- Email: [email protected]
On the oscillation of fractional-order delay differential equations with constant coefficients
Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
On the oscillation of fractional-order delay differential equations with constant coefficients Yasßar Bolat ⇑ Department of Mathematics, Faculty of Art & Science, Kastamonu University, Kastamonu, Turkey
a r t i c l e
i n f o
Article history: Received 29 January 2012 Received in revised form 11 June 2013 Accepted 8 January 2014 Available online xxxx Keywords: Oscillation Delay differential equation Fractional-order differential equation Fractional-order derivative
a b s t r a c t In this manuscript, some oscillation results are given including sufficient conditions or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients. For this, a-exponential function which is a kind of functions that play the same role of the classical exponential functions and Laplace transformation formulations of fractional-order derivatives are used. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The popularity of fractional calculus have started in 1695 with Marquis de L’Hopital’s (1661–1704) question to Gottfried n Wilhelm Leibniz (1646–1716), which is what if n ¼ 12 ? in the Leibniz’s (currently popular) notation ddxny for the derivative of order n 2 N 0 ¼ f0; 1; 2; . . .g. In his reply, dated 30 September 1695, Leibniz wrote to L’Hopital following: ‘‘. . .This is an apparent paradox from which, one day, useful consequence will be drawn.. . .’’. In course of time, mention of fractional derivatives was made, one can found it in [21–23]. Differential and integral equations with fractional-order derivatives have found many applications in various problems in science and engineering such as electrode–electrolyte polarization [1,2], electrochemistry of corrosion [3–5], optics and signal processing [6,7], electro-thermoelasticity [8], circuit systems [9], diffusion wave [10,11], heat conduction [12,13], fluid flow [14,15], probability and statistics [16,17], control theory of dynamical systems [18], and so on. It is a well-recognized belief that fractional calculus leads to better results than classical calculus. The concept of order is a key point to understand a differential equation. Mathematically, the order of a differential equation can be defined as the degree of highest derivative in the differential equations. It is a reality that qualitative investigation of the solutions of any differential equation is more useful then solving these equations. Because some times solving an equation can take too long time and can be difficult. At the same time some known methods can not suffice to solve the equation. Therefore recently, particularly, for last three decades this concept has an increasing interest. For the oscillation of ordinary differential equations many authors obtained a great number of results including sufficient conditions or necessary and sufficient conditions. However, to the best of author’s knowledge very little is known regarding the oscillatory behavior of fractional differential equations up to now, we refer to [24]. In particular, nothing is known regarding the oscillation properties of (1)–(3) up to now. ⇑ Tel.: +90 5062348577; fax: +90 2154969. E-mail address: [email protected] http://dx.doi.org/10.1016/j.cnsns.2014.01.005 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
2
Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
In this study some criteria are obtained including sufficient conditions, and necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients. As is customary, a solution of a differential equation is said to be oscillatory if it has arbitrarily many zeros. Otherwise the solution is called nonoscillatory. If all solutions of an equation are oscillatory, then this equation is said to be oscillatory. The paper is organized as follows: Section 2 recalls some popular definitions of fractional derivatives. Main results are given in Section 3. In Section 3.1 for the first order the delay differential equation including fractional-order derivative of the form
_ þ pxða ðtÞ þ qxðt sÞ ¼ 0; xðtÞ
ð1Þ
where p; q; s 2 R; 0 < a < 1, and its general form
_ þ pxða ðtÞ þ xðtÞ
m X qi xðt si Þ ¼ 0;
ð2Þ
i¼1
where p; qi ; si 2 R; i ¼ 1; 2; . . . ; m; a = odd integer/odd integer such that 0 < a < 1, some oscillation results are given. In Section 3.2 we give some oscillation results for the second-order delay differential equation with fractional-order derivative of the form
€xðtÞ þ pxða ðtÞ þ qxðt sÞ ¼ 0;
ð3Þ
where p; q; s 2 R; a = odd integer/odd integer such that 0 < a < 1. For this purpose, known methods are expanded for our results. 2. Some known definitions for fractional-order derivatives The fractional-order derivative 0 Da can be defined in different ways [20–22], Riemann–Liouville’s definition and Caputo’s definition are two populars ones. Let a 2 R; n 1 < a 6 n; n 2 N, and f be a continuous function, then Riemann–Liouville fractional-order derivative is defined by
Z
n
RL a 0 D f ðtÞ
¼
1 d Cðn aÞ dtn
t
0
f ð sÞ ðt sÞ1þan
ds;
where CðzÞ is a Gamma function defined by
CðzÞ ¼
Z
1
et tz1 dt;
ðRðzÞ > 0Þ
0
a satisfying Cðz þ 1Þ ¼ zCðzÞ. The derivative RL 0 D f ðtÞ requires that f ðtÞ is continuous only, so it is widely used for problem description. Caputo’s fractional-order derivative has the following definition: C a 0 D f ðtÞ
¼
1 Cðn aÞ
Z 0
t
f ðnÞ ðsÞ ðt sÞ1þan
ds;
f ðtÞ 2 C n :
For function f ðtÞ, with n-order continuous derivative and starting from standstill, however, the Caputo fractional derivative gives the same value of Riemann–Liouville fractional derivative. In addition, the Laplace transformation of Caputo’s derivative has a similar formulation as that of integer-order derivatives. Caputo’s derivative is widely used in the problems of control theory. In classical calculus, the function ekt plays an important role in solving ordinary differential equations with constant coefficients, and it is satisfies
d kt e ¼ kekt : dt In fractional calculus, the following a-exponential function [19]: a1 ekt a ¼t
1 X
kk tak ; Cððk þ 1ÞaÞ k¼0
ðt > 0Þ
plays the same role as ekt in solving ordinary differential equations with constant coefficients, and ekt a satisfies the following differential equation: a xa ðtÞ ¼ RL 0 D xðtÞ ¼ kxðtÞ;
where derivative na
RL 0 D
xðtÞ ¼
RL a D xðtÞ 0
ðt > 0Þ;
ð4Þ
has the property of
ðn1Þa RL a ðRL ð0 D ÞxðtÞ: 0 D
Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
3
As a is a positive rational number, getting a ¼ k=n ¼ kb satisfying 0 < a < 1, where 1 6 k < n; k – n, we can rewrite Eq. (1) of the form RL nb 0 D xðtÞ
kb þ pRL 0 D xðtÞ þ qxðt sÞ ¼ 0:
Moreover, for Eq. (1) with constant coefficients using (4), one has
RL nb 0 D
kb ks kt þ pRL ekt 0 D þ qekb kb ¼ f1 ðkÞekb ;
where s f1 ðkÞ ¼ kn þ pk þ qek kb
ð5Þ
is called the characteristic polinomial of Eq. (1), and for Eq. (2) with constant coefficients RL nb 0 D
þ
kb pRL 0 D
þ
m X
! ks qi ekb i
kt ekt kb ¼ f2 ðkÞekb ;
i¼1
where
f2 ðkÞ ¼ kn þ pk þ
n X ks qi ekb i
ð6Þ
i¼1
is called the characteristic polinomial of Eq. (2), and for Eq. (3) with constant coefficients
RL 2nb 0 D
kb ks kt þ pRL ekt 0 D þ qekb kb ¼ f3 ðkÞekb ;
where s f3 ðkÞ ¼ k2n þ pk þ qek a
ð7Þ
is called the characteristic polinomial of Eq. (3). Assume that Eqs. (1)–(3) have a solution of the form cekt a in the sense of Riemann–Liouville derivative, then k must be a root of fi ðkÞ; i ¼ 1; 2; 3 in (5)–(7). Thus, Eqs. (1)–(3) have a solution cekt a if and only if k is a root of fi ðkÞ; i ¼ 1; 2; 3. Similarly, in the sense of Caputo’s derivative, let
^ekðtaÞ ¼ a
1 X kk ðt aÞak k¼0
Cðka þ 1Þ
;
ðt P aÞ
or
^ekt a ¼
1 X
kk t ak ; Cðka þ 1Þ k¼0
ðt > 0Þ;
then it satisfies C0 Da xðtÞ ¼ kxðtÞ, ðt > 0Þ. Eqs. (1)–(3) with constant coefficients has a solution c^ekt a if and only if k is a root of fi ðkÞ; i ¼ 1; 2; 3, given in (5)–(7). In this manuscript, the Caputo’s Laplace transform of the formulation of the fractional derivative will be used. Caputo derivative is prefered over the Riemann–Liouville derivative for physical reasons. Consider the Laplace transform of the two formulations of the fractional derivative for 0 < a < 1 and ^ekt a
RL 0
Da f ðtÞ ¼ sFðsÞ RL 0 Df ðtÞj0 ;
C
0D
a
f ðtÞ ¼ sFðsÞ f ðtÞj0 :
Fractional derivative of some special functions are as follows [23] a d sin x ap a ¼ sin x þ 2 dx
and a d cos x ap ¼ cos x þ : a 2 dx
3. Main results 3.1. First-order delay differential equations with fractional-order derivative In this subsection we give some oscillation results for the first-order delay differential equation with fractional-order derivative and constant coefficients of the form (1) and (2). Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
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Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
Theorem 1. Assume that p; q; s 2 Rþ and a is the quotient of odd integers such that 0 < a < 1. Then the following statements are equivalent. (a) Every solution of Eq. (1) oscillates. (b) The characteristic equation (5) has no real roots. Proof. ðaÞ ) ðbÞ. The proof is elementary. As the statement ðaÞ holds, if the characteristic equation (5) has a real root k0 then ek0 t is a non-oscillatory solution of Eq. (1). This contradicts to the statement ðbÞ. ðbÞ ) ðaÞ. We will make the proof using Caputo’s Laplace transform formulation of the fractional derivative. Assume, on the contrary, that ðbÞ holds and that Eq. (1) has an eventually positive solution xðtÞ. Since Eq. (1) is autonomous, we may assume that xðtÞ > 0 for t P s. As a is rational number, we can write a ¼ k=n ¼ kb satisfying 0 < a < 1, where 1 6 k < n; k – n. In this case, Eq. (1) can be recast into the following delay differential equation with fractional-order derivatives RL nb 0 D xðtÞ
kb þ pRL 0 D xðtÞ þ qxðt sÞ ¼ 0:
One can show that there exist M and
XðsÞ ¼
Z
l such that jxðtÞj 6
ð8Þ Mela t ,
t P s. Thus the Laplace transform
1
est a xðtÞdt
0
ð9Þ
exists for RðsÞ > l. Let r0 be the abscissa of convergence of XðsÞ, that is r0 ¼ inffr 2 R : xðrÞ existsg. Then by taking the Laplace transforms directly as done in the traditional way of both sides of Eq. (8) we obtain the characteristic equation
sn XðsÞ xð0Þ þ pðsXðsÞ xð0ÞÞ þ qess XðsÞ þ qhðssÞ ¼ 0; ss
where hðssÞ ¼ e
R0
s
FðsÞXðsÞ ¼ Q ðsÞ;
st
e
RðsÞ > r0 ;
xðtÞdt, or
RðsÞ > r0 ;
ð10Þ
where
FðsÞ ¼ sn þ ps þ qess
ð11Þ
QðsÞ ¼ ð1 þ pÞxð0Þ qhðssÞ:
ð12Þ
Clearly, FðsÞ and Q ðsÞ are entire functions. Also by hypothesis, FðsÞ – 0 for all real s. Therefore we can write from (10)
XðsÞ ¼
Q ðsÞ ; FðsÞ
RðsÞ > r0 :
ð13Þ
ðsÞ We now claim that r0 ¼ 1. Otherwise r0 > 1 and the point s ¼ r0 must be a singularity of the quotient QFðsÞ . But this quotient has no singularity on the real axis. Because of the numerator and dominator are entire functions and by hypothesis the dominator has no real zeros. Thus r0 ¼ 1 and (13) becomes
XðsÞ ¼
Q ðsÞ for all s 2 R: FðsÞ
ð14Þ
Now we can see that as s ! 1, trough real values, (14) leads to a contraction because XðsÞ and FðsÞ are always positive (These could seen from (9), (11) and (12)) while Q ðsÞ becomes eventually negative. The proof is complete. h Example 1. We consider the first-order delay differential equation with fractional-order derivative of the form
_ þ x1=3 ðtÞ þ xðtÞ
pffiffiffi 2p ¼ 0; 3x t 3
ð15Þ
pffiffiffi where a ¼ 1=3; p ¼ 1; q ¼ 3 and s ¼ 23p. All the conditions of Theorem 1 are satisfied. Hence all the solutions of Eq. (15) are oscillatory. One of the such solutions is xðtÞ ¼ sin t. Theorem 2. Assume that p; qi ; si 2 Rþ ; i ¼ 1; 2; . . . ; m, and a ¼ odd integer/odd integer such that 0 < a < 1. Then the following statements are equivalent. (a) Every solution of Eq. (2) oscillates. (b) The characteristic equation (6) has no real roots.
Proof. The proof can be made by the same way as in the proof of Theorem 1.
h
Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
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Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
Example 2. We consider first-order two delays differential equation with fractional-order derivatives and constant coefficients of the form
pffiffiffi pffiffiffi 93 3 _ þ 2x1=3 ðtÞ þ 3 3 5 xðt p=6Þ þ pffiffiffi xðt 3p=4Þ ¼ 0; xðtÞ 2
ð16Þ
pffiffi 93 pffiffi 3 2
pffiffiffi where a ¼ 1=3; p ¼ 2; q1 ¼ 3 3 5; q2 ¼ and s1 ¼ p=6; s2 ¼ 3p=4. All the conditions of Theorem 2 are satisfied. Hence all the solutions of Eq. (16) are oscilatory. One of the such solutions is xðtÞ ¼ sin t. 3.2. Second-order delay differential equations with fractional-order derivative
Theorem 3. Assume that p; q; s 2 Rþ and a ¼ odd odd
integer integer
such that 0 < a < 1. Then the following statements are equivalent.
(a) Every solution of Eq. (3) oscillates. (b) The characteristic equation (7) has no real roots.
Proof. ðaÞ ) ðbÞ. The proof is obvious. ðbÞ ) ðaÞ. As a = odd integer/odd integer such that 0 < a < 1, we can write a ¼ k=n ¼ kb, where k; n 2 N such that 1 6 k < n. In this case, Eq. (3) can be recast into the following differential equation with fractional-order derivatives RL 2nb xðtÞ 0 D
kb þ pRL 0 D xðtÞ þ qxðt sÞ ¼ 0:
ð17Þ
By taking the Laplace transforms directly as done in the traditional way of both sides of Eq. (17) we obtain the characteristic equation s s2n þ ps þ ðq þ hðssÞÞes kb ¼ 0;
where hðssÞ ¼
R0
s
est xðtÞdt. The rest of proof can be made as in the proof of Theorem 2. We omit it in here.
ð18Þ h
Example 3. We consider the fractional-order delay differential equation of the form
€xðtÞ þ px3=5 ðtÞ þ qxðt 17p=10Þ ¼ 0:
ð19Þ
pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffi p . All the conditions of Theorem 3 are satisfied. Hence all the soluwhere a ¼ 3=5; p ¼ 2 5 5; q ¼ 14 5 þ 14 and s ¼ 17 10 tions of Eq. (19) are oscilatory. One of the such solutions is xðtÞ ¼ cos t. 1 4
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Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
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[18] Bhalekar Sachin, Daftardar-Gejji Varsha. Synchronization of different fractional order chaotic systems using active control. Commun Nonlinear Sci Numer Simul 2010;15(11):3536–46. [19] Bonilla B, Rivero M, Trujillo JJ. Linear differential equations of fractional orders. In: Sabatier J, Agrawal OP, Tenreiro Machado JA, editors. Advances in fractional calculus. Dordrecht: Springer; 2007. p. 77–91. [20] Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999. [21] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier; 2006. [22] Das S. Functional fractional calculus for system identification and controls. Berlin: Springer-Verlag; 2008. [23] Dalir M, Bashour M. Applications of fractional calculus. Appl Math Sci 2010;4(21):1021–32. [24] Chen Da-Xue. Oscillation criteria of fractional differential equations. Adv Differ Equ 2012;1(33).
Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
On the oscillation of fractional-order delay differential equations with constant coefficients Yasßar Bolat ⇑ Department of Mathematics, Faculty of Art & Science, Kastamonu University, Kastamonu, Turkey
a r t i c l e
i n f o
Article history: Received 29 January 2012 Received in revised form 11 June 2013 Accepted 8 January 2014 Available online xxxx Keywords: Oscillation Delay differential equation Fractional-order differential equation Fractional-order derivative
a b s t r a c t In this manuscript, some oscillation results are given including sufficient conditions or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients. For this, a-exponential function which is a kind of functions that play the same role of the classical exponential functions and Laplace transformation formulations of fractional-order derivatives are used. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The popularity of fractional calculus have started in 1695 with Marquis de L’Hopital’s (1661–1704) question to Gottfried n Wilhelm Leibniz (1646–1716), which is what if n ¼ 12 ? in the Leibniz’s (currently popular) notation ddxny for the derivative of order n 2 N 0 ¼ f0; 1; 2; . . .g. In his reply, dated 30 September 1695, Leibniz wrote to L’Hopital following: ‘‘. . .This is an apparent paradox from which, one day, useful consequence will be drawn.. . .’’. In course of time, mention of fractional derivatives was made, one can found it in [21–23]. Differential and integral equations with fractional-order derivatives have found many applications in various problems in science and engineering such as electrode–electrolyte polarization [1,2], electrochemistry of corrosion [3–5], optics and signal processing [6,7], electro-thermoelasticity [8], circuit systems [9], diffusion wave [10,11], heat conduction [12,13], fluid flow [14,15], probability and statistics [16,17], control theory of dynamical systems [18], and so on. It is a well-recognized belief that fractional calculus leads to better results than classical calculus. The concept of order is a key point to understand a differential equation. Mathematically, the order of a differential equation can be defined as the degree of highest derivative in the differential equations. It is a reality that qualitative investigation of the solutions of any differential equation is more useful then solving these equations. Because some times solving an equation can take too long time and can be difficult. At the same time some known methods can not suffice to solve the equation. Therefore recently, particularly, for last three decades this concept has an increasing interest. For the oscillation of ordinary differential equations many authors obtained a great number of results including sufficient conditions or necessary and sufficient conditions. However, to the best of author’s knowledge very little is known regarding the oscillatory behavior of fractional differential equations up to now, we refer to [24]. In particular, nothing is known regarding the oscillation properties of (1)–(3) up to now. ⇑ Tel.: +90 5062348577; fax: +90 2154969. E-mail address: [email protected] http://dx.doi.org/10.1016/j.cnsns.2014.01.005 1007-5704/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
2
Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
In this study some criteria are obtained including sufficient conditions, and necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients. As is customary, a solution of a differential equation is said to be oscillatory if it has arbitrarily many zeros. Otherwise the solution is called nonoscillatory. If all solutions of an equation are oscillatory, then this equation is said to be oscillatory. The paper is organized as follows: Section 2 recalls some popular definitions of fractional derivatives. Main results are given in Section 3. In Section 3.1 for the first order the delay differential equation including fractional-order derivative of the form
_ þ pxða ðtÞ þ qxðt sÞ ¼ 0; xðtÞ
ð1Þ
where p; q; s 2 R; 0 < a < 1, and its general form
_ þ pxða ðtÞ þ xðtÞ
m X qi xðt si Þ ¼ 0;
ð2Þ
i¼1
where p; qi ; si 2 R; i ¼ 1; 2; . . . ; m; a = odd integer/odd integer such that 0 < a < 1, some oscillation results are given. In Section 3.2 we give some oscillation results for the second-order delay differential equation with fractional-order derivative of the form
€xðtÞ þ pxða ðtÞ þ qxðt sÞ ¼ 0;
ð3Þ
where p; q; s 2 R; a = odd integer/odd integer such that 0 < a < 1. For this purpose, known methods are expanded for our results. 2. Some known definitions for fractional-order derivatives The fractional-order derivative 0 Da can be defined in different ways [20–22], Riemann–Liouville’s definition and Caputo’s definition are two populars ones. Let a 2 R; n 1 < a 6 n; n 2 N, and f be a continuous function, then Riemann–Liouville fractional-order derivative is defined by
Z
n
RL a 0 D f ðtÞ
¼
1 d Cðn aÞ dtn
t
0
f ð sÞ ðt sÞ1þan
ds;
where CðzÞ is a Gamma function defined by
CðzÞ ¼
Z
1
et tz1 dt;
ðRðzÞ > 0Þ
0
a satisfying Cðz þ 1Þ ¼ zCðzÞ. The derivative RL 0 D f ðtÞ requires that f ðtÞ is continuous only, so it is widely used for problem description. Caputo’s fractional-order derivative has the following definition: C a 0 D f ðtÞ
¼
1 Cðn aÞ
Z 0
t
f ðnÞ ðsÞ ðt sÞ1þan
ds;
f ðtÞ 2 C n :
For function f ðtÞ, with n-order continuous derivative and starting from standstill, however, the Caputo fractional derivative gives the same value of Riemann–Liouville fractional derivative. In addition, the Laplace transformation of Caputo’s derivative has a similar formulation as that of integer-order derivatives. Caputo’s derivative is widely used in the problems of control theory. In classical calculus, the function ekt plays an important role in solving ordinary differential equations with constant coefficients, and it is satisfies
d kt e ¼ kekt : dt In fractional calculus, the following a-exponential function [19]: a1 ekt a ¼t
1 X
kk tak ; Cððk þ 1ÞaÞ k¼0
ðt > 0Þ
plays the same role as ekt in solving ordinary differential equations with constant coefficients, and ekt a satisfies the following differential equation: a xa ðtÞ ¼ RL 0 D xðtÞ ¼ kxðtÞ;
where derivative na
RL 0 D
xðtÞ ¼
RL a D xðtÞ 0
ðt > 0Þ;
ð4Þ
has the property of
ðn1Þa RL a ðRL ð0 D ÞxðtÞ: 0 D
Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
3
As a is a positive rational number, getting a ¼ k=n ¼ kb satisfying 0 < a < 1, where 1 6 k < n; k – n, we can rewrite Eq. (1) of the form RL nb 0 D xðtÞ
kb þ pRL 0 D xðtÞ þ qxðt sÞ ¼ 0:
Moreover, for Eq. (1) with constant coefficients using (4), one has
RL nb 0 D
kb ks kt þ pRL ekt 0 D þ qekb kb ¼ f1 ðkÞekb ;
where s f1 ðkÞ ¼ kn þ pk þ qek kb
ð5Þ
is called the characteristic polinomial of Eq. (1), and for Eq. (2) with constant coefficients RL nb 0 D
þ
kb pRL 0 D
þ
m X
! ks qi ekb i
kt ekt kb ¼ f2 ðkÞekb ;
i¼1
where
f2 ðkÞ ¼ kn þ pk þ
n X ks qi ekb i
ð6Þ
i¼1
is called the characteristic polinomial of Eq. (2), and for Eq. (3) with constant coefficients
RL 2nb 0 D
kb ks kt þ pRL ekt 0 D þ qekb kb ¼ f3 ðkÞekb ;
where s f3 ðkÞ ¼ k2n þ pk þ qek a
ð7Þ
is called the characteristic polinomial of Eq. (3). Assume that Eqs. (1)–(3) have a solution of the form cekt a in the sense of Riemann–Liouville derivative, then k must be a root of fi ðkÞ; i ¼ 1; 2; 3 in (5)–(7). Thus, Eqs. (1)–(3) have a solution cekt a if and only if k is a root of fi ðkÞ; i ¼ 1; 2; 3. Similarly, in the sense of Caputo’s derivative, let
^ekðtaÞ ¼ a
1 X kk ðt aÞak k¼0
Cðka þ 1Þ
;
ðt P aÞ
or
^ekt a ¼
1 X
kk t ak ; Cðka þ 1Þ k¼0
ðt > 0Þ;
then it satisfies C0 Da xðtÞ ¼ kxðtÞ, ðt > 0Þ. Eqs. (1)–(3) with constant coefficients has a solution c^ekt a if and only if k is a root of fi ðkÞ; i ¼ 1; 2; 3, given in (5)–(7). In this manuscript, the Caputo’s Laplace transform of the formulation of the fractional derivative will be used. Caputo derivative is prefered over the Riemann–Liouville derivative for physical reasons. Consider the Laplace transform of the two formulations of the fractional derivative for 0 < a < 1 and ^ekt a
RL 0
Da f ðtÞ ¼ sFðsÞ RL 0 Df ðtÞj0 ;
C
0D
a
f ðtÞ ¼ sFðsÞ f ðtÞj0 :
Fractional derivative of some special functions are as follows [23] a d sin x ap a ¼ sin x þ 2 dx
and a d cos x ap ¼ cos x þ : a 2 dx
3. Main results 3.1. First-order delay differential equations with fractional-order derivative In this subsection we give some oscillation results for the first-order delay differential equation with fractional-order derivative and constant coefficients of the form (1) and (2). Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
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Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
Theorem 1. Assume that p; q; s 2 Rþ and a is the quotient of odd integers such that 0 < a < 1. Then the following statements are equivalent. (a) Every solution of Eq. (1) oscillates. (b) The characteristic equation (5) has no real roots. Proof. ðaÞ ) ðbÞ. The proof is elementary. As the statement ðaÞ holds, if the characteristic equation (5) has a real root k0 then ek0 t is a non-oscillatory solution of Eq. (1). This contradicts to the statement ðbÞ. ðbÞ ) ðaÞ. We will make the proof using Caputo’s Laplace transform formulation of the fractional derivative. Assume, on the contrary, that ðbÞ holds and that Eq. (1) has an eventually positive solution xðtÞ. Since Eq. (1) is autonomous, we may assume that xðtÞ > 0 for t P s. As a is rational number, we can write a ¼ k=n ¼ kb satisfying 0 < a < 1, where 1 6 k < n; k – n. In this case, Eq. (1) can be recast into the following delay differential equation with fractional-order derivatives RL nb 0 D xðtÞ
kb þ pRL 0 D xðtÞ þ qxðt sÞ ¼ 0:
One can show that there exist M and
XðsÞ ¼
Z
l such that jxðtÞj 6
ð8Þ Mela t ,
t P s. Thus the Laplace transform
1
est a xðtÞdt
0
ð9Þ
exists for RðsÞ > l. Let r0 be the abscissa of convergence of XðsÞ, that is r0 ¼ inffr 2 R : xðrÞ existsg. Then by taking the Laplace transforms directly as done in the traditional way of both sides of Eq. (8) we obtain the characteristic equation
sn XðsÞ xð0Þ þ pðsXðsÞ xð0ÞÞ þ qess XðsÞ þ qhðssÞ ¼ 0; ss
where hðssÞ ¼ e
R0
s
FðsÞXðsÞ ¼ Q ðsÞ;
st
e
RðsÞ > r0 ;
xðtÞdt, or
RðsÞ > r0 ;
ð10Þ
where
FðsÞ ¼ sn þ ps þ qess
ð11Þ
QðsÞ ¼ ð1 þ pÞxð0Þ qhðssÞ:
ð12Þ
Clearly, FðsÞ and Q ðsÞ are entire functions. Also by hypothesis, FðsÞ – 0 for all real s. Therefore we can write from (10)
XðsÞ ¼
Q ðsÞ ; FðsÞ
RðsÞ > r0 :
ð13Þ
ðsÞ We now claim that r0 ¼ 1. Otherwise r0 > 1 and the point s ¼ r0 must be a singularity of the quotient QFðsÞ . But this quotient has no singularity on the real axis. Because of the numerator and dominator are entire functions and by hypothesis the dominator has no real zeros. Thus r0 ¼ 1 and (13) becomes
XðsÞ ¼
Q ðsÞ for all s 2 R: FðsÞ
ð14Þ
Now we can see that as s ! 1, trough real values, (14) leads to a contraction because XðsÞ and FðsÞ are always positive (These could seen from (9), (11) and (12)) while Q ðsÞ becomes eventually negative. The proof is complete. h Example 1. We consider the first-order delay differential equation with fractional-order derivative of the form
_ þ x1=3 ðtÞ þ xðtÞ
pffiffiffi 2p ¼ 0; 3x t 3
ð15Þ
pffiffiffi where a ¼ 1=3; p ¼ 1; q ¼ 3 and s ¼ 23p. All the conditions of Theorem 1 are satisfied. Hence all the solutions of Eq. (15) are oscillatory. One of the such solutions is xðtÞ ¼ sin t. Theorem 2. Assume that p; qi ; si 2 Rþ ; i ¼ 1; 2; . . . ; m, and a ¼ odd integer/odd integer such that 0 < a < 1. Then the following statements are equivalent. (a) Every solution of Eq. (2) oscillates. (b) The characteristic equation (6) has no real roots.
Proof. The proof can be made by the same way as in the proof of Theorem 1.
h
Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005
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Y. Bolat / Commun Nonlinear Sci Numer Simulat xxx (2014) xxx–xxx
Example 2. We consider first-order two delays differential equation with fractional-order derivatives and constant coefficients of the form
pffiffiffi pffiffiffi 93 3 _ þ 2x1=3 ðtÞ þ 3 3 5 xðt p=6Þ þ pffiffiffi xðt 3p=4Þ ¼ 0; xðtÞ 2
ð16Þ
pffiffi 93 pffiffi 3 2
pffiffiffi where a ¼ 1=3; p ¼ 2; q1 ¼ 3 3 5; q2 ¼ and s1 ¼ p=6; s2 ¼ 3p=4. All the conditions of Theorem 2 are satisfied. Hence all the solutions of Eq. (16) are oscilatory. One of the such solutions is xðtÞ ¼ sin t. 3.2. Second-order delay differential equations with fractional-order derivative
Theorem 3. Assume that p; q; s 2 Rþ and a ¼ odd odd
integer integer
such that 0 < a < 1. Then the following statements are equivalent.
(a) Every solution of Eq. (3) oscillates. (b) The characteristic equation (7) has no real roots.
Proof. ðaÞ ) ðbÞ. The proof is obvious. ðbÞ ) ðaÞ. As a = odd integer/odd integer such that 0 < a < 1, we can write a ¼ k=n ¼ kb, where k; n 2 N such that 1 6 k < n. In this case, Eq. (3) can be recast into the following differential equation with fractional-order derivatives RL 2nb xðtÞ 0 D
kb þ pRL 0 D xðtÞ þ qxðt sÞ ¼ 0:
ð17Þ
By taking the Laplace transforms directly as done in the traditional way of both sides of Eq. (17) we obtain the characteristic equation s s2n þ ps þ ðq þ hðssÞÞes kb ¼ 0;
where hðssÞ ¼
R0
s
est xðtÞdt. The rest of proof can be made as in the proof of Theorem 2. We omit it in here.
ð18Þ h
Example 3. We consider the fractional-order delay differential equation of the form
€xðtÞ þ px3=5 ðtÞ þ qxðt 17p=10Þ ¼ 0:
ð19Þ
pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffi p . All the conditions of Theorem 3 are satisfied. Hence all the soluwhere a ¼ 3=5; p ¼ 2 5 5; q ¼ 14 5 þ 14 and s ¼ 17 10 tions of Eq. (19) are oscilatory. One of the such solutions is xðtÞ ¼ cos t. 1 4
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Please cite this article in press as: Bolat Y. On the oscillation of fractional-order delay differential equations with constant coefficients. Commun Nonlinear Sci Numer Simulat (2014), http://dx.doi.org/10.1016/j.cnsns.2014.01.005