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Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 272198, 7 pages
The Oscillation of a Class of the Fractional-Order Delay Differential Equations
Department of Control Science and Engineering, Tongji University, Shanghai 200092, China
Received 5 June 2014; Accepted 26 June 2014; Published 3 August 2014
Academic Editor: Sanling Yuan
Copyright © 2014 Qianli Lu and Feng Cen. 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.
Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this, α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.
For the past three centuries, fractional calculus has been dealt with by mathematicians. In the recent decades, it is widely used in the fields of engineering, science, and economy. Recently, fractional differential systems have gained scholar’s attention [1–3]. Many researchers demonstrated applications of fractional calculus in the frequency dependent damping behavior of many viscoelastic materials, dynamics of interfaces between nanoparticles and substrates, the nonlinear oscillation of earthquakes, bioengineering, continuum and statistical mechanics, signal processing, filter design, circuit theory, and robotics. In some practical systems, such as economic systems, biological systems, and space-light industry systems, due to the transmission of the signal or the mechanical transmission, the research on fractional differential systems with delay is desired.
In , the authors described the application of the background and significance of the differential delay equations oscillation and fractional differential equations. The delay differential equation is different from ordinary differential equations. Moreover, compared with the periodic solution, oscillatory solution has more extensive application prospect and theory value in terms of the general solution of equation. In this paper, the necessary or sufficient conditions for the oscillation of several fractional-order delay differential equations are analyzed and discussed. Some oscillation results are proposed for the fractional-order linear delay differential equation as follows: where . Some oscillation results are also given for the delay parabolic differential equation as follows:
For the purpose, known methods are expanded for our results.
We first give some definitions and make some preliminaries. The fractional-order derivative can be defined in different ways, wherein Riemann-Liouville’s definition and Caputo’s definition are two main expressions as follows.
Definition 2. -exponential function is as follows :
where satisfies the differential equation , .
The derivative has the property of
Mittag-Lefflerfunction [3, 4] can be defined by
The first order delay differential equation including fractional-order derivative  is expressed by where , . Since is a positive rational number, let satisfy . The expression for (1) with constant coefficients is given by where
Equation (10) is the characteristic polynomial of (8). The contribution in  is considered.
In this paper, the main results are derived from the application of the basic theory in . The oscillation or nonoscillation of solutions for several fractional-order delay differential equations can be obtained based on these main results.
3. Main Result
3.1. The Fractional-Order Delay Differential Equation with Constant Coefficients
The general form of the fractional-order linear delay differential equation is
Since , the characteristic polynomial of (11) is as follows:
Proof. The equation has a solution , if and only if is a root of characteristic of equation . Using the conditions for (11), we have the following.(1)If or , (12) has a real root, and the solution of (11) is nonoscillatory.(2)If , (12) does not have a real root, and the solution is oscillatory. So we can obtain , and the solutions of (11) are oscillatory. Therefore, the proof of the theorem can be completed.
Example 7. We consider delay differential equation with fractional-order derivatives
We consider the following fractional-order linear delay differential equation:
: , .
Theorem 8. Assume that holds, and are satisfied. Every solution of the delay differential equation (17) oscillates.
Proof. We may assume that (17) has eventually positive solution. Then, there exists , and if , , , and .
We consider the eventually positive solution of the following inequality:
Assume the inequality (18) has a solution :
If , the inequality does not have a real root. That means every solution of the inequality (18) is oscillatory. Thus, if , every solution of the delay differential equation (17) oscillates.
3.2. The Fractional-Order Delay Differential Equation with Oscillating Coefficients
We consider oscillation of solutions of the fractional-order delay differential equation with oscillating coefficients:
Let . If is sufficiently large,
We have the sufficient conditions for oscillation of all solutions of (20) as follows.
Theorem 9. If is sufficiently large, . Then, the solution of (20) is oscillatory.
Proof. Assume that (20) has eventually positive solution. If there exists such that , then , ,
Similar to the proof process of Theorem 8, every solution of the inequality (24) is oscillatory due to . Thus, the inequality (24) does not have eventually positive solution. Therefore, if is sufficiently large, , and every solution of (20) is oscillatory. The proof of the theorem is complete.
We consider the following fractional-order linear delay differential equations:
4. The Fractional-Order Delay Partial Differential Equation
To research the oscillation properties of solutions for delay partial differential equations, we consider the delay parabolic differential equations : where . is the bounded region with piecewise smooth in .
The hypotheses are always true as follows. , and is a nonnegative constant. are nonnegative constants.
Consider the boundary conditions as follows: where is the unit exterior normal vector in .
If there are no special instructions in this section, we only discuss the oscillation of solutions of the fractional-order delay partial differential equation with respect to time variable .
Proof. Let . Equation (29) on both sides of the integral of in is expressed by
By using Green’s formula and boundary conditions ,
Let . Substituting (30) with (31),
Based on Corollary 10, (1) and (2) in Theorem 14 are proved.
Example 12. Consider the parabolic equation:
Because of , all the conditions in Theorem 11 (1) are satisfied. By using Theorem 11, every solution of (33), with oscillates in . One of such solutions is .
For the more general equation of (29) with coefficients,
There are the boundary conditions as follows: where is the unit exterior normal vector in .
Assume that , , , are nonnegative constants.
Lemma 13. For the Dirichlet problem, where is a constant. The smallest eigenvalue of problem (36) is positive, and the corresponding eigenfunction is also positive on . With each solution of problem (35) with the boundary condition , the function is defined as
Proof. Let . If we multiply both sides of (35) by and integrate in , we obtain
where , which satisfies Lemma 13.
By using Green’s formula and boundary condition , we obtain where is the surface element on . Substituting with (37), we obtain
Based on Corollary 10, the proof of the theorem can be immediately completed. Let be defined as , , where denotes a solution of problem (35) with the boundary condition or . Then, we obtain
Theorem 15. Assume that hold and is satisfied. Every solution of the delay differential equation with several coefficients (25) oscillates. Then, every solution of (35) with the boundary condition oscillates.
Proof. Let on the be the definite integral in . Assume that (35) with the boundary condition has a nonoscillatory solution and (35) has an eventually positive solution:
By using Green’s formula and boundary condition , we obtain where is the surface element on .
Let () be substituted into (42). We obtain
The detail of the proof process is similar to the proof process of Theorem 15.
Theorem 16. Assume that and hold and is satisfied. Every solution of the delay differential equation with the coefficients (25) oscillates. Then, every solution of (35) with the boundary condition oscillates.
Proof. Let on the be the definite integral in . For the sake of contradiction, assume that (35) with the boundary condition has a nonoscillatory solution and (35) with the boundary condition has an eventually positive solution:
By using Green’s formula and boundary condition , we obtain where is the surface element on . Let () be substituted into (45). We obtain
The detail of the proof process is similar to the proof process of Theorem 15.
The delay parabolic differential equations are expressed by
Theorem 17. Assume that , , and hold and is satisfied. If , the solution of (48) with the boundary condition is oscillatory.
Proof. Let . Assume that (48) with the boundary condition has no oscillation solutions, and (48) has eventually positive solution. If there exists such that , then , , . If is satisfied, (48) can be transformed into the inequality as follows:
The inequality (49) on both sides of the integral of in is expressed by
By using Green’s formula and boundary conditions , we obtain
Substituting (50) with (51) and , we obtain
By using Theorem 9, the proof of the theorem can be completed.
A. Maximum of
Figure 1 illustrates the curves of the function with different . From bottom to top, , , , , , and 1, respectively. From Figure 1, it can be observed that there exists such that . In detail, the values of with different are reported in Table 1. is the maximum for a fixed in Table 1.
B. Supplementary Comment of
The curves of function and () are showed in Figure 2. The function is a supplementary comment of the maximum value of . From top to bottom, , , , , , and , respectively.
C. The Characteristic Equation
The Laplace transforms for two fractional derivatives with  are expressed as follows:
In the most general form , we can obtain and . and are both entire functions. Assume for all real . can be expressed as , . Therefore, is the characteristic polynomial of (11).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Mr. Weisheng Xu for useful discussions on the proof process in the paper.
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