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Analysis of differential equations of fractional order
Applied Mathematical Modelling 36 (2012) 4356–4364
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Analysis of differential equations of fractional order K. Sayevand a,⇑, A. Golbabai b, Ahmet Yildirim c,d a
Department of Mathematics, Faculty of Science, Malayer University, Malayer, Iran School of Mathematics, Iran University of Science and Technology, Tehran, Iran Ege University, Science Faculty, Department of Mathematics, 35100 Bornova, Izmir, Turkey d University of South Florida, Department of Mathematics and Statistics, Tampa, FL 33620-5700, USA b c
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 8 November 2011 Accepted 15 November 2011 Available online 25 November 2011 Keywords: Caputo fractional derivative Existence and uniqueness of solution Homotopy perturbation technique Nonlinear fractional differential equations
a b s t r a c t This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate. 2011 Elsevier Inc. All rights reserved.
1. Introduction The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) was planted over 300 years ago. Since that time the fractional calculus has drawn the attention of many researchers. In recent years, fractional calculus has played a significant role in many areas of science and engineering. For an interesting history and more scientific applications of fractional calculus, see [1–5]. Finding approximate or exact solutions of fractional differential equations is an important task. Except in a limited number of these equations, we have difficulty finding their analytical solutions. Therefore, there have been attempts to develop the new methods for obtaining analytical solutions which reasonably approximate the exact solutions. Recently, several such techniques have drawn special attention, such as Hirtoa’s bilinear method, the Adomian’s decomposition method, the homogeneous balance method, inverse scatting method, the homotopy analysis method, the variational iteration method, parameter-expanding method, differential transform method and homotopy perturbation technique. The homotopy perturbation technique [6–8] is a promising analytic technique which has successfully been applied to solve linear, nonlinear, initial and boundary value problems of fractional order [9,10]. Considerable research work has recently been conducted in applying this method to Navier–Stokes equation, nonlinear Schrödinger equation, Volterra’s integro-differential equation, nonlinear oscillators, fractional KdV equations, quadratic Riccati differential equation of fractional order and many other problems. For more details about homotopy perturbation technique and its applications, the reader is advised to consult the results of the research works presented in [11–15]. In this paper, nonlinear fractional differential equations are investigated by means of the homotopy perturbation technique. Some theorems about uniqueness and existence of the solutions are presented. Moreover, we consider how the theoretical results may be applied in homotopy perturbation technique. ⇑ Corresponding author. E-mail address: [email protected] (K. Sayevand). 0307-904X/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.11.061
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2. Preliminaries and notations This section deals with some preliminaries and notations regarding fractional calculus. For more details see [1–5]. Definition 1. A real function n(t), t > 0 is said to be in the space C a ; a 2 R if there exists a real number p(>a), such that S (m) n(t) = tpn1(t), where n1(t) 2 C[0, 1), and it is said to be in the space C m a ; m 2 N f0g if and only if n (t) 2 Ca. Definition 2. The (left sided) Riemann–Liouville fractional integral of order a > 0 of a function n(t) 2 Ca, a P 1 is defined as
( a
1
CðaÞ
It nðtÞ ¼
Rt
nðsÞ 0 ðtsÞ1a
ds;
a > 0; t > 0;
ð1Þ
nðtÞ;
Iat nðx; tÞ ¼
1 CðaÞ
Z
t
nðx; sÞ ðt sÞ1a
0
ds;
a > 0; t > 0;
ð2Þ
where C() is the well-known Gamma function. Definition 3. The (left sided) Riemann–Liouville fractional derivative of nðtÞ; nðtÞ 2 C m 1 ; m 2 N
S
f0g, of order a is defined as
m
Dat nðtÞ ¼
d ma nðtÞ; mI dt t
m 1 < a 6 m;
m 2 N:
ð3Þ
Definition 4. The (left sided) Caputo fractional derivative of nðtÞ; nðtÞ 2 C m 1 ; m 2 N
( a
Dt nðtÞ ¼
a ðmÞ n ðtÞ m 1 < a < m; ½Im t dm dt m
S f0g, is defined as
m 2 N;
ð4Þ
a ¼ m;
nðtÞ
@ m nðx; tÞ ; m 1 < a < m; @t m a m aþm Dt Dt nðtÞ ¼ Dt nðtÞ; m ¼ 0; 1; . . . ; n 1 < a < n:
a Dat nðx; tÞ ¼ Im t
ð5Þ ð6Þ
Property. Similar to integer-order differentiation, fractional differentiation is a linear operation
Dat ðcnðtÞ þ gvðtÞÞ ¼ cDat nðtÞ þ gDat vðtÞ:
ð7Þ
Definition 5. A two-parameter Mittag–Leffler function is defined by the series expansion [1]
Ea;b ðzÞ ¼
1 X n¼0
zn ; Cðan þ bÞ
z 2 C:
ð8Þ
2.1. The relation between fractional derivative and fractional integral
Theorem 2.1.1. Assume that the continuous function n(t) has a fractional derivative of order a, then we have
8 ba > a < b; < It nðtÞ Dat Ibt nðtÞ ¼ nðtÞ a ¼ b; > : bþa Dt nðtÞ a > b; Iat Dat nðtÞ ¼ nðtÞ
m 1 X k¼0
Dat Iat nðtÞ ¼
nðkÞ ð0þ Þ
tk ; k!
ð9Þ
m 1 < a 6 m;
nðtÞ; m 1 < a 6 m; m 2 N ; Iat Dat nðtÞ þ nð0Þ; 0 < a < 1:
m 2 N;
ð10Þ
ð11Þ
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3. Existence and uniqueness of the solution This section, is devoted to proving the existence and uniqueness of solutions for fractional initial value problems under a global Lipschitz condition on a finite interval of the real axis in spaces of continuous functions. Theorem 3.1 (Generalized Banach fixed point theorem). Let U be a nonempty closed subset of a Banach space E, and let bn P 0 P for every nand such that 1 n¼0 bn converges. Moreover, let the mapping A : U ? U satisfy the inequality
kAn u An v k 6 bn ku v k
ð12Þ
for every n 2 N and for every u, v 2 U. Then, A has a uniquely defined fixed point u . Furthermore, for any u0 2 U, the sequence ⁄ ðAn u0 Þ1 n¼1 converges to this fixed point u . ⁄
Proof. See [16]. h Now, consider the following fractional initial value problems under a global Lipschitz condition
Dat nðtÞ ¼ f ðt; nðtÞÞ; k
n ð0Þ ¼
ðkÞ n0 ;
m 1 < a < m;
ð13Þ
k ¼ 0; 1; . . . ; m 1:
According to (4), Eq. (13) can be written in the following form
Dat ðn T m1 ðnÞÞðtÞ ¼ f ðt; nðtÞÞ;
ð14Þ
ðkÞ
nk ð0Þ ¼ n0 ; where Tm1(n) is the Taylor polynomial of order (m 1) for n, centered at 0 (see [16]). ðkÞ
ðkÞ
Theorem 3.2 (Existence). Assume that K ¼ ½0; v ½n0 b; n0 þ b with some v⁄, b > 0, and let the function f : K ! R be a1 continuous. Furthermore, define v ¼ minfv ; bCkfðakþ1Þ g. Then, there exists a function n : ½0; v ! R solving the initial value 1
problem (14). Proof. See [16]. h ðkÞ
ðkÞ
Theorem 3.3 (Uniqueness). Assume that K ¼ ½0; v ½n0 b; n0 þ b with some v⁄,b > 0. Furthermore, let the function f : K ! R be bounded on K and fulfil a Lipschitz condition with respect to the second variable, i.e.,
jf ðx; yÞ f ðx; zÞj 6 Ljy zj
ð15Þ
with some constant L > 0 independent of x, y and z. Then, denoting v as in Theorem 3.2, there exists at most one function n : ½0; v ! R solving the initial value problem (14). Proof. See [16]. h a
Theorem 3.4. In equation Dtj nðtÞ ¼ f ðt; nðtÞ; Dati nðtÞÞ, if f be a continuous function which satisfy in a uniform Lipschitz condition, then this equation, subject to the given initial conditions, has a unique continuous solution on interval [0, T]. Proof. See [17]. h
4. Multi-order fractional differential equations A multi-order fractional differential equations (M-OFDEs) can be presented in the following form [14,17,18]
Datn nðtÞ ¼ f ðt; nðtÞ; Dat1 nðtÞ; Dat2 nðtÞ; . . . ; Datn1 nðtÞÞ; ðkÞ
n ð0Þ ¼ ck ;
n 2 N;
ð16Þ
k ¼ 0; 1; 2; . . . ;
where ai 2 R, a1 < a2 < < an1 < an and Dati is used to represent the Caputo fractional derivative of order ai. 5. M-OFDEs as a system of FDEs In this section, we present a theorem to convert the initial value problem (16) into a system of fractional differential equations.
K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364
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Theorem 5.1. Eq. (16) is equivalent to the system of equations in the following form
Dati ni ðtÞ ¼ niþ1 ðtÞ;
i ¼ 1; 2; . . . ; n 1;
Datn ni ðtÞ ¼ f ðt; n1 ðtÞ; n2 ðtÞ; . . . ; nn ðtÞÞ; ðkÞ ni ð0Þ
¼
cik ;
k ¼ 0; 1; . . . ; ai 2 R;
ð17Þ i ¼ 1; 2; . . . ; n:
Proof. In Eq. (16), set n(t) = n1(t) and define Dat ni ðtÞ ¼ niþ1 ðtÞ; i ¼ 1; 2; . . . ; n 1. Hence, by substituting these relations into Eq. (16), we can convert the M-OFDEs into a system of FDEs. h
6. Performance evaluation of homotopy perturbation technique In Eq. (16), let us set
f ðt; n1 ðtÞ; . . . ; nn ðtÞÞ ¼ gðtÞ þ
n X
/j ðtÞnj ðtÞ þ Nðt; n1 ðtÞ; . . . ; nn ðtÞÞ;
ð18Þ
j¼1
where g(t) and /j(t) are arbitrary functions. Also, N is a nonlinear part of f. To solve Eq. (17), we can construct the following homotopy
Dati ni ðtÞ pðniþ1 ðtÞÞ ¼ 0;
p 2 ½0; 1
ð19Þ
and
Datn ni ðtÞ p gðtÞ þ
n X
! /j ðtÞnj ðtÞ þ Nðt; n1 ; . . . ; nn Þ
¼ 0;
p 2 ½0; 1:
ð20Þ
j¼1
Homotopy perturbation technique yields the solution ni(t) by the series
ni ðtÞ ¼
1 X
pm ni;m ðtÞ;
p 2 ½0; 1;
i ¼ 1; 2; . . . :
ð21Þ
m¼0
Assume now that
N t;
1 X
m
p n1;m ðtÞ;
m¼0
1 X
m
p n2;m ðtÞ; . . . ;
m¼0
1 X
! m
p nn;m ðtÞ
m¼0
¼
1 X
pk Nk ;
p 2 ½0; 1;
ð22Þ
k¼0
where
N0 ¼ N 0 ðt; n1;0 ðtÞ; . . . ; nn;0 ðtÞÞ; N1 ¼ N 1 ðt; n1;0 ðtÞ; . . . ; nn;0 ðtÞ; n1;1 ðtÞ; . . . ; nn;1 ðtÞÞ;
ð23Þ
N2 ¼ N 2 ðt; n1;0 ðtÞ; . . . ; nn;0 ðtÞ; n1;1 ðtÞ; . . . ; nn;1 ðtÞ; n1;2 ðtÞ; n2;2 ðtÞ; . . . ; nn;2 ðtÞÞ; .. .: In view of homotopy perturbation technique and in order to generate the necessary data, we obtain
8 ai Dt ni ;0 ðtÞ ¼ 0; > > > > > Dai ni ;1 ðtÞ niþ1;0 ðtÞ ¼ 0; > > < t Dati ni ;2 ðtÞ niþ1;1 ðtÞ ¼ 0; > >. > .. > > > > : ai Dt ni ;n ðtÞ niþ1;n1 ðtÞ ¼ 0
ð24Þ
and
8 an Dt nn ;0 ðtÞ ¼ 0; > > > > n > P > > Datn nn ;1 ðtÞ /j ðtÞnj ;0 ðtÞ N0 gðtÞ ¼ 0; > > > j¼1 > > > > n < an P Dt nn ;2 ðtÞ /j ðtÞnj ;1 ðtÞ N1 ¼ 0; j¼1 > > > > > . > > > .. > > > a n P > > n > : Dt nn ;m ðtÞ /j ðtÞnj ;m1 ðtÞ Nm1 ¼ 0: j¼1
ð25Þ
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Consequently, in view of Theorem 2.1.1 we have
ni;0 ðtÞ ¼
mi X
tk ; k!
cik
k¼0
ni;mþ1 ðtÞ ¼ Iat i ðniþ1 ;m ðtÞÞ; mn X tk cnk ; nn;0 ðtÞ ¼ k! k¼0 n X
nn;1 ðtÞ ¼ Iat n
m ¼ 0; 1; 2; . . . ;
ð26Þ !
/j ðtÞnj ;0 ðtÞ þ N0 þ gðtÞ ;
j¼1 n X
an
nn;mþ1 ðtÞ ¼ It
i ¼ 1; 2; . . . ; n;
! /j ðtÞnj ;m ðtÞ þ Nm ;
m ¼ 1; 2; . . . :
j¼1
Thus, we get the approximate solution of Eq. (16) in the following form
nðtÞ ¼
1 X 1 X
pm ni;m ðtÞjp¼1 :
ð27Þ
i¼0 m¼0
Theorem 6.1. Consider the following equation
Dat nðtÞ ¼ f ðt; nðtÞÞ; m 1 < a < m; m 2 N ;
ð28Þ
a
where Dt denotes, the fractional derivative in the Caputo sense, subject to the initial conditions ðkÞ
nðkÞ ð0Þ ¼ n0 ;
k ¼ 0; 1; . . . ; m 1:
ð29Þ
The homotopy perturbation technique, yields that the initial value problem (28) be equivalent to the nonlinear Volterra integral equation of the second kind
nðtÞ ¼
m1 X
ðkÞ
n0
k¼0
tk 1 þ k! CðaÞ
Z 0
t
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð30Þ
Proof. Suppose that
f ðt; nðtÞÞ ¼ gðtÞ LnðtÞ NnðtÞ;
ð31Þ
where L is a linear operator while N is a nonlinear operator and g is a known analytical function. In view of homotopy perturbation method, we construct the following homotopy
ð1 pÞDat nðtÞ þ pðDat nðtÞ þ LnðtÞ þ NnðtÞ gðtÞ ¼ 0;
p 2 ½0; 1:
ð32Þ
In order to generate the necessary data, we use the homotopy parameter p to expand the solution in the following form
nðtÞ ¼ n0 ðtÞ þ pn1 ðtÞ þ p2 n2 ðtÞ þ p3 n3 ðtÞ þ :
ð33Þ
For the nonlinear terms, let us set Nn(t) = S(t). Hence, we have
0
1
0
1
0
1
1 0 0 C B n ðtÞ C B n ðtÞ C B B S0 ðn0 ðtÞÞ C B C B gðtÞ C B 1 C B 0 C C C B C B B B C C B C B0 C B B S1 ðn0 ðtÞ; n1 ðtÞÞ C; Dat B n2 ðtÞ C ¼ LB n1 ðtÞ C B CþB C C B. B . C B . C B C .. C @ .. B . C B . C B A . . . A A @ A @ @ 0 nj ðtÞ nj1 ðtÞ Sj1 ðn0 ðtÞ; n1 ðtÞ; . . . ; nj1 ðtÞÞ n0 ðtÞ
0
0
ð34Þ
where j = 3, 4, 5, . . . and also, the functions S0, S1, S2, . . . are similar with (22). Consequently, we have
8 m1 P ðkÞ tk > > n0 k! ; > < n0 ðtÞ ¼ k¼0
> n1 ðtÞ ¼ Iat ðLn0 ðtÞÞ Iat S0 ðn0 ðtÞÞ þ Iat gðtÞ; > > : nj ðtÞ ¼ Iat ðLnj1 ðtÞÞ Iat Sj1 ðn0 ðtÞ; n1 ðtÞ; . . . ; nj1 ðtÞÞ;
ð35Þ j ¼ 2; 3; . . . :
Hence, we get an accurate approximation in the following form
K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364
nðtÞ ¼
m1 X
ðkÞ
n0
k¼0
! 1 1 X X tk Lnk ðtÞ þ Sk gðtÞ : Iat k! k¼0 k¼0
4361
ð36Þ
Finally, we have a nonlinear Volterra integral equation of the second kind in the following form
nðtÞ ¼
m 1 X
ðkÞ
n0
k¼0
tk 1 þ k! CðaÞ
Z
t
0
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð37Þ
Looking at the questions of existence and uniqueness of the homotopy perturbation technique solution, we present the following results. Lemma 6.1. Eq. (37) is weakly singular if 0 < a < 1 and regular for a P 1. Proof. See [19]. h Lemma 6.2. If 0 < a < 1, Eq. (37) can be written in the following form ð0Þ
nðtÞ ¼ n0 þ
1 CðaÞ
Z
t
0
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð38Þ
Proof. It is an immediate consequence of (37). Thus, we only discuss the case 0 < a < 1. Now, let us define U as a close subset of the Banach space of all continues function on [0, v], equipped with the Chebyshev norm in the following form
ð0Þ U ¼ fn 2 C½0; v : n n0 6 bg:
ð39Þ
1
U is a nonempty set. On U we define the operator A by ð0Þ
ðAnÞðtÞ ¼ n0 þ
1 CðaÞ
Z
t
0
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð40Þ
Lemma 6.3. Under the assumptions of the relations (38)–(40), we have
AnðtÞ ¼ nðtÞ:
ð41Þ
Lemma 6.4. In Eq. (40) for "t, t1, t2 2 [0, v] we have [16]
8 1 jðAnÞðt 1 Þ ðAnÞðt2 Þj 6 Ckfðakþ1Þ ð2ðt 2 t 1 Þa þ ðt 1 Þa ðt2 Þa Þ; > > < ð0Þ jðAnÞðtÞ n0 j 6 b; > > a n : n kA nðtÞ An ~nðtÞk 6 ðLv Þ knðtÞ ~nðtÞk : L1 ½0;t
ð42Þ
1
Cðanþ1Þ
Thus, A has a unique fixed point. Consequently, the proof of the existence and uniqueness of homotopy perturbation technique solution for Eq. (28) is thoroughly investigated. 6.1. Structural stability Let n(t) and v(t) be the solutions, respectively, of the initial value problems
(
(
Dat nðtÞ ¼ f ðt; nðtÞÞ; ðkÞ
nðkÞ ð0Þ ¼ n0 ;
k ¼ 0; 1; . . . ; m 1;
Dat vðtÞ ¼ f ðt; vðtÞÞ; ~
vðkÞ ð0Þ ¼ nðkÞ k ¼ 0; 1; . . . ; m 1; 0 ;
ð43Þ
~ j < . Thus, under natural Lipschitz conditions imposed on f we have [16] where ja a
knðtÞ vðtÞkL1 ½0;T ¼ OðÞ;
! 0;
T < 1:
ð44Þ
7. Applications In this section to demonstrate the effectiveness of our approach, some examples are presented. It is to be noted that in all examples n 2 Ck[0, T < 1], k = 1, 2, . . . , where
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K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364
a ðmÞ Dat nðtÞ ¼ Im n ðtÞ; t
m 1 < a < m;
m 2 N:
Example 1. Consider the following initial value problem [20]
MD2t nðtÞ þ 2S
pffiffiffiffiffiffiffi 3=2 lqDt nðtÞ þ KnðtÞ ¼ #ðtÞ;
0 6 t 6 T;
ð45Þ
n0 ð0Þ ¼ 1:
nð0Þ ¼ 1;
This problem described the motion of a large plate of the surface Sand mass Min a Newtonian fluid viscosity l and density q. The plate is hanging on a massless spring of stiffness K. The function #(t) represents the loading force. The Eq. (45) is so called inhomogeneous Bagley–Torvik equation [20]. The Bagley–Torvik equation is known as a prototype of fractional differential equation with pffiffiffiffiffiffiffi two derivatives. In order to make comparison with the numerical solution of [20] we choose M ¼ 2S lq ¼ K ¼ 1; T ¼ 5 and #(t) = K(t + 1). By the same manipulation as Section 6 we set 3=2 Dt n1 ðtÞ ¼ n2 ðtÞ; 1=2 Dt n2 ðtÞ ¼ n2 ðtÞ n1 ðtÞ þ 1 þ t;
ð46Þ
n1 ð0Þ ¼ n01 ð0Þ ¼ 1; n2 ð0Þ ¼ 0: It is to be noted that since ni is continuous on the compact set [0, T], it is uniformly continuous there. Whence absolute maximum theorem implies that
jni ðtÞj 6 di ;
t 2 ½0; T;
di 2 Rþ :
ð47Þ
Hence
jni ðtÞ ni ð~tÞj 6 jni ðtÞj þ jni ð~tÞj 6 di þ ~di ;
t; ~t 2 ½0; T:
ð48Þ
Thus, by the Archimedean property of real number we will have
9L > 0;
s:t: d 6 Ljt ~tj;
ð49Þ
where
d ¼ di þ ~di :
ð50Þ
Consequently, in view of (4) we have
n1;0 ðtÞ ¼ 1 þ t; n1;mþ1 ðtÞ ¼ I1:5 n2;m ;
m ¼ 0; 1; . . . ;
ð51Þ
n2;0 ðtÞ ¼ 0; n2;1 ðtÞ ¼ 0; n2;mþ1 ðtÞ ¼ I0:5 ðn1;m ðtÞ þ n2;m ðtÞÞ;
m ¼ 1; 2; . . . :
Therefore we will obtain
n1;mþ1 ðtÞ ¼ 0; n2;mþ1 ðtÞ ¼ 0;
m ¼ 0; 1; . . . :
ð52Þ
Hence, n1(t) = 1 + t and n2(t) = 0. So, n(t) = 1 + t is the solution of Eq. (45). It is easily verified that t + 1 is the exact solution of Eq. (45). Table 1 shows the resulting error at t = 5 obtained by numerical method in [17] and compared with the solution obtained by the proposed scheme. Example 2. Consider the following initial value problem [21]
Dat nðtÞ ¼
3 3 40320 8a C 5 þ a2 4a 9 3 a 4 2 þ 2 t t t C ð a þ 1Þ þ n2 ðtÞ; t 4 2 Cð9 aÞ C 5 a2
nð0Þ ¼ 0;
0 < a 6 2;
n0 ð0Þ ¼ 0ða > 1Þ:
ð53Þ
Assume now that
3 3 40320 8a C 5 þ a2 4a 9 3 a t 2 t 4 n2 ðtÞ; f ðtÞ ¼ t t 2 þ Cða þ 1Þ þ a 4 2 Cð9 aÞ Cð5 2Þ Consequently, we will have
ð54Þ
4363
K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364 Table 1 Error at t = 5 for Eq. (45).
jf ðtÞ f ð~tÞj 6 lkgk1 jt ~tj;
Error at t = 5 by proposed approach
Error at t = 5 by [17]
(step size)
0 0 0 0
0.15131473519232 0.04684102179946 0.01602947553912 0.00562770408881
(0.5000) (0.2500) (0.1250) (0.0625)
t; ~t 2 ½0; T;
ð55Þ
where
l ¼ sup I2t n00 ðtÞ ; g ¼ gða; t; ~tÞ; s:t:kgk1 < 1: 1
ð56Þ
t2½0;T
The exact solution of Eq. (53) by homotopy perturbation technique was derived in [21] and is
nðtÞ ¼
2 3 a t2 t4 : 2
ð57Þ
~ þ . In view of (44) we have Now suppose that a < a
knðtÞ vðtÞkL1 ½0;T 6
a~ þ8 9 a~ T jT 1j þ 3T 2 T 2 1 : 4
ð58Þ
Hence by our assumption on a, we find that
sup jnðtÞ vðtÞj ! 0;
as
! 0;
t 2 ½0; T:
ð59Þ
Example 3. Consider the following fractional nonlinear Schrödinger equation
1 @ 2 nðx; tÞ þ jnðx; tÞj2 nðx; tÞ ¼ 0; 2 @x2 jnðx; tÞj2 ¼ nðx; tÞnðx; tÞ; a
iDt nðx; tÞ þ
nðx; 0Þ ¼ eix ;
ð60Þ
ðx; tÞ 2 R ½0; 2;
where nðx; tÞ is the conjugate of n(x, t) and i2 = 1. The Schrödinger equation is the fundamental equation of physics for describing non-relativistic quantum mechanical behavior. The nonlinear Schrödinger equation was studied by many researchers [11,22]. By the same manipulation as Section 6, the solution n(x, t) is given as
nðx; tÞ ¼
1 X
a
ðit Þn eix : 2 Cðna þ 1Þ
ð61Þ
n
n¼0
As mentioned in previous examples (Eqs. (48) and (55)) one will set
a a X 1 1 X ðit Þn ðit Þn ix iz jnðt; xÞ nðt; zÞj ¼ e e n n¼0 2n Cðna þ 1Þ 2 C ðn a þ 1Þ n¼0 a a X 1 ðit Þn Ea;1 it jx zj 6 #jx zj; jðcos x cos zÞ þ iðsin x sin zÞj 6 2 ¼ n n¼0 2 Cðna þ 1Þ 2
# 2 Rþ :
ð62Þ
Consequently, in view of homotopy perturbation technique we have [22]
! j jk 1 @ 2 nj ðx; tÞ X X t þ n ðx; tÞn ðx; tÞ n ðx; tÞ dt nðx; tÞÞ 6 kn n0 ðx; tÞ eiðxþ2Þ ; jli i k 2 2 @x k¼0 l¼0
X n Z t i j¼0 0
ð63Þ
where n0(x, t) = n(x, 0) and k 2 (0, 1). Thus, we have
nn ðx; tÞ n¼0
1 X
t
! eiðxþ2Þ :
ð64Þ
a¼1 t
Therefore we deduce that for a ¼ 1; eiðxþ2Þ is the exact solution of Eq. (60).
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8. Discussion and conclusion In this study, we developed a homotopy perturbation scheme to solve the nonlinear fractional differential equations based on the Caputo definition. The uniqueness and existence of the homotopy perturbation technique solution was examined. The test we performed to evaluate the performances of the proposed analytical approach is encouraging. Moreover, the results show that the solutions of proposed scheme are stable. Finally, we point out that this new strategy has its own limitations and should be generalized and verified for more complicated linear and nonlinear problems. In other words, the present paper is only an introduction to the topic, and there remains a lot of work to do. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus Appl. Anal. 5 (2002) 367–386. J. Tenreiro Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1140–1153. K.B. Oldham, J. Spainer, The fractional calculus, Academic Press, New York, 1974. G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, Yverdon, 1993. J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (3/4) (1999) 257–262. M.A. Noor, S.T. Mohyud-Din, Homotopy perturbation method for solving Thomas–Fermi equation using Pade approximants, Int. J. Nonlinear Sci. 8 (2009) 27–31. Y. Khan, Q. Wu, Homotopy perturbation transform method for nonlinear equations using Hes polynomials, Comput. Math. Appl. 61 (2011) 1963–1967. S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A 365 (2007) 345–350. O. Abdulaziz, I. Hashim, S. Momani, Solving systems of fractional differential equations by homotopy-perturbation method, Phys. Lett. A 372 (4) (2008) 451–459. A. Yildirim, An algorithm for solving the fractional nonlinear Schördinger equation by means of the homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009) 445–450. D.D. Ganji, S.S. Ganji, S. Karimpur, Z.Z. Ganji, Numerical study of homotopy-perturbation method to burgers equation in fluid, Numer. Methods Partial Differ. Eqn. 26 (4) (2010) 114–124. M. Dehghan, F. Shakeri, The numerical solution of the second Painlevé equation, Numer. Methods Partial Differ. Eqn. 25 (5) (2009) 1238–1259. A. Golbabai, K. Sayevand, The homotopy perturbation method for multi-order time fractional differential equations, Nonlinear Sci. Lett. A 1 (2) (2010) 147–154. A. Golbabai, K. Sayevand, Analytical treatment of differential equations with fractional coordinate derivatives, Comput. Math. Appl. 62 (2011) 1003– 1012. K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. K. Diethelm, N.J. Ford, Numerical solution of the Bagley–Torvik equation, BIT 42 (2002) 490–507. V. Daftardar, H. Jafari, Solving a multi-order fractional differential equation using Adomian decomposition, Appl. Math. Comput. 189 (2007) 541–548. P. Linz, Analytical and numerical methods for Volterra equations, SIAM, Philadelphia, 1985. R.L. Bagly, P.J. Torvik, On the appearance of the fractional derivative in the behaviour of real materials, J. Appl. Mech. 51 (1984) 294–298. O. Abdulaziz, I. Hashim, S. Momani, Application of homotopy perturbation method to fractional IVPs, J. Comput. Appl. Math. 216 (2008) 574–584. J. Biazar, H. Ghazvini, Convergence of the homotopy perturbation method for partial differential equation, Nonlinear Anal. RWA 10 (2009) 2633–2640.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Analysis of differential equations of fractional order K. Sayevand a,⇑, A. Golbabai b, Ahmet Yildirim c,d a
Department of Mathematics, Faculty of Science, Malayer University, Malayer, Iran School of Mathematics, Iran University of Science and Technology, Tehran, Iran Ege University, Science Faculty, Department of Mathematics, 35100 Bornova, Izmir, Turkey d University of South Florida, Department of Mathematics and Statistics, Tampa, FL 33620-5700, USA b c
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 8 November 2011 Accepted 15 November 2011 Available online 25 November 2011 Keywords: Caputo fractional derivative Existence and uniqueness of solution Homotopy perturbation technique Nonlinear fractional differential equations
a b s t r a c t This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate. 2011 Elsevier Inc. All rights reserved.
1. Introduction The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) was planted over 300 years ago. Since that time the fractional calculus has drawn the attention of many researchers. In recent years, fractional calculus has played a significant role in many areas of science and engineering. For an interesting history and more scientific applications of fractional calculus, see [1–5]. Finding approximate or exact solutions of fractional differential equations is an important task. Except in a limited number of these equations, we have difficulty finding their analytical solutions. Therefore, there have been attempts to develop the new methods for obtaining analytical solutions which reasonably approximate the exact solutions. Recently, several such techniques have drawn special attention, such as Hirtoa’s bilinear method, the Adomian’s decomposition method, the homogeneous balance method, inverse scatting method, the homotopy analysis method, the variational iteration method, parameter-expanding method, differential transform method and homotopy perturbation technique. The homotopy perturbation technique [6–8] is a promising analytic technique which has successfully been applied to solve linear, nonlinear, initial and boundary value problems of fractional order [9,10]. Considerable research work has recently been conducted in applying this method to Navier–Stokes equation, nonlinear Schrödinger equation, Volterra’s integro-differential equation, nonlinear oscillators, fractional KdV equations, quadratic Riccati differential equation of fractional order and many other problems. For more details about homotopy perturbation technique and its applications, the reader is advised to consult the results of the research works presented in [11–15]. In this paper, nonlinear fractional differential equations are investigated by means of the homotopy perturbation technique. Some theorems about uniqueness and existence of the solutions are presented. Moreover, we consider how the theoretical results may be applied in homotopy perturbation technique. ⇑ Corresponding author. E-mail address: [email protected] (K. Sayevand). 0307-904X/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.11.061
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2. Preliminaries and notations This section deals with some preliminaries and notations regarding fractional calculus. For more details see [1–5]. Definition 1. A real function n(t), t > 0 is said to be in the space C a ; a 2 R if there exists a real number p(>a), such that S (m) n(t) = tpn1(t), where n1(t) 2 C[0, 1), and it is said to be in the space C m a ; m 2 N f0g if and only if n (t) 2 Ca. Definition 2. The (left sided) Riemann–Liouville fractional integral of order a > 0 of a function n(t) 2 Ca, a P 1 is defined as
( a
1
CðaÞ
It nðtÞ ¼
Rt
nðsÞ 0 ðtsÞ1a
ds;
a > 0; t > 0;
ð1Þ
nðtÞ;
Iat nðx; tÞ ¼
1 CðaÞ
Z
t
nðx; sÞ ðt sÞ1a
0
ds;
a > 0; t > 0;
ð2Þ
where C() is the well-known Gamma function. Definition 3. The (left sided) Riemann–Liouville fractional derivative of nðtÞ; nðtÞ 2 C m 1 ; m 2 N
S
f0g, of order a is defined as
m
Dat nðtÞ ¼
d ma nðtÞ; mI dt t
m 1 < a 6 m;
m 2 N:
ð3Þ
Definition 4. The (left sided) Caputo fractional derivative of nðtÞ; nðtÞ 2 C m 1 ; m 2 N
( a
Dt nðtÞ ¼
a ðmÞ n ðtÞ m 1 < a < m; ½Im t dm dt m
S f0g, is defined as
m 2 N;
ð4Þ
a ¼ m;
nðtÞ
@ m nðx; tÞ ; m 1 < a < m; @t m a m aþm Dt Dt nðtÞ ¼ Dt nðtÞ; m ¼ 0; 1; . . . ; n 1 < a < n:
a Dat nðx; tÞ ¼ Im t
ð5Þ ð6Þ
Property. Similar to integer-order differentiation, fractional differentiation is a linear operation
Dat ðcnðtÞ þ gvðtÞÞ ¼ cDat nðtÞ þ gDat vðtÞ:
ð7Þ
Definition 5. A two-parameter Mittag–Leffler function is defined by the series expansion [1]
Ea;b ðzÞ ¼
1 X n¼0
zn ; Cðan þ bÞ
z 2 C:
ð8Þ
2.1. The relation between fractional derivative and fractional integral
Theorem 2.1.1. Assume that the continuous function n(t) has a fractional derivative of order a, then we have
8 ba > a < b; < It nðtÞ Dat Ibt nðtÞ ¼ nðtÞ a ¼ b; > : bþa Dt nðtÞ a > b; Iat Dat nðtÞ ¼ nðtÞ
m 1 X k¼0
Dat Iat nðtÞ ¼
nðkÞ ð0þ Þ
tk ; k!
ð9Þ
m 1 < a 6 m;
nðtÞ; m 1 < a 6 m; m 2 N ; Iat Dat nðtÞ þ nð0Þ; 0 < a < 1:
m 2 N;
ð10Þ
ð11Þ
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3. Existence and uniqueness of the solution This section, is devoted to proving the existence and uniqueness of solutions for fractional initial value problems under a global Lipschitz condition on a finite interval of the real axis in spaces of continuous functions. Theorem 3.1 (Generalized Banach fixed point theorem). Let U be a nonempty closed subset of a Banach space E, and let bn P 0 P for every nand such that 1 n¼0 bn converges. Moreover, let the mapping A : U ? U satisfy the inequality
kAn u An v k 6 bn ku v k
ð12Þ
for every n 2 N and for every u, v 2 U. Then, A has a uniquely defined fixed point u . Furthermore, for any u0 2 U, the sequence ⁄ ðAn u0 Þ1 n¼1 converges to this fixed point u . ⁄
Proof. See [16]. h Now, consider the following fractional initial value problems under a global Lipschitz condition
Dat nðtÞ ¼ f ðt; nðtÞÞ; k
n ð0Þ ¼
ðkÞ n0 ;
m 1 < a < m;
ð13Þ
k ¼ 0; 1; . . . ; m 1:
According to (4), Eq. (13) can be written in the following form
Dat ðn T m1 ðnÞÞðtÞ ¼ f ðt; nðtÞÞ;
ð14Þ
ðkÞ
nk ð0Þ ¼ n0 ; where Tm1(n) is the Taylor polynomial of order (m 1) for n, centered at 0 (see [16]). ðkÞ
ðkÞ
Theorem 3.2 (Existence). Assume that K ¼ ½0; v ½n0 b; n0 þ b with some v⁄, b > 0, and let the function f : K ! R be a1 continuous. Furthermore, define v ¼ minfv ; bCkfðakþ1Þ g. Then, there exists a function n : ½0; v ! R solving the initial value 1
problem (14). Proof. See [16]. h ðkÞ
ðkÞ
Theorem 3.3 (Uniqueness). Assume that K ¼ ½0; v ½n0 b; n0 þ b with some v⁄,b > 0. Furthermore, let the function f : K ! R be bounded on K and fulfil a Lipschitz condition with respect to the second variable, i.e.,
jf ðx; yÞ f ðx; zÞj 6 Ljy zj
ð15Þ
with some constant L > 0 independent of x, y and z. Then, denoting v as in Theorem 3.2, there exists at most one function n : ½0; v ! R solving the initial value problem (14). Proof. See [16]. h a
Theorem 3.4. In equation Dtj nðtÞ ¼ f ðt; nðtÞ; Dati nðtÞÞ, if f be a continuous function which satisfy in a uniform Lipschitz condition, then this equation, subject to the given initial conditions, has a unique continuous solution on interval [0, T]. Proof. See [17]. h
4. Multi-order fractional differential equations A multi-order fractional differential equations (M-OFDEs) can be presented in the following form [14,17,18]
Datn nðtÞ ¼ f ðt; nðtÞ; Dat1 nðtÞ; Dat2 nðtÞ; . . . ; Datn1 nðtÞÞ; ðkÞ
n ð0Þ ¼ ck ;
n 2 N;
ð16Þ
k ¼ 0; 1; 2; . . . ;
where ai 2 R, a1 < a2 < < an1 < an and Dati is used to represent the Caputo fractional derivative of order ai. 5. M-OFDEs as a system of FDEs In this section, we present a theorem to convert the initial value problem (16) into a system of fractional differential equations.
K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364
4359
Theorem 5.1. Eq. (16) is equivalent to the system of equations in the following form
Dati ni ðtÞ ¼ niþ1 ðtÞ;
i ¼ 1; 2; . . . ; n 1;
Datn ni ðtÞ ¼ f ðt; n1 ðtÞ; n2 ðtÞ; . . . ; nn ðtÞÞ; ðkÞ ni ð0Þ
¼
cik ;
k ¼ 0; 1; . . . ; ai 2 R;
ð17Þ i ¼ 1; 2; . . . ; n:
Proof. In Eq. (16), set n(t) = n1(t) and define Dat ni ðtÞ ¼ niþ1 ðtÞ; i ¼ 1; 2; . . . ; n 1. Hence, by substituting these relations into Eq. (16), we can convert the M-OFDEs into a system of FDEs. h
6. Performance evaluation of homotopy perturbation technique In Eq. (16), let us set
f ðt; n1 ðtÞ; . . . ; nn ðtÞÞ ¼ gðtÞ þ
n X
/j ðtÞnj ðtÞ þ Nðt; n1 ðtÞ; . . . ; nn ðtÞÞ;
ð18Þ
j¼1
where g(t) and /j(t) are arbitrary functions. Also, N is a nonlinear part of f. To solve Eq. (17), we can construct the following homotopy
Dati ni ðtÞ pðniþ1 ðtÞÞ ¼ 0;
p 2 ½0; 1
ð19Þ
and
Datn ni ðtÞ p gðtÞ þ
n X
! /j ðtÞnj ðtÞ þ Nðt; n1 ; . . . ; nn Þ
¼ 0;
p 2 ½0; 1:
ð20Þ
j¼1
Homotopy perturbation technique yields the solution ni(t) by the series
ni ðtÞ ¼
1 X
pm ni;m ðtÞ;
p 2 ½0; 1;
i ¼ 1; 2; . . . :
ð21Þ
m¼0
Assume now that
N t;
1 X
m
p n1;m ðtÞ;
m¼0
1 X
m
p n2;m ðtÞ; . . . ;
m¼0
1 X
! m
p nn;m ðtÞ
m¼0
¼
1 X
pk Nk ;
p 2 ½0; 1;
ð22Þ
k¼0
where
N0 ¼ N 0 ðt; n1;0 ðtÞ; . . . ; nn;0 ðtÞÞ; N1 ¼ N 1 ðt; n1;0 ðtÞ; . . . ; nn;0 ðtÞ; n1;1 ðtÞ; . . . ; nn;1 ðtÞÞ;
ð23Þ
N2 ¼ N 2 ðt; n1;0 ðtÞ; . . . ; nn;0 ðtÞ; n1;1 ðtÞ; . . . ; nn;1 ðtÞ; n1;2 ðtÞ; n2;2 ðtÞ; . . . ; nn;2 ðtÞÞ; .. .: In view of homotopy perturbation technique and in order to generate the necessary data, we obtain
8 ai Dt ni ;0 ðtÞ ¼ 0; > > > > > Dai ni ;1 ðtÞ niþ1;0 ðtÞ ¼ 0; > > < t Dati ni ;2 ðtÞ niþ1;1 ðtÞ ¼ 0; > >. > .. > > > > : ai Dt ni ;n ðtÞ niþ1;n1 ðtÞ ¼ 0
ð24Þ
and
8 an Dt nn ;0 ðtÞ ¼ 0; > > > > n > P > > Datn nn ;1 ðtÞ /j ðtÞnj ;0 ðtÞ N0 gðtÞ ¼ 0; > > > j¼1 > > > > n < an P Dt nn ;2 ðtÞ /j ðtÞnj ;1 ðtÞ N1 ¼ 0; j¼1 > > > > > . > > > .. > > > a n P > > n > : Dt nn ;m ðtÞ /j ðtÞnj ;m1 ðtÞ Nm1 ¼ 0: j¼1
ð25Þ
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K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364
Consequently, in view of Theorem 2.1.1 we have
ni;0 ðtÞ ¼
mi X
tk ; k!
cik
k¼0
ni;mþ1 ðtÞ ¼ Iat i ðniþ1 ;m ðtÞÞ; mn X tk cnk ; nn;0 ðtÞ ¼ k! k¼0 n X
nn;1 ðtÞ ¼ Iat n
m ¼ 0; 1; 2; . . . ;
ð26Þ !
/j ðtÞnj ;0 ðtÞ þ N0 þ gðtÞ ;
j¼1 n X
an
nn;mþ1 ðtÞ ¼ It
i ¼ 1; 2; . . . ; n;
! /j ðtÞnj ;m ðtÞ þ Nm ;
m ¼ 1; 2; . . . :
j¼1
Thus, we get the approximate solution of Eq. (16) in the following form
nðtÞ ¼
1 X 1 X
pm ni;m ðtÞjp¼1 :
ð27Þ
i¼0 m¼0
Theorem 6.1. Consider the following equation
Dat nðtÞ ¼ f ðt; nðtÞÞ; m 1 < a < m; m 2 N ;
ð28Þ
a
where Dt denotes, the fractional derivative in the Caputo sense, subject to the initial conditions ðkÞ
nðkÞ ð0Þ ¼ n0 ;
k ¼ 0; 1; . . . ; m 1:
ð29Þ
The homotopy perturbation technique, yields that the initial value problem (28) be equivalent to the nonlinear Volterra integral equation of the second kind
nðtÞ ¼
m1 X
ðkÞ
n0
k¼0
tk 1 þ k! CðaÞ
Z 0
t
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð30Þ
Proof. Suppose that
f ðt; nðtÞÞ ¼ gðtÞ LnðtÞ NnðtÞ;
ð31Þ
where L is a linear operator while N is a nonlinear operator and g is a known analytical function. In view of homotopy perturbation method, we construct the following homotopy
ð1 pÞDat nðtÞ þ pðDat nðtÞ þ LnðtÞ þ NnðtÞ gðtÞ ¼ 0;
p 2 ½0; 1:
ð32Þ
In order to generate the necessary data, we use the homotopy parameter p to expand the solution in the following form
nðtÞ ¼ n0 ðtÞ þ pn1 ðtÞ þ p2 n2 ðtÞ þ p3 n3 ðtÞ þ :
ð33Þ
For the nonlinear terms, let us set Nn(t) = S(t). Hence, we have
0
1
0
1
0
1
1 0 0 C B n ðtÞ C B n ðtÞ C B B S0 ðn0 ðtÞÞ C B C B gðtÞ C B 1 C B 0 C C C B C B B B C C B C B0 C B B S1 ðn0 ðtÞ; n1 ðtÞÞ C; Dat B n2 ðtÞ C ¼ LB n1 ðtÞ C B CþB C C B. B . C B . C B C .. C @ .. B . C B . C B A . . . A A @ A @ @ 0 nj ðtÞ nj1 ðtÞ Sj1 ðn0 ðtÞ; n1 ðtÞ; . . . ; nj1 ðtÞÞ n0 ðtÞ
0
0
ð34Þ
where j = 3, 4, 5, . . . and also, the functions S0, S1, S2, . . . are similar with (22). Consequently, we have
8 m1 P ðkÞ tk > > n0 k! ; > < n0 ðtÞ ¼ k¼0
> n1 ðtÞ ¼ Iat ðLn0 ðtÞÞ Iat S0 ðn0 ðtÞÞ þ Iat gðtÞ; > > : nj ðtÞ ¼ Iat ðLnj1 ðtÞÞ Iat Sj1 ðn0 ðtÞ; n1 ðtÞ; . . . ; nj1 ðtÞÞ;
ð35Þ j ¼ 2; 3; . . . :
Hence, we get an accurate approximation in the following form
K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364
nðtÞ ¼
m1 X
ðkÞ
n0
k¼0
! 1 1 X X tk Lnk ðtÞ þ Sk gðtÞ : Iat k! k¼0 k¼0
4361
ð36Þ
Finally, we have a nonlinear Volterra integral equation of the second kind in the following form
nðtÞ ¼
m 1 X
ðkÞ
n0
k¼0
tk 1 þ k! CðaÞ
Z
t
0
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð37Þ
Looking at the questions of existence and uniqueness of the homotopy perturbation technique solution, we present the following results. Lemma 6.1. Eq. (37) is weakly singular if 0 < a < 1 and regular for a P 1. Proof. See [19]. h Lemma 6.2. If 0 < a < 1, Eq. (37) can be written in the following form ð0Þ
nðtÞ ¼ n0 þ
1 CðaÞ
Z
t
0
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð38Þ
Proof. It is an immediate consequence of (37). Thus, we only discuss the case 0 < a < 1. Now, let us define U as a close subset of the Banach space of all continues function on [0, v], equipped with the Chebyshev norm in the following form
ð0Þ U ¼ fn 2 C½0; v : n n0 6 bg:
ð39Þ
1
U is a nonempty set. On U we define the operator A by ð0Þ
ðAnÞðtÞ ¼ n0 þ
1 CðaÞ
Z
t
0
f ðs; nðsÞÞ ðt sÞ1a
ds:
ð40Þ
Lemma 6.3. Under the assumptions of the relations (38)–(40), we have
AnðtÞ ¼ nðtÞ:
ð41Þ
Lemma 6.4. In Eq. (40) for "t, t1, t2 2 [0, v] we have [16]
8 1 jðAnÞðt 1 Þ ðAnÞðt2 Þj 6 Ckfðakþ1Þ ð2ðt 2 t 1 Þa þ ðt 1 Þa ðt2 Þa Þ; > > < ð0Þ jðAnÞðtÞ n0 j 6 b; > > a n : n kA nðtÞ An ~nðtÞk 6 ðLv Þ knðtÞ ~nðtÞk : L1 ½0;t
ð42Þ
1
Cðanþ1Þ
Thus, A has a unique fixed point. Consequently, the proof of the existence and uniqueness of homotopy perturbation technique solution for Eq. (28) is thoroughly investigated. 6.1. Structural stability Let n(t) and v(t) be the solutions, respectively, of the initial value problems
(
(
Dat nðtÞ ¼ f ðt; nðtÞÞ; ðkÞ
nðkÞ ð0Þ ¼ n0 ;
k ¼ 0; 1; . . . ; m 1;
Dat vðtÞ ¼ f ðt; vðtÞÞ; ~
vðkÞ ð0Þ ¼ nðkÞ k ¼ 0; 1; . . . ; m 1; 0 ;
ð43Þ
~ j < . Thus, under natural Lipschitz conditions imposed on f we have [16] where ja a
knðtÞ vðtÞkL1 ½0;T ¼ OðÞ;
! 0;
T < 1:
ð44Þ
7. Applications In this section to demonstrate the effectiveness of our approach, some examples are presented. It is to be noted that in all examples n 2 Ck[0, T < 1], k = 1, 2, . . . , where
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K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364
a ðmÞ Dat nðtÞ ¼ Im n ðtÞ; t
m 1 < a < m;
m 2 N:
Example 1. Consider the following initial value problem [20]
MD2t nðtÞ þ 2S
pffiffiffiffiffiffiffi 3=2 lqDt nðtÞ þ KnðtÞ ¼ #ðtÞ;
0 6 t 6 T;
ð45Þ
n0 ð0Þ ¼ 1:
nð0Þ ¼ 1;
This problem described the motion of a large plate of the surface Sand mass Min a Newtonian fluid viscosity l and density q. The plate is hanging on a massless spring of stiffness K. The function #(t) represents the loading force. The Eq. (45) is so called inhomogeneous Bagley–Torvik equation [20]. The Bagley–Torvik equation is known as a prototype of fractional differential equation with pffiffiffiffiffiffiffi two derivatives. In order to make comparison with the numerical solution of [20] we choose M ¼ 2S lq ¼ K ¼ 1; T ¼ 5 and #(t) = K(t + 1). By the same manipulation as Section 6 we set 3=2 Dt n1 ðtÞ ¼ n2 ðtÞ; 1=2 Dt n2 ðtÞ ¼ n2 ðtÞ n1 ðtÞ þ 1 þ t;
ð46Þ
n1 ð0Þ ¼ n01 ð0Þ ¼ 1; n2 ð0Þ ¼ 0: It is to be noted that since ni is continuous on the compact set [0, T], it is uniformly continuous there. Whence absolute maximum theorem implies that
jni ðtÞj 6 di ;
t 2 ½0; T;
di 2 Rþ :
ð47Þ
Hence
jni ðtÞ ni ð~tÞj 6 jni ðtÞj þ jni ð~tÞj 6 di þ ~di ;
t; ~t 2 ½0; T:
ð48Þ
Thus, by the Archimedean property of real number we will have
9L > 0;
s:t: d 6 Ljt ~tj;
ð49Þ
where
d ¼ di þ ~di :
ð50Þ
Consequently, in view of (4) we have
n1;0 ðtÞ ¼ 1 þ t; n1;mþ1 ðtÞ ¼ I1:5 n2;m ;
m ¼ 0; 1; . . . ;
ð51Þ
n2;0 ðtÞ ¼ 0; n2;1 ðtÞ ¼ 0; n2;mþ1 ðtÞ ¼ I0:5 ðn1;m ðtÞ þ n2;m ðtÞÞ;
m ¼ 1; 2; . . . :
Therefore we will obtain
n1;mþ1 ðtÞ ¼ 0; n2;mþ1 ðtÞ ¼ 0;
m ¼ 0; 1; . . . :
ð52Þ
Hence, n1(t) = 1 + t and n2(t) = 0. So, n(t) = 1 + t is the solution of Eq. (45). It is easily verified that t + 1 is the exact solution of Eq. (45). Table 1 shows the resulting error at t = 5 obtained by numerical method in [17] and compared with the solution obtained by the proposed scheme. Example 2. Consider the following initial value problem [21]
Dat nðtÞ ¼
3 3 40320 8a C 5 þ a2 4a 9 3 a 4 2 þ 2 t t t C ð a þ 1Þ þ n2 ðtÞ; t 4 2 Cð9 aÞ C 5 a2
nð0Þ ¼ 0;
0 < a 6 2;
n0 ð0Þ ¼ 0ða > 1Þ:
ð53Þ
Assume now that
3 3 40320 8a C 5 þ a2 4a 9 3 a t 2 t 4 n2 ðtÞ; f ðtÞ ¼ t t 2 þ Cða þ 1Þ þ a 4 2 Cð9 aÞ Cð5 2Þ Consequently, we will have
ð54Þ
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K. Sayevand et al. / Applied Mathematical Modelling 36 (2012) 4356–4364 Table 1 Error at t = 5 for Eq. (45).
jf ðtÞ f ð~tÞj 6 lkgk1 jt ~tj;
Error at t = 5 by proposed approach
Error at t = 5 by [17]
(step size)
0 0 0 0
0.15131473519232 0.04684102179946 0.01602947553912 0.00562770408881
(0.5000) (0.2500) (0.1250) (0.0625)
t; ~t 2 ½0; T;
ð55Þ
where
l ¼ sup I2t n00 ðtÞ ; g ¼ gða; t; ~tÞ; s:t:kgk1 < 1: 1
ð56Þ
t2½0;T
The exact solution of Eq. (53) by homotopy perturbation technique was derived in [21] and is
nðtÞ ¼
2 3 a t2 t4 : 2
ð57Þ
~ þ . In view of (44) we have Now suppose that a < a
knðtÞ vðtÞkL1 ½0;T 6
a~ þ8 9 a~ T jT 1j þ 3T 2 T 2 1 : 4
ð58Þ
Hence by our assumption on a, we find that
sup jnðtÞ vðtÞj ! 0;
as
! 0;
t 2 ½0; T:
ð59Þ
Example 3. Consider the following fractional nonlinear Schrödinger equation
1 @ 2 nðx; tÞ þ jnðx; tÞj2 nðx; tÞ ¼ 0; 2 @x2 jnðx; tÞj2 ¼ nðx; tÞnðx; tÞ; a
iDt nðx; tÞ þ
nðx; 0Þ ¼ eix ;
ð60Þ
ðx; tÞ 2 R ½0; 2;
where nðx; tÞ is the conjugate of n(x, t) and i2 = 1. The Schrödinger equation is the fundamental equation of physics for describing non-relativistic quantum mechanical behavior. The nonlinear Schrödinger equation was studied by many researchers [11,22]. By the same manipulation as Section 6, the solution n(x, t) is given as
nðx; tÞ ¼
1 X
a
ðit Þn eix : 2 Cðna þ 1Þ
ð61Þ
n
n¼0
As mentioned in previous examples (Eqs. (48) and (55)) one will set
a a X 1 1 X ðit Þn ðit Þn ix iz jnðt; xÞ nðt; zÞj ¼ e e n n¼0 2n Cðna þ 1Þ 2 C ðn a þ 1Þ n¼0 a a X 1 ðit Þn Ea;1 it jx zj 6 #jx zj; jðcos x cos zÞ þ iðsin x sin zÞj 6 2 ¼ n n¼0 2 Cðna þ 1Þ 2
# 2 Rþ :
ð62Þ
Consequently, in view of homotopy perturbation technique we have [22]
! j jk 1 @ 2 nj ðx; tÞ X X t þ n ðx; tÞn ðx; tÞ n ðx; tÞ dt nðx; tÞÞ 6 kn n0 ðx; tÞ eiðxþ2Þ ; jli i k 2 2 @x k¼0 l¼0
X n Z t i j¼0 0
ð63Þ
where n0(x, t) = n(x, 0) and k 2 (0, 1). Thus, we have
nn ðx; tÞ n¼0
1 X
t
! eiðxþ2Þ :
ð64Þ
a¼1 t
Therefore we deduce that for a ¼ 1; eiðxþ2Þ is the exact solution of Eq. (60).
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8. Discussion and conclusion In this study, we developed a homotopy perturbation scheme to solve the nonlinear fractional differential equations based on the Caputo definition. The uniqueness and existence of the homotopy perturbation technique solution was examined. The test we performed to evaluate the performances of the proposed analytical approach is encouraging. Moreover, the results show that the solutions of proposed scheme are stable. Finally, we point out that this new strategy has its own limitations and should be generalized and verified for more complicated linear and nonlinear problems. In other words, the present paper is only an introduction to the topic, and there remains a lot of work to do. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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