝑥 (𝑠))

𝑋

𝑑𝑠

≤𝑥0

𝑋

+ 2𝑙1+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓1(𝑠, 𝑥 (𝑠))

𝑋

𝑑𝑠

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓2(𝑠, 𝑥 (𝑠))

𝑑𝑠

≤𝑥0

𝑋

+ 2𝑙1+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


≤𝑥0

𝑋

+ 2𝑙1+

1

Γ (𝛼)

(∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

×(∫

𝑡

0

𝑀1/𝑝

1(𝑠) 𝑑𝑠)

𝑝

+

1

Γ (𝛼)

× (∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

(∫

𝑡

0

𝑀1/𝑝

2(𝑠) 𝑑𝑠)

𝑝

≤𝑥0

𝑋

+ 2𝑙1+

𝑐 (𝑀1

1/𝑝

+𝑀2

1/𝑝

)

Γ (𝛼)

𝑇𝛼−𝑝

= 𝜆2.

(13)

For 𝑡 ∈ (𝑡𝑚, 𝑇],

‖𝐹𝑥 (𝑡)‖𝑋

≤𝑥0

𝑋

+

𝑚

∑

𝑘=0


𝑘))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠 ⋅ 𝑥 (𝑠))

𝑋

𝑑𝑠

≤𝑥0

𝑋

+ 𝑚𝑙1

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓1(𝑠, 𝑥 (𝑠))

𝑋

𝑑𝑠

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓2(𝑠, 𝑥 (𝑠))

𝑑𝑠

≤𝑥0

𝑋

+ 𝑚𝑙1+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


≤𝑥0

𝑋

+ 𝑚𝑙1

+

1

Γ (𝛼)

(∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

× (∫

𝑡

0

𝑀1/𝑝

1(𝑠) 𝑑𝑠)

𝑝

+

1

Γ (𝛼)

× (∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

(∫

𝑡

0

𝑀1/𝑝

2(𝑠) 𝑑𝑠)

𝑝

≤𝑥0

𝑋

+ 𝑚𝑙1+

𝑐 (𝑀1

1/𝑝

+𝑀2

1/𝑝

)

Γ (𝛼)

𝑇𝛼−𝑝

= 𝜆𝑚,

(14)

and thus 𝐹(𝐵𝜆) ⊂ 𝐵𝜆.

Step 2 (𝐹1is continuous and 𝛾-contraction). To prove the

continuity of 𝐹1for 𝑡 ∈ [0, 𝑇], let us consider a sequence 𝑥

𝑛

converging to 𝑥. Taking the norm of 𝐹1𝑥𝑛(𝑡)−𝐹

1𝑥(𝑡), we have

𝐹1𝑥𝑛(𝑡) − 𝐹

1𝑥 (𝑡)

𝑋

≤ ∑

0


+

1

Γ𝛼

(∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

× (∫

𝑡

0

𝐿1/𝑝

1(𝑠) 𝑑𝑠)

𝑝

𝑥𝑛− 𝑥

≤ (𝑚𝑙2+

𝑐𝐿1

1/𝑝

𝑇𝛼−𝑝

Γ𝛼

)𝑥𝑛− 𝑥

.

(15)

From the pervoius analysis, we obtain

𝐹1𝑥𝑛− 𝐹1𝑥≤ (𝑚𝑙

2+

𝑐𝐿1

1/𝑝

𝑇𝛼−𝑝

Γ (𝛼)

)𝑥𝑛− 𝑥

. (16)

To prove that 𝐹1is 𝛾1-contraction, let us consider that, for

𝑥, 𝑦 ∈ 𝐵𝑟,

𝐹1𝑥 (𝑡) − 𝐹

1𝑦 (𝑡)

𝑋

≤ ∑

0


The right-hand side of the pervoiusly expression does notdepend on 𝑥. Thus using Arzela-Ascoli theorem for equicon-tinuous functions (Diethelm, Theorem D.10 [10]), we con-clude that 𝐹

2(𝐵𝑟) is relatively compact, and hence 𝐹

2is

completely continuous on 𝐼− {𝑡1, 𝑡2, . . . , 𝑡

𝑚}. In a similar way,

it can be prove the equi-continuity of 𝐹 on 𝑡 = 𝑡−𝑘and 𝑡 = 𝑡+

𝑘,

𝑘 = 1, 2, . . . , 𝑚. And thus 𝐹2is compact on [0, 𝑇].

Step 4 (𝐹 is condensing). As 𝐹 = 𝐹1+ 𝐹2, 𝐹1is continuous

contraction, 𝐹2is compact, so by using Lemma 6, 𝐹 is

condensing map on 𝐵𝑟.

And hence by using the Theorem 5, we conclude that (4)has a solution in 𝐵

𝑟.

Theorem 9. If 𝑓 is bounded and Lipschitz, in particular,‖𝑓(𝑡, 𝑥) − 𝑓(𝑡, 𝑦)‖

𝑋≤ 𝐿∗

1(𝑡)‖𝑥 − 𝑦‖

𝑋for all (𝑡, 𝑥), (𝑡, 𝑦) ∈ 𝑅

and 𝐿∗1∈ 𝐿1/𝑝

([0, 𝑇],R+), then the problem (4) has a uniquesolution in 𝐵

𝜆, provided that

𝛾∗

1= 𝑚𝑙2+

𝑐𝐿∗

1

(1/𝑝) 𝑇

𝛼−𝑝

Γ (𝛼 + 1)

< 1, (20)

where 𝑐 = ((1 − 𝑝)/(𝛼 − 𝑝))1−𝑝.

3. Impulsive Fractional Differential Equationswith Finite Delay

In this section, we discuss existence and uniqueness ofsolutions of the following impulsive fractional differentialequations of Caputo type with finite delay in a Banach space𝑋 with norm | ⋅ |

𝐶

𝐷𝛼

𝑥 (𝑡) = 𝑓 (𝑡, 𝑥𝑡) , 𝑡 ∈ 𝐼 = [0, 𝑇] , 𝑡 ̸= 𝑡

𝑘,



𝑘)) , 𝑘 = 1, 2, . . . , 𝑚,

𝑥 (𝑡) = 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] ,

(21)

where 𝑓 : 𝐼 × C → 𝑋, C = 𝐶([−𝑟, 0], 𝑋) with norm ‖𝑥‖𝑟=

sup{‖𝑥(𝑡)‖𝑋

: 𝑡 ∈ [−𝑟, 0]}. 𝐼𝑘∈ 𝐶(𝑋,𝑋), (𝑘 = 1, 2, . . . , 𝑚), and

𝑋 is a Banach space with a norm | ⋅ |𝑋. For any 𝑥 : [−𝑟, 𝑇] →

𝑋 and 𝑡 ∈ 𝐼, 𝑥𝑡∈ C and is defined by 𝑥

𝑡(𝑠) = 𝑥(𝑡 + 𝑠), 𝑠 ∈

[−𝑟, 0]. Here, our tools are Banach and Schaefer fixed pointtheorems.

Define a new Banach space PC([−𝑟, 𝑇], 𝑋)

PC ([−𝑟, 𝑇] , 𝑋) = {𝑥 : [−𝑟, 𝑇] → 𝑋 | 𝑥 ∈ 𝐶 ((𝑡𝑘, 𝑡𝑘+1

] , 𝑋)

∪ 𝐶 ([−𝑟, 0] , 𝑋) , 𝑘 = 0, 1, 2, . . . , 𝑚,

𝑥 (𝑡+

𝑘) and 𝑥 (𝑡−

𝑘) exist,

𝑥 (𝑡𝑘) = 𝑥 (𝑡

−

𝑘)}

(22)

with sup-norm ‖ ⋅ ‖, defined by ‖𝑥‖ = sup{‖𝑥(𝑡)‖𝑋

: 𝑡 ∈

[−𝑟, 𝑇]}.

Definition 10. A solution of fractional differential equation(21) is a piecewise continuous function 𝑥 ∈ PC([−𝑟, 𝑇], 𝑋)which satisfies (21).

Consider the initial value problem (21) on 𝐼 ×C for somefixed 𝑇 > 0 and assume that there exist 𝑝 ∈ (0, 𝛼), 𝛼 ∈(0, 1),𝑀

3,𝑀4, 𝐿2

∈ 𝐿1/𝑝

([0, 𝑇],R+) such that the followingassumptions are satisfied:(A.4) 𝑓 ∈ 𝐶(𝐼 × C, 𝑋),(A.5) 𝑓 bounded, in particular, ‖𝑓(𝑡, 𝜙)‖

𝑋≤ 𝑀3(𝑡) for all

(𝑡, 𝜙) ∈ 𝐼 × C,(A.6) 𝑓 is Lipschitz, in particular, ‖𝑓(𝑡, 𝜙) − 𝑓(𝑡, 𝜓)‖

𝑋≤

𝐿2(𝑡) ‖𝜙 − 𝜓‖

𝑟for all (𝑡, 𝜙), (𝑡, 𝜓) ∈ 𝐼 × C.


𝑥 (𝑡) = 𝜙 (0) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠, 𝑥𝑠) 𝑑𝑠,

𝑡 ∈ [0, 𝑡1]

= 𝜙 (0) + 𝐼1(𝑥 (𝑡−

1))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠, 𝑥𝑠) 𝑑𝑠, 𝑡 ∈ (𝑡

1, 𝑡2]

= 𝜙 (0) +

2

∑

𝑘=1


𝑘))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


2, 𝑡3]

= 𝜙 (0) +

𝑚

∑

𝑘=1


𝑘))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


𝑚, 𝑇]

= 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] .

(23)

In other words, every solution of the integral equation (23)is also solution of our original initial value problem (21) andconversely.

Remark 12. Since history part/initial condition 𝑥(𝑡) =𝜙(𝑡), 𝑡 ∈ [−𝑟, 0], is known, sowewill investigate the existenceand uniqueness of solution in 𝐼 = [0, 𝑇].

Theorem 13 (existence and uniqueness of solution). Underthe assumptions (A.3)–(A.6), the problem (21) has a uniquesolution in [0, 𝑇], provided that

𝛾2= 𝑚𝑙2+

𝑐𝐿2

1/𝑝

𝑇𝛼−𝑝

Γ (𝛼)

< 1, (24)

where 𝑐 = ((1 − 𝑝)/(𝛼 − 𝑝))1−𝑝.

Proof. In this case, we define the operator 𝐹 : PC(𝐼, 𝑋) →PC(𝐼, 𝑋) by

𝐹𝑥 (𝑡) =

𝑚

∑

0


Step 1. To prove that 𝐹 is self mapping, we need to prove thatfor each 𝑥 ∈ PC(𝐼, 𝑋), 𝐹

𝑥∈ PC(𝐼, 𝑋).

The proof is similar to the proof of continuity of 𝐹1in

Step 2 of Theorem 8, and hence we omit it.Step 2 (𝐹 is continuous and 𝛾

2-contraction). The proof of

this step is also similar to the proof of continuous and 𝛾1-

contraction of 𝐹1in Step 2 of Theorem 8.

Now by applying Banach’s fixed point theorem, we getthat the operator 𝐹 has an unique fixed point in PC(𝐼, 𝑋),and hence the problem (21) has a unique solution inPC([−𝑟, 𝑇], 𝑋).

Our next result is based on Schaefer’s fixed point theorem.In this case, we replace assumption (A.3) with the followinglinear growth condition:

(A.3) 𝐼𝑘bounded, in particular, ‖𝐼

𝑘(𝑥)‖𝑋

≤ 𝑙∗

1,

(A.5) 𝑓 bounded, in particular, ‖𝑓(𝑡, 𝜙)‖𝑋

≤ 𝑀4(𝑡)(1 +

‖𝜙‖𝑟) for all (𝑡, 𝜙) ∈ 𝐼 × C.

Theorem 14. Under the assumptions (A.3) and (A.5), prob-lem (21) has at least one solution.

Proof. We transform the problem into a fixed point problem.For this purpose, consider the operator 𝐹 : PC(𝐼, 𝑋) →PC(𝐼, 𝑋) defined by

𝐹𝑥 (𝑡) =

𝑚

∑

0


× (∫

𝑡

0

(𝑡 − 𝑠)(𝛼−1)/(1−𝑝)

𝑑𝑠)

1−𝑝

(∫

𝑡

0

𝑀1/𝑝

4(𝑠) 𝑑𝑠)

𝑝

≤𝜙 (0)

𝑟

+ 𝑚𝑙∗

1+

𝑐 (1 + ‖𝑥‖) (𝑀4

1/𝑝

)

Γ (𝛼)

𝑇𝛼−𝑝

,

(29)

and hence

‖𝑥‖ ≤

𝜙 (0)

𝑟

+ 𝑚𝑙∗

1+ (𝑐 (

𝑀4

1/𝑝

) /Γ (𝛼)) 𝑇𝛼−𝑝

1 − (𝑐 (𝑀4

1/𝑝

) /Γ (𝛼)) 𝑇𝛼−𝑝

. (30)

This shows that 𝐸(𝐹) is bounded.As a consequence of Schaefer’s fixed point theorem, the

problem (21) has at least one solution in PC([−𝑟, 𝑇], 𝑋).

Theorem 15. If 𝑓 is bounded and Lipschitz, in particular,‖𝑓(𝑡, 𝜙)−𝑓(𝑡, 𝜓)‖

𝑋≤ 𝐿∗

2(𝑡) ‖𝜙−𝜓‖

𝑟, for all (𝑡, 𝜙), (𝑡, 𝜓) ∈ 𝐼×C

and 𝐿∗2

∈ 𝐿1/𝑝

([0, 𝑇],R+), then problem (21) has a uniquesolution in PC([−𝑟, 𝑇], 𝑋), provided that

𝛾∗

2= 𝑚𝑙2+

𝑐𝐿∗

2

1/𝑝

𝑇𝛼−𝑝

Γ (𝛼 + 1)

< 1, (31)

where 𝑐 = ((1 − 𝑝)/(𝛼 − 𝑝))1−𝑝.

Further, we consider the following more general Caputofractional differential equation

𝐶

𝐷𝛼

𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑥𝑡) , 𝑡 ∈ [0, 𝑇] , 𝑡 ̸= 𝑡

𝑘,



𝑘)) , 𝑘 = 1, 2, . . . , 𝑚,

𝑥 (𝑡) = 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] ,

(32)

where 𝑓 : 𝐼 × 𝑋 × C → 𝑋 and C = 𝐶([−𝑟, 0], 𝑋) withnorm ‖𝑥‖

𝑟= sup{‖𝑥(𝑡)‖

𝑋: 𝑡 ∈ [−𝑟, 0]}, 𝐼

𝑘∈ 𝐶(𝑋,𝑋), (𝑘 =

1, 2, . . . , 𝑚), and 𝑋 is a separable real Banach space with thenorm ‖ ⋅ ‖

𝑋. Here, our tools will be Banach and Schaefer fixed

point theorems.

Definition 16. A solution of fractional differential equation(32) is a piecewise continuous function 𝑥 ∈ PC([−𝑟, 𝑇], 𝑋)which satisfies (32).

Consider the initial value problem (32) on 𝐼 × C × 𝑋 forsome fixed 𝑇 > 0 and assume that there exist 𝑝 ∈ (0, 𝛼),𝑀5,𝑀6, 𝐿3, 𝐿4

∈ 𝐿1/𝑝

([0, 𝑇],R+) such that the followingassumptions are satisfied:

(A.7) 𝑓 ∈ 𝐶(𝐼 × 𝑋 × C, 𝑋),(A.8) 𝑓 bounded, in particular, ‖𝑓(𝑡, 𝑥, 𝜙)‖

𝑋≤ 𝑀5(𝑡) for

all (𝑡, 𝑥, 𝜙) ∈ 𝐼 × 𝑋 × C,(A.9) 𝑓 is Lipschitz, in particular, ‖𝑓(𝑡, 𝑥, 𝜙) −

𝑓(𝑡, 𝑦, 𝜓)‖𝑋

≤ 𝐿3(𝑡)‖𝑥 − 𝑦‖

𝑋+ 𝐿4(𝑡)‖𝜙 − 𝜓‖

𝑟

for all (𝑡, 𝑥, 𝜙), (𝑡, 𝑦, 𝜓) ∈ 𝐼 × 𝑋 × C,(A.8) ‖𝑓(𝑡, 𝑥, 𝜙)‖

𝑋≤ 𝑀6(𝑡)(1 + ‖𝑥‖

𝑋+ ‖𝜙‖𝑟).

Lemma 17 (Fečken et al. [24], Lemma 2). The initial valueproblem (32) is equivalent to the following nonlinear integralequation

𝑥 (𝑡) = 𝜙 (0) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠, 𝑥 (𝑠) , 𝑥𝑠) 𝑑𝑠,

𝑡 ∈ [0, 𝑡1]

= 𝜙 (0) + 𝐼1(𝑥 (𝑡−

1))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠, 𝑥 (𝑠) , 𝑥𝑠) 𝑑𝑠, 𝑡 ∈ (𝑡

1, 𝑡2]

= 𝜙 (0) +

2

∑

𝑘=1


𝑘))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠, 𝑥 (𝑠) , 𝑥𝑠) 𝑑𝑠, 𝑡 ∈ (𝑡

2, 𝑡3]

= 𝜙 (0) +

𝑚

∑

𝑘=1


𝑘))

+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑓 (𝑠, 𝑥 (𝑠) , 𝑥𝑠) 𝑑𝑠, 𝑡 ∈ (𝑡

𝑚, 𝑇]

= 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] .

(33)

In other words, every solution of the integral equation (33)is also solution of our original initial value problem (32) andconversely.

Theorem 18. Under the assumptions (A.3), (A.7), (A.8), and(A.9), the problem (32) has a unique solution, provided that

𝛾3= 𝑚𝑙2+

𝑐 (𝐿3

1/𝑝

+𝐿4

1/𝑝

) 𝑇𝛼−𝑝

Γ (𝛼)

< 1.(34)

Proof. The proof is similar to the proof of Theorem 13.

Theorem 19. Under the assumptions (A.3), and (A.8), theproblem (32) has at least one solution in PC([−𝑟, 𝑇], 𝑋).

Proof. The proof is similar to the proof of Theorem 14.

Theorem 20. If 𝑓 is bounded and Lipschitz, in particular,‖𝑓(𝑡, 𝑥, 𝜙) −𝑓(𝑡, 𝑦, 𝜓)‖

𝑋≤ 𝐿∗

3(𝑡)‖𝑥 −𝑦‖

𝑋+𝐿∗

4(𝑡)‖𝜙 −𝜓‖

𝑟, for

all (𝑡, 𝑥, 𝜙), (𝑡, 𝑦, 𝜓) ∈ 𝐼×𝑋×C and 𝐿∗3, 𝐿∗

4∈ 𝐿1/𝑝

([0, 𝑇],R+),then the problem (32) has a unique solution in PC([−𝑟, 𝑇], 𝑋),provided that

𝛾∗

3= 𝑚𝑙2+

𝑐 (𝐿∗

3

1/𝑝

+𝐿∗

2

1/𝑝

) 𝑇𝛼−𝑝

Γ (𝛼 + 1)

< 1,(35)

where 𝑐 = ((1 − 𝑝)/(𝛼 − 𝑝))1−𝑝.


4. Examples

Example 21 (fractional impulsive logistic equation). Con-sider the following class of fractional logistic equations inbanach space 𝑋 with norm ‖ ⋅ ‖

𝑋:

𝐶

𝐷𝛼

𝑥 (𝑡) = 𝑥 (𝑡) (𝑎 (𝑡) − 𝑏 (𝑡) 𝑥 (𝑡)) , 𝑡 ∈ [0, 𝑇] , 𝑡 ̸= 𝑡𝑘,



𝑘)) , 𝑘 = 1, 2, . . . , 𝑚,

𝑥 (0) = 𝑥0,

(36)

where 𝑎(𝑡) ∈ [𝑎∗, 𝑎∗

] and 𝑏(𝑡) ∈ [𝑏∗, 𝑏∗

] with 𝑎∗, 𝑏∗

> 0.


𝑥 (𝑡) = 𝑥0+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

(𝑥 (𝑠) (𝑎 (𝑠) − 𝑏 (𝑠) 𝑥 (𝑠))) 𝑑𝑠,

𝑡 ∈ [0, 𝑡1]

= 𝑥0+ 𝐼1(𝑥 (𝑡−

1)) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑥 (𝑠) 𝑎 (𝑠) 𝑑𝑠

−

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑏 (𝑠) 𝑥2

(𝑠) 𝑑𝑠, 𝑡 ∈ (𝑡1, 𝑡2]

= 𝑥0+

2

∑

𝑘=1


𝑘)) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


−

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑏 (𝑠) 𝑥2

(𝑠) 𝑑𝑠,

𝑡 ∈ (𝑡2, 𝑡3]

= 𝑥0+

𝑚

∑

𝑘=1


𝑘)) +

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1


−

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑏 (𝑠) 𝑥2

(𝑠) 𝑑𝑠, 𝑡 ∈ [𝑡𝑚, 𝑇] .

(37)

In other words, every solution of the integral equation (37) isalso solution of our original initial value problem (36) andconversely.

We can easily see that for the problem (36), our functionsare 𝑓1(𝑡, 𝑥) = 𝑎(𝑡)𝑥 and 𝑓

2(𝑡, 𝑥) = −𝑏(𝑡)𝑥

2. It is not difficultto deduce that

𝑓1(𝑡, 𝑥)

𝑋

≤ 𝑎∗

‖𝑥‖𝑋

+ 𝑚𝑙1,

𝑓1(𝑡, 𝑥) − 𝑓

1(𝑡, 𝑦)

𝑋

≤ 𝑎∗𝑥 − 𝑦

𝑋

.

(38)

Also ‖𝑓2(𝑡, 𝑥)‖

𝑋≤ 𝑏∗

‖𝑥‖2. From the integral representa-

tion of problem (36), we get

‖𝑥 (𝑡)‖𝑋

≤𝑥0

𝑋

+ 𝑚𝑙1+

1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑎∗

‖𝑥 (𝑠)‖𝑋𝑑𝑠.

(39)

Using Gronwall’s inequality (Diethelm, Lemma 6.19 [10,31]), we get

‖𝑥 (𝑡)‖𝑋

≤ (𝑥0

𝑋

+ 𝑚𝑙1) exp( 1

Γ (𝛼)

∫

𝑡

0

(𝑡 − 𝑠)𝛼−1

𝑎∗

𝑑𝑠)

≤ (𝑥0

𝑋

+ 𝑚𝑙1) exp( 𝑎

∗

Γ (𝛼 + 1)

) .

(40)

Thus 𝑥 is bounded which implies that all the assumptions ofTheorem 8 are satisfied, and hence there exists a solution ofthe problem (36).

We give some more examples which are inspired by [24].

Example 23. Consider the following Caputo impulsive delayfractional differential equations

𝐶

𝐷𝛼

𝑥 (𝑡) =

𝑒−𝜈𝑡

𝑥𝑡

(1 + 𝑒𝑡) (1 +

𝑥𝑡

)

, 𝑡 ∈ [0, 1] , 𝑡 ̸= 𝑡1, 𝜈 > 0,

Δ𝑥 (𝑡) |𝑡=𝑡1

=

1

2

,

𝑥 (𝑡) = 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] .

(41)

Set 𝐶1

= 𝐶([−𝑟, 0],R+), 𝑓(𝑡, 𝜙) = 𝑒−𝜈𝑡𝜙/((1 + 𝑒𝑡)(1 +

𝜙)), (𝑡, 𝜙) ∈ [0, 1] × 𝐶1.

Let 𝜙1, 𝜙2∈ 𝐶1and let 𝑡 ∈ [0, 1]. Then, we have

𝑓 (𝑡, 𝜙

1) − 𝑓 (𝑡, 𝜙

2)=

𝑒−𝜈𝑡

1 + 𝑒𝑡

𝜙1

1 + 𝜙1

−

𝜙2

1 + 𝜙2

=

𝑒−𝜈𝑡

𝜙1− 𝜙2

𝑟

(1 + 𝑒𝑡) (1 + 𝜙

1) (1 + 𝜙

2)

≤

𝑒−𝜈𝑡

𝜙1− 𝜙2

𝑟

(1 + 𝑒𝑡)

≤ 𝐿∗

(𝑡)𝜙1− 𝜙2

𝑟

,

(42)

where 𝐿∗(𝑡) = 𝑒−𝜈𝑡/2.Again, for all 𝜙 ∈ 𝐶

1and each 𝑡 ∈ [0, 𝑇],

𝑓 (𝑡, 𝜙)

=

𝑒−𝜈𝑡

1 + 𝑒𝑡

𝜙

1 + 𝜙

≤

𝑒−𝜈𝑡

(1 + 𝑒𝑡)

< 𝑚1(𝑡) ,

(43)

where 𝑚1(𝑡) = 𝑒

−𝜈𝑡

/2.For 𝑡 ∈ [0, 1] and some 𝑝 ∈ (0, 𝛼), 𝐿∗(𝑡) = 𝑚

1(𝑡) =

𝑒−𝜈𝑡

/2 ∈ 𝐿1/𝑝

([0, 1],R+) with 𝑀∗1

= ‖𝑒−𝜈𝑡

/10‖1/𝑝

, wecan arrive at the inequality 1/4 + 𝑐𝑀∗

1/Γ(𝛼) < 1. We

can see that all the assumptions of Theorem 13 are satis-fied, and hence the problem (41) has a unique solution in[0, 1].


Example 24. Consider the following Caputo impulsive delayfractional differential equation

𝐶

𝐷𝛼

𝑥 (𝑡) =

𝑥𝑡

(1 + 𝑒𝑡) (1 +

𝑥𝑡

)

, 𝑡 ∈ [0, 1] , 𝑡 ̸= 𝑡1, 𝜈 > 0,

Δ𝑥 (𝑡) |𝑡=𝑡1

=

1

2

,

𝑥 (𝑡) = 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] .

(44)

Set 𝐶2

= 𝐶([0, 1],R+), 𝑓(𝑡, 𝜙) = 𝜙/((1 + 𝑒𝑡)(1 +

𝜙)), (𝑡, 𝜙) ∈ [−𝑟, 0] × 𝐶2.

Again, for all 𝜙 ∈ 𝐶2and each 𝑡 ∈ [0, 𝑇],

𝑓 (𝑡, 𝜙)

=

𝑒−𝑡

1 + 𝑒𝑡

𝜙

1 + 𝜙

≤

𝑒−𝑡

(1 + 𝑒𝑡)

< 𝑚2(𝑡) (1 +

𝜙) ,

(45)

where 𝑚2(𝑡) = 𝑒

−𝑡

/4.For 𝑡 ∈ [0, 1] and some 𝑝 ∈ (0, 𝛼), 𝑚

2(𝑡) = 𝑒

−𝑡

/4 ∈

𝐿1/𝑝

([0, 1],R+) with 𝑀∗2

= ‖𝑒−𝑡

/4‖1/𝑝

, we can arrive atthe inequality 1/4 + 𝑐𝑀/Γ(𝛼) < 1. We can see that allthe assumptions of Theorem 14 are satisfied, and hence theproblem (44) has a solution in [0, 1].

Acknowledgments

The authors are thankful to the anonymous reviewerfor his/her valuable comments and suggestions. S. Abbasacknowledge “Erasmus Mundus Lot13, India 4EU” for pro-viding him fellowship to visit University of Bologna, Italy.He also acknowledges Professor Stefan Siegmund, Centerfor Dynamics of TU Dresden, Germany, for his fruitfuldiscussion on this topic and for hosting his short visit.

References

[1] V. D. Milman and A. D. Myškis, “On the stability of motion inthe presence of impulses,” Siberial Mathematical Journal, vol. 1,pp. 233–237, 1960.

[2] A. M. Samoilenko and N. A. Perestyuk, Differential EquationsWith Impulses, Viska Scola, Kiev, Ukraine, 1987.

[3] V. Lakshmikantham, D. D. Baı̆nov, and P. S. Simeonov, Theoryof Impulsive Differential Equations, vol. 6 of Series in ModernApplied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.

[4] D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effects,Horwood, Chichister, UK, 1989.

[5] D. D. Bainov and P. S. Simeonov, Impulsive DiffErential Equa-tions: Periodic Solutions and Its Applications, LongmanScientificand Technical Group, Harlow, UK, 1993.

[6] D. Bainov and V. Covachev, Impulsive Differential Equationswith a Small Parameter, vol. 24 of Series on Advances inMathematics for Applied Sciences, World Scientific, River Edge,NJ, USA, 1994.

[7] M. Benchohra, J. Henderson, and S. K. Ntonyas, ImpulsiveDiffrential Equations and Inclusions, vol. 2, Hindawi PublishingCorporation, New York, NY, USA, 2006.

[8] J. Hale, Theory of Functional Differential Equations, Springer,New York, NY, USA, 2nd edition, 1977.

[9] F. Mainardi, “Fractional calculus: some basic problems incontinuum and statistical mechanics,” in Fractals and FractionalCalculus in ContinuumMechanics, vol. 378 ofCISMCourses andLectures, pp. 291–348, Springer, Vienna, Austria, 1997.

[10] K. Diethelm, The Analysis of Fractional Differential Equations,vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin,Germany, 2010.

[11] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theoryand Applications of Fractional Differential Equations, vol. 204,North-Holland Mathematics Studies, 2006.

[12] K. S.Miller and B. Ross,An Introduction to the Fractional Calcu-lus and Fractional Differential Equations, A Wiley-IntersciencePublication, John Wiley & Sons, New York, NY, USA, 1993.

[13] I. Podlubny, Fractional Differential Equations, vol. 198 ofMath-ematics in Science and Engineering, Academic Press, San Diego,Calif, USA, 1999.

[14] J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advancesin Fractional Calculus, Springer, Dordrecht, The Netherlands,2007.

[15] S. G. Samko, A. A. Kilbas, and O. I. Marichev, FractionalIntegrals and Derivatives, Gordon and Breach Science, Yverdon,Switzerland, 1993.

[16] S. Abbas, “Existence of solutions to fractional order ordinaryand delay differential equations and applications,” ElectronicJournal of Differential Equations, vol. 2011, no. 9, pp. 1–11, 2011.

[17] S. Abbas, M. Banerjee, and S. Momani, “Dynamical analysisof fractional-order modified logistic model,” Computers &Mathematics with Applications, vol. 62, no. 3, pp. 1098–1104,2011.

[18] E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “Equi-librium points, stability and numerical solutions of fractional-order predator-prey and rabies models,” Journal of Mathemati-cal Analysis and Applications, vol. 325, no. 1, pp. 542–553, 2007.

[19] S. B.Hadid, “Local and global existence theorems on differentialequations of non-integer order,” Journal of Fractional Calculus,vol. 7, pp. 101–105, 1995.

[20] R. W. Ibrahim and S. Momani, “On the existence and unique-ness of solutions of a class of fractional differential equations,”Journal of Mathematical Analysis and Applications, vol. 334, no.1, pp. 1–10, 2007.

[21] V. Lakshmikantham and A. S. Vatsala, “Basic theory of frac-tional differential equations,” Nonlinear Analysis: Theory, Meth-ods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.

[22] R. P. Agarwal, M. Benchohra, and B. A. Slimani, “Exis-tence results for differential equations with fractional orderand impulses,” Georgian Academy of Sciences. A. RazmadzeMathematical Institute. Memoirs on Differential Equations andMathematical Physics, vol. 44, pp. 1–21, 2008.

[23] M. Benchohra and B. A. Slimani, “Existence and uniquenessof solutions to impulsive fractional differential equations,”Electronic Journal of Differential Equations, vol. 2009, no. 10, pp.1–11, 2009.

[24] M. Fečkan, Y. Zhou, and J.Wang, “On the concept and existenceof solution for impulsive fractional differential equations,”Com-munications inNonlinear Science andNumerical Simulation, vol.17, no. 7, pp. 3050–3060, 2012.

[25] D. Xu, Y. Hueng, and L. Ling, “Existence of positive solutionsof an Impulsive Delay Fishing model,” Bulletin of MathematicalAnalysis and Applications, vol. 3, no. 2, pp. 89–94, 2011.


[26] C. Giannantoni, “The problemof the initial conditions and theirphysical meaning in linear differential equations of fractionalorder,”AppliedMathematics and Computation, vol. 141, no. 1, pp.87–102, 2003.

[27] N. Heymans and I. Podlubny, “Physical interpretation of initialconditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5,pp. 765–771, 2006.

[28] I. Podlubny, “Geometric and physical interpretation of frac-tional integration and fractional differentiation,” FractionalCalculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002,Dedicated to the 60th anniversary of Prof. Francesco Mainard.

[29] E. Zeidler, Non-Linear Functional Analysis and Its Application:Fixed Point-Theorems, vol. 1, Springer,NewYork,NY,USA, 1986.

[30] B. N. Sadovskĭı, “On a fixed point principle,” Functional Analysisand Its Applications, vol. 1, no. 2, pp. 74–76, 1967.

[31] H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequalityand its application to a fractional differential equation,” Journalof Mathematical Analysis and Applications, vol. 328, no. 2, pp.1075–1081, 2007.

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2012, Article ID 472030, 14 pagesdoi:10.1155/2012/472030

Research ArticleSolving the Fractional Rosenau-HymanEquation via Variational Iteration Method andHomotopy Perturbation Method

R. Yulita Molliq1 and M. S. M. Noorani2

1 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri MedanUNIMED 20221, Medan, Sumatera Utara, Indonesia

2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan MalaysiaUKM, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to R. Yulita Molliq, [email protected]

Received 31 May 2012; Accepted 8 November 2012

Academic Editor: Shaher Momani

Copyright q 2012 R. Yulita Molliq and M. S. M. Noorani. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

In this study, fractional Rosenau-Hynam equations is considered. We implement relatively newanalytical techniques, the variational iteration method and the homotopy perturbation method,for solving this equation. The fractional derivatives are described in the Caputo sense. The twomethods in applied mathematics can be used as alternative methods for obtaining analytic andapproximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solutiontakes the form of a convergent series with easily computable components. The present methodsperform extremely well in terms of efficiency and simplicity.

1. Introduction

Recent advances of fractional differential equations are stimulated by new examples ofapplications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, andphysics. For example, the nonlinear oscillation of earthquake can be modeled with fractionalderivatives �1�, and the fluid-dynamic traffic model with fractional derivatives �2� caneliminate the deficiency arising from the assumption of continuum traffic flow. Based onexperimental data fractional partial differential equations for seepage flow in porous mediaare suggested in �3�, and differential equations with fractional order have recently provedto be valuable tools to the modeling of many physical phenomena �4�. Fractional partialdifferential equations also have studied and successfully solved such as the space-timefractional diffusion-wave equation �5–7�, the fractional advection-dispersion equation �8, 9�,


the fractional KdV equation �10�, and the linear inhomogeneous fractional partial differentialequations �11�.

Most nonlinear differential equations are usually arising frommathematical modelingof many physical systems. In most cases, it is very difficult to achieve analytic solutionsof these equations. Perturbation techniques are widely used in science and engineering tohandle nonlinear problems and do great contribution to help us understand many nonlinearphenomena. However, perturbation techniques are based on the existence of small/largeparameter. Therefore, these techniques are not valid for strongly nonlinear problems.

The homotopy perturbation method �HPM� is the new approach for finding theapproximate analytical solution of linear and nonlinear problems. The method was firstproposed byHe �12, 13� andwas successfully applied to solve nonlinear wave equation byHe�14–16�. The convergence of Homotopy perturbation series to the exact solution is consideredin �17�. Similarly, applying the variational iteration method, created by He �3, 18, 19�,consists in constructing the appropriate correction functional connected with the consideredequation. The correction functional contains a Lagrange multiplier, the determination ofwhich leads to a recurrence formula. Convergence of the VIM method is discussed by Tatariand Dehghan in �20�. Both of the methods examined have found application in determiningthe approximate solutions of different technical problems �21�. Adaptation of the VIMmethod for solving fractional heat-wave-like equation and fractional Zakharov-Kuznetsovequation were discussed by Yulita and colleagues in �22, 23�. Whereas, Chun �24� obtainedthe numerical solution of heat conduction problem by VIM. Recently, the application of theVIMmethod for solving kuramoto and Sivashinsky equations was presented by Porshokouhiand Ghanbari in �25�. For the application of HPM, this method used for solving fractionalvibration equation �26� and partial differential equations of fractional order in finite domains�27�.

In the present paper, VIM andHPMwill be applied for solving fractional Rosenau-Haynam equation which written as

Dαt u � uDxxx�u� uDx�u� 3Dx�u�Dxx�u�, t > 0, �1.1�

subject to the initial condition

u�x, 0� � −83c cos2

(x4

), �1.2�

where u � u�x, t�, α is a parameter describing the order of the fractional derivative �0 < α ≤1�, t is the time, and x is the spatial coordinate. Fractional RH equation when α � 1.0 hasappeared in the study of the formation of patterns in liquid drops �28�.

2. Basic Definitions

Fractional calculus unifies and generalizes the notions of integer-order differentiation and n-fold integration �4, 29�. We give some basic definitions and properties of fractional calculustheory which will be used in this paper:


Definition 2.1. A real function f�x�, x > 0 is said to be in the space Cμ, μ ∈ R if there existsa real number p�> μ�, such that f�x� � xpf1�x�, where f1�x� ∈ C�0,∞�, and it is said to be inthe space Cmμ if and only if f �m� ∈ Cm, m ∈ N.

The Riemann-Liouville fractional integral operator is defined as follows.

Definition 2.2. The Riemann-Liouville fractional integral operator of order α ≥ 0, of a functionf ∈ Cμ, μ ≥ −1, is defined as

Jαf�x� �1

Γ�α�

∫x0�x − t�α−1f�t�dt, α > 0, x > 0,

J0f�x� � f�x�

�2.1�

In this paper only real and positive values of α will be considered.Properties of the operator Jα can be found in �29� and we mention only the following:

for f ∈ Cμ, μ ≥ −1, α, β ≥ 0, and γ ≥ −1:

�1� JαJβf�x� � Jαβf�x�,

�2� JαJβf�x� � JβJαf�x�,

�3� Jαxγ � �Γ�γ 1�/Γ�α γ 1��xαγ .

The Reimann-Liouville derivative has certain disadvantages when trying to model real-world phenomena with FDEs. Therefore, we will introduce a modified fractional differentialoperator Dα∗ proposed by Caputo in his work on the theory of viscoelasticity �30�.

Definition 2.3. The fractional derivative of f�x� in Caputo sense is defined as

Dα∗f�x� � Jm−αDm∗ f�x� �

1Γ�m�

∫x0�x − s�m−α−1f �m��s�ds,

for m − 1 < α ≤ m,m ∈ N, x > 0, f ∈ Cm−1.�2.2�

In addition, we also need the following property.

Lemma 2.4. Ifm − 1 < α ≤ m, m ∈ N and f ∈ Cmμ , μ ≥ −1, then

Dα∗ Jαf�x� � f�x�,

JαDα∗f�x� � f�x� −m−1∑i�0

f �i��0�xi

i!, x > 0.

�2.3�

The Caputo differential derivative is considered here because the initial and boundary conditions canbe included in the formulation of the problems [4]. The fractional derivative is taken in the Caputosense as follows.


Definition 2.5. Form to be the smallest integer that exceeds α, the Caputo fractional derivativeoperator of order α > 0 is defined as

Dαt u�x, t� �

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1Γ�m − α�

∫ t0�t − s�m−α−1�∂mu�x,s�/∂sm�ds, for m − 1 < α ≤ m,

∂mu�x, t�∂tm

, for α � m ∈ N.�2.4�

For more information on the mathematical properties of fractional derivatives and integrals,one can consult �4, 29�.

3. Basic Idea of He’s Variational Iteration Method

To clarify the basic ideas of VIM, we consider the following differential equation:

Dα∗t�u� � f�u, ux, uxx� g�x, t�, m − 1 < α < m, �3.1�

where u � u�x, t�, Dα∗t � ∂α/∂tα is the Caputo fractional derivative of order α, m ∈ N, f is

a nonlinear function, and g is the source function. According to VIM, we can write down acorrection functional as follows:

un1�x, t� � un�x, t� ∫ t0λ�s�

[∂m

∂sm�un� − f�ũn, ũn,x, ũn,xx� − g�x, s�

]ds, �3.2�

where λ�s� is a general Lagrangian multiplier which can be optimally identified via thevariational theory �31�. ũn is considered as a restricted variation �32�, that is, δũn � 0 andthe subscript n indicates the nth approximation. We have

δun1 � δun δ∫ t0λ�s�

[∂m

∂sm�un� − f�ũn, ũn,x, ũn,xx� − g�x, s�

]ds, �3.3�

where ũn is considered as restricted variations, that is, δũn � 0. For m � 1, we have

δun1 � δun δ∫ t0λ�s�

[∂

∂sun

]ds,

δun1 �[1 − λ′�t�]δun δλ�t� ∂

∂tun

∫10δλ′′�s�unds.

�3.4�

Thus, we obtain the following stationary conditions:

δun�t� :[1 − λ′�t�]|s�t � 0,

δu′n�t� : λ�t�|s�t � 0,δun�s� : λ′′�t� � 0.

�3.5�


Solving this system of equations yields

λ�s� � −1. �3.6�

Furthermore, substituting �3.6� to �3.2�, the iteration formula of VIM can be written asfollows:

un1 � un −∫ t0

[∂m

∂sm�un� − f�un, un,x, un,xx� − g�x, s�

]ds. �3.7�

In this case, we begin with the initial approximation:

u0�x, t� � h�x�. �3.8�

The correction functional �3.8� will give several approximations, and therefore the exactsolution is obtained as

u�x, t� � limn→∞

un�x, t�. �3.9�

4. Basic Idea of Homotopy Perturbation Method

Consider the following nonlinear differential equation:

A�u� − f�r� � 0, �4.1�

with boundary conditions

B

(u,

∂u

∂n

)� 0, r ∈ Γ, �4.2�

where A is a general differential operator, B is a boundary operator, f�r� is a known analyticfunction, and Γ is the boundary of the domain Ω.

The operator A can, generally speaking, be divided into two parts, L and N, where Lis linear and N is nonlinear, therefore �4.1� can be written as

L�u� N�u� − f�r� � 0. �4.3�

By using homotopy technique, one can construct a homotopy v�r, p� : Ω × �0, 1� → whichsatisfies

H(v, p

)�(1 − p)�L�v� − L�u0�� p

[A�v� − f�r�] � 0, �4.4�

or

H(v, p

)� L�v� − L�u0� pL�u0� p

[N�v� − f�r�] � 0, �4.5�


where p ∈ �0, 1� is an embedding parameter, and u0 is the initial approximation of �4.1�whichsatisfies the boundary conditions. Clearly, we have

H�v, 0� � L�v� − L�u0� � 0, �4.6�

or

H�v, 1� � A�v� − f�r� � 0, �4.7�

the changing process of p from zero to unity is just that of v�r, p� changing from u0�r� tou�r�. This is called deformation, and also, L�v� − L�u0� and A�v� − f�r� are called homotopicin topology. If the embedding parameter p�0 ≤ p ≤ 1� is considered as a small parameter,applying the classical perturbation technique, we can assume that the solution of �4.3� and�4.4� can be given as a power series in p, that is,

v � v0 pv1 p2v2 · · · , �4.8�

and setting p � 1 results in the approximate solution of �4.1� as

u � limp→ 1

v � v0 v1 v2 · · · . �4.9�

5. Application of HPM and VIM Methods

We consider the application of VIM to fractional Rosenau-Hynam equations which isrewritten as follows:

Dαt u � uDxxx�u� uDx�u� 3Dx�u�Dxx�u�, �5.1�

where u � u�x, t� with the initial conditions of

u�x, 0� � −83c cos2

(x4

). �5.2�

5.1. VIM Implement for Fractional Rosenau-Hynam Equation

According to the formula �3.8�, the iteration formula for �4.9� is given by

un1 � un −∫ t0�Dαs �un� − unDxxx�un� − unDx�un� − 3Dx�un�Dxx�un��ds. �5.3�


Table

1:The

approx

imatesolution

offraction

alRosen

au-H

ynam

equa

tion

usingfifth

iterates

ofVIM

andfifth

term

sof

HPM

indifferen

tvalue

sof

α,tha

tis,

α�0.7,0.8,

and0.9an

dα�1.0whe

nc�0.5.

xt

VIM

HPM

VIM

HPM

VIM

HPM

VIM

HPM

Exa

ctα�0.7

α�0.8

α�0.9

α�1.0

π/4

0.2

−1.3027037

−1.3020263

−1.2995666

−1.2994063

−1.2968978

−1.2968865

−1.1139217

−1.2945674

−1.2945674

0.6

−1.3163161

−1.3160219

−1.3159022

−1.3158706

−1.3150924

−1.3150940

−1.3137953

−1.3137952

−1.3137952

1.0

−1.3194666

−1.3204884

−1.3228476

−1.3231121

−1.3251352

−1.3251648

−1.3265574

−1.3265572

−1.3265572

π/2

0.2

−1.1781765

−1.1775095

−1.1715349

−1.1713677

−1.1658775

−1.1658583

1.1610425

−1.1610425

−1.1610425

0.6

−1.2168471

−1.2161259

−1.2124978

−1.2123777

−1.2080227

−1.2079404

−1.2032236

−1.2032236

−1.2032236

1.0

−1.2404292

−1.2410937

−1.2415934

−1.2417625

−1.2413894

−1.2412100

−1.2400439

−1.2400437

−1.2400437

3π/4

0.2

−0.4677769

−0.9752216

−0.4579952

−0.9664930

−0.4496547

−1.1658655

−0.4426464

−0.2194117

−0.4426464

0.6

−0.5359648

−1.0325797

−0.5255989

−1.0258054

−0.5158600

−1.0185228

−0.5064510

−0.2710013

−0.5064510

1.0

−0.5899819

−1.0742478

−0.5848618

−1.0728597

−0.5788060

−1.0701462

−0.5718564

−0.3265443

−0.5718565

π0.2

−0.7263173

−0.7259589

−0.7160797

−0.7159725

−0.7073528

−0.7073345

−0.6999861

−0.6999861

−0.6999861

0.6

−0.7945239

−0.7933266

−0.7847888

−0.7845575

−0.7754812

−0.7753489

−0.7662921

−0.7662921

−0.7662921

1.0

−0.8456349

−0.8453513

−0.8421984

−0.8421178

−0.8374790

−0.8371566

−0.8316027

−0.8316026

−0.8316026


Table

2:The

approx

imatesolution

offraction

alRosen

au-H

ynam

usingfifthiterateof

VIM

andfifth

term

sof

HPM

indifferen

tvalue

sof

α,tha

tis,α�0.7,0.8,

and0.9an

dα�1.0whe

nc�1.0.

xt

VIM

HPM

VIM

HPM

VIM

HPM

VIM

HPM

Exa

ctα�0.7

α�0.8

α�0.9

α�1.0

π/4

0.2

−2.6345127

−2.6293263

−2.6251671

−2.6239643

−2.6172579

−2.6171710

−2.6099581

−2.6099581

−2.6099581

0.6

−2.6345745

−2.6342711

−2.6474590

−2.6477297

−2.6563851

−2.6450822

−2.6609433

−2.6609420

−2.6609420

1.0

−2.5722042

−2.5854045

−2.6080979

−2.6122085

−2.6365766

−2.6372049

−2.6590258

−2.6589984

−2.6589984

π/2

0.2

−2.4277172

−2.4226797

−2.4036916

−2.4024670

−2.3832715

−2.4054513

−2.3655561

−2.3655561

−2.3655561

0.6

−2.5350335

−2.5308256

−2.5303337

−2.5295597

−2.5232013

−2.5409052

−2.5126533

−2.5126523

−2.5126523

1.0

−2.5563768

−2.5641211

−2.5831444

−2.5854712

−2.6015950

−2.6030469

−2.6127547

−2.6127328

−2.6127328

3π/4

0.2

−0.5752642

−2.0501900

−0.5416828

−2.0182039

−0.5130518

−1.9893536

−0.4893583

−0.4893583

−1.9640076

0.6

−0.8341872

−2.5308256

−0.7891721

−2.2292752

−0.7486279

−2.1055837

−0.7112517

−0.7112526

−2.1848220

1.0

−1.0486958

−2.3554614

−1.0195238

−2.3681073

−0.9896491

−2.3733581

−0.9579284

−0.9579473

−2.3716903

π0.2

−1.5711438

−2.4226797

−1.5304096

−1.5296754

−1.4957251

−1.5301673

−1.4664446

−1.4664446

−1.4664446

0.6

−1.8301141

−2.5308256

−1.7949603

−1.7925916

−1.7612479

−1.7971174

−1.7273603

−1.7273603

−1.7273603

1.0

−1.9968865

−2.5641211

−1.9951864

−1.9932084

−1.9871083

−2.0042754

−1.9725695

−1.9725674

−1.9725674


Table 3: Errors of the approximate solution of fractional Rosenau-Hynam equation using using fifth termof HPM and fifth iterate of VIM when α � 1.0 and c � 0.5.

tx � π/2 x � π x � 3π/2

VIM HPM VIM HPM VIM HPM0.1 1.0000E − 11 1.0000E − 11 5.0000E − 10 1.0000E − 11 2.0000E − 10 1.0000E − 110.2 1.0000E − 11 1.2173E − 09 5.0000E − 10 1.7360E − 09 3.0000E − 10 1.2378E − 090.3 1.0000E − 11 9.2044E − 09 5.0000E − 10 1.3182E − 08 3.0000E − 10 9.4375E − 090.4 1.0000E − 10 3.8620E − 08 1.0000E − 10 5.5542E − 08 9.0000E − 10 3.9929E − 080.5 3.0000E − 10 1.1734E − 07 4.0000E − 10 1.6948E − 07 1.9000E − 09 1.2233E − 070.6 1.0000E − 09 2.9070E − 07 7.0000E − 10 4.2165E − 07 6.1000E − 09 3.0561E − 070.7 2.6000E − 09 6.2550E − 07 1.2000E − 09 9.1117E − 07 1.8300E − 08 6.6309E − 070.8 5.4000E − 09 1.2140E − 06 2.1000E − 09 1.7761E − 06 3.9300E − 08 1.2978E − 060.9 1.1110E − 07 2.1777E − 06 4.0000E − 09 3.1998E − 06 8.2200E − 08 2.3474E − 061.0 2.1110E − 07 3.6709E − 06 8.6000E − 09 5.4173E − 06 1.5230E − 07 3.9903E − 06

Table 4: Errors of the approximate solution of fractional Rosenau-Hynam equation using fifth term of HPMand fifth iterate of VIM when α � 1.0 and c � 0.5.

tx � π/2 x � π x � 3π/2

VIM HPM VIM HPM VIM ADM0.1 1.0000E − 11 2.4346E − 09 5.0000E − 10 3.4720E − 09 2.0000E − 10 2.4756E − 090.2 1.0000E − 11 7.7240E − 08 5.0000E − 10 1.1109E − 07 3.0000E − 10 7.9858E − 080.3 1.0000E − 11 5.8140E − 07 5.0000E − 10 8.4330E − 07 3.0000E − 10 6.1121E − 070.4 1.0000E − 10 2.4280E − 06 1.0000E − 10 3.5522E − 06 9.0000E − 10 2.5955E − 060.5 3.0000E − 10 7.3419E − 06 4.0000E − 10 1.0835E − 05 1.9000E − 09 7.9805E − 060.6 1.0000E − 09 1.8100E − 05 7.0000E − 10 2.6942E − 05 6.1000E − 09 2.0004E − 050.7 2.6000E − 09 3.8743E − 05 1.2000E − 09 5.8188E − 05 1.8300E − 08 4.3547E − 050.8 5.4000E − 09 7.4800E − 05 2.1000E − 09 1.1335E − 04 3.9300E − 08 8.5496E − 050.9 1.1110E − 07 1.3340E − 04 4.0000E − 09 2.0405E − 04 8.2200E − 08 1.5512E − 041.0 2.1110E − 07 2.2370E − 04 8.6000E − 09 3.4516E − 04 1.5230E − 07 2.6444E − 04

The iteration starts with an initial approximation which is initial condition in �5.2�.Furthermore, using the iteration formula in �5.3�, we can directly obtain other components as

u1�x, t� � − 23c2 sin

(x2

)t − 4

3c[1 cos

(x2

)],

u2�x, t� � − 43c2 sin

(x2

)[16c2

43c

](1 cos

(x2

))

− 16c3t2

23c2 sin

(x2

) t�2−α�Γ�3 − α� ,

u3�x, t� �76c3t3 cos

(x2

)− 9636

ccos2(x4

)− 4836

c2t sin(x2

)− 23cos

(x2

) t3−αΓ�4 − α�

�6 − 2α�c2 sin(x2

) t2−αΓ�4 − α� −

23c2 sin

(x2

) t3−2αΓ�4 − 2α� ,

�5.4�

and so on.


xt

0−0.4−0.8−1.2

0−0.4−0.8−1.2

654321010.80.60.40.2

0

u(x,t)

�a�

x

t

−0.6−0.8−1

−1.2−1.4

654321010.80.60.40.2

0

−0.6−0.8−1−1.2−1.4u

(x,t)

�b�

x

t

−0.6−0.8−1

−1.2−1.4

654321010.80.60.4

0.20

−0.6−0.8−1−1.2−1.4u(

x,t)

�c�

Figure 1: The approximate solution of fractional Rosenau-Hynam equation when α � 1.0 and c � 0.5 usingsuch method: �a� HPM, �b� VIM and �c� Exact.

The exact solution of this equation is given by �33�

u�x, t� � −83c cos2

(14�x − ct�

), |x − ct| ≤ 2π, �5.5�

where c is arbitrary constant �28�.

6. Analysis of the Homotopy Perturbation Method (HPM)

Now applying the classical perturbation technique for solving �5.1� with initial conditionin �5.2�. To solve �5.1� by the homotopy perturbation method, we construct the followinghomotopy:

(1 − p)Dαt u�x, t� p

[Dαt u�x, t� − �u�x, t�Dxxxu�x, t� u�x, t�Dxu�x, t�

3Dxu�x, t�Dxxu�x, t��]� 0,

�6.1�

or

Dαt u�x, t� � p�u�x, t�Dxxxu�x, t� u�x, t�Dxu�x, t� 3Dxu�x, t�Dxxu�x, t��, �6.2�

where p ∈ �0, 1� is an embedding parameter. If p � 0, then �6.2� becomes a linear equation,

Dαt u�x, t� � 0. �6.3�


x

t

0−0.5−1

−1.5−2

0−0.5−1−1.5−2

654321010.8

0.60.40.20

u(x,t)

�a�

x

−0.8−1.2−1.6−2−2.4−2.8

−0.8−1.2−1.6−2−2.4−2.8

65432101t 0.80.60.4

0.20

u(x,t)

�b�

x

65432101

−1.2−1.6−2−2.4−2.8

−1.2−1.6−2

−2.4−2.8

t0.80.6

0.40.20

u(x,t)

�c�

Figure 2: The approximate solution of fractional Rosenau-Hynam equation when α � 1.0 and c � 1.0 usingsuch method: �a� HPM, �b� VIM, and �c� Exact.

And when p � 1, then �6.2� turns out to be �5.1�. Assume the solution of �6.2� to be in theform

u�x, t� � v0 pv1 p2v2 p3v3 · · · . �6.4�

Substituting �6.4� into �6.2� and equating the terms with identical powers of p, we obtain thefollowing set of linear differential equations:

p0 : Dαt u0 � 0, �6.5�

p1 : Dαt u1 � v0∂v0∂x

3∂v0∂x

∂2v0∂x2

v0∂3v0∂x3

, �6.6�

p2 : Dαt u2 � v1∂v0∂x

v0∂v1∂x

3∂v1∂x

∂2v0∂x2

3∂v0∂x

∂2v1∂x2

v1∂3v0∂x3

v0∂3v1∂x3

, �6.7�

and so on. Equations �6.6� and �6.7� can be solved by applying the operator Jα, which isthe inverse of the operator Dα and then by simple computation, Thus, the solution reads asfollows

v1�x, t� � − 2c2tα sin�x/4�3Γ�1 α�

,

v2�x, t� � − c3

3πt2α cos

(x2

)Γ�−2α� sin 1πα,


v3�x, t� � − c4πt3α csc πα sin�x/2�6Γ�1 − α�Γ�α�Γ�1 3α� ,

v4�x, t� � − c5πt4α cos�x/2� csc απ

12Γ�1 − α�Γ�α�Γ�1 4α� ,

�6.8�

and so on.In this manner, the rest of the components of the homotopy perturbation series can be

obtained. Finally, we approximate the analytical solutions of u�x, t� by the truncated series

u�x, t� � limN→∞

u5�x, t�, �6.9�

where u5�x, t� �∑4

n�0 vk�x, t�.

7. Numerical Results and Discussion

Tables 1 and 2 show the approximate solutions for �5.1� obtained for different values of αusing the decomposition method and the variational iteration method in different values ofc, that is, c � 1 and c � 0.5, respectively. Tables 3 and 4 show the absolute error of �5.1�whenα � 1.0 in different value of c, that is, c � 1.0 and c � 0.5, respectively. Figures 1 and 2 showthe approximate solutions for �5.1� in different values of c using the fifth iterates of VIM, thefifth terms of HPM when α � 1, and exact solution, respectively. From Tables 3 and 4 showthat the approximate solution using the VIM is more accurate than the approximate solutionobtained using the HPM. It is to be noted that only the fifth iterates of the variational iterationsolution and only fifth terms of the homotopy perturbation series were used in evaluating.

8. Conclusions

The fundamental goal of this work has been to construct an approximate solution of nonlinearpartial differential equations of fractional order. For computations and plots, theMathematicaandMaple packages were used. The goal has been achieved by using the variational iterationmethod �VIM� and the homotopy perturbation method �HPM�. The methods were used in adirect way without using linearization or restrictive assumptions. There are four importantpoints that were gotten. First, the VIM and the HPM provide the solutions in terms ofconvergent series with easily computable components. Second, the approximate solution in�5.1� using the VIM converges faster than the approximate solution using the HPM. Third,the variational iteration method handles nonlinear equations without any need for the so-called He’s polynomials. Finally, the recent appearance of fractional differential equations asmodels in some fields of applied mathematics makes it necessary to investigate methods ofsolution for such equations �analytical and numerical� and we hope that this work is a stepin this direction.


Acknowledgment

The financial support received from UKM Grant ERGS/1/2011/STG/UKM/01/13 is grate-fully acknowledged.

References

�1� J. H. He, “Semi-inverse method of establishing generalized variational principles for fluid mechanicswith emphasis on turbomachinery aerodynamics,” International Journal of Turbo and Jet Engines, vol.14, no. 1, pp. 23–28, 1997.

�2� J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,”Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86–90, 1999.

�3� J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porousmedia,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.

�4� I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,Academic Press, San Diego, Calif, USA, 1999.

�5� K. Al-Khaled and S. Momani, “An approximate solution for a fractional diffusion-wave equationusing the decomposition method,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 473–483,2005.

�6� F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractionaldiffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.

�7� A. Hanyga, “Multidimensional solutions of time-fractional diffusion-wave equations,” Proceedings ofthe Royal Society of London, Series A, vol. 458, no. 2020, pp. 933–957, 2002.

�8� F. Huang and F. Liu, “The time fractional diffusion equation and the advection-dispersion equation,”The Australian & New Zealand Industrial and Applied Mathematics Journal, vol. 46, no. 3, pp. 317–330,2005.

�9� F. Huang and F. Liu, “The fundamental solution of the space-time fractional advection-dispersionequation,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 339–350, 2005.

�10� S. Momani, “An explicit and numerical solutions of the fractional KdV equation,” Mathematics andComputers in Simulation, vol. 70, no. 2, pp. 110–118, 2005.

�11� L. Debnath and D. D. Bhatta, “Solutions to few linear fractional inhomogeneous partial differentialequations in fluid mechanics,” Fractional Calculus & Applied Analysis, vol. 7, no. 1, pp. 21–36, 2004.

�12� J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering,vol. 178, no. 3-4, pp. 257–262, 1999.

�13� J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linearproblems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.

�14� J. H. He, “Periodic solutions and bifurcations of delay-differential equations,” Physics Letters A, vol.347, no. 4–6, pp. 228–230, 2005.

�15� J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos,Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005.

�16� J. H. He, “Limit cycle and bifurcation of nonlinear problems,” Chaos, Solitons and Fractals, vol. 26, no.3, pp. 827–833, 2005.

�17� J. H. He, Non-perturbative methods for strongly nonlinear problems [Dissertation], de-Verlag im InternetGmbH, Berlin, Germany, 2006.

�18� J. H. He, “A new approach to nonlinear partial differential equations,” Communications in NonlinearScience and Numerical Simulation, vol. 2, no. 4, pp. 230–235, 1997.

�19� J. H. He, “Variational iteration method for delay differential equations,” Communications in NonlinearScience and Numerical Simulation, vol. 2, no. 4, pp. 235–236, 1997.

�20� M. Tatari and M. Dehghan, “On the convergence of He’s variational iteration method,” Journal ofComputational and Applied Mathematics, vol. 207, no. 1, pp. 121–128, 2007.

�21� Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations offractional order,” Applied Mathematical Modelling, vol. 32, no. 1, pp. 28–39, 2008.

�22� R. Yulita Molliq, M. S. M. Noorani, and I. Hashim, “Variational iteration method for fractional heat-and wave-like equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1854–1869,2009.


�23� R. Yulita Molliq, M. S. M. Noorani, I. Hashim, and R. R. Ahmad, “Approximate solutions of fractionalZakharov-Kuznetsov equations by VIM,” Journal of Computational and Applied Mathematics, vol. 233,no. 2, pp. 103–108, 2009.

�24� C. Chun, “Variational iteration method for a reliable treatment of heat equations with ill-definedinitial data,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 435–440, 2008.

�25� M. G. Porshokouhi and B. Ghanbari, “Application of He’s variational iteration method for solution ofthe family of Kuramoto-Sivashinsky equations,” Journal of King Saud University - Science, vol. 23, no.4, pp. 407–411, 2011.

�26� S. T. Mohyud-Din and A. Yildirim, “An algorithm for solving the fractional vibration equation,”Computational Mathematics and Modeling, vol. 23, no. 2, pp. 228–23

Fractional Differential Equations 2012downloads.hindawi.com/journals/specialissues/714979.pdf ·...

Documents

Transcript of Fractional Differential Equations 2012downloads.hindawi.com/journals/specialissues/714979.pdf ·...