A detailed analysis for the fundamental solution of fractional vibration equation

Abstract In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.


Introduction
The fractional calculus has been extensively applied to various science and engineering problems to describe the memory and hereditary properties of various materials and processes [1][2][3][4][5][6][7][8][9]. In particular, the fractional calculus has been applied to the mathematical modelling of viscoelastic materials. For some viscoelastic materials the stress-strain constitutive relation can be more accurately described by introducing the fractional derivatives [9][10][11][12][13].
Scott-Blair [14,15] introduced the fractional calculus to characterize a viscoelastic material whose mechanical properties are intermediate between those of a pure elastic solid (Hooke model) and a pure viscous fluid (Newton model) [6,16]. In [11], a fractional calculus element whose constitutive law obeys stress is proportional to a fractional derivative of strain is said to be a spring-pot.
Fractional oscillation or vibration was discussed by Gorenflo and Mainardi [2], Bagley and Torvik [17], Beyer and Kempfle [18], and others [19][20][21][22][23][24][25][26][27][28]. In [2], the fractional relaxation and oscillation equations with the Caputo derivative were considered by means of the Laplace transform and Mittag-Leffler function. In [17], the damping term was described by using the Riemann-Liouville fractional derivative, and the Laplace transform was applied to analyze the behavior of the oscillator. In [18], the causal condition for solution of the fractional vibration equation was investigated with the help of the Fourier transformation, operator theory and complex analysis.
Afterwards, Achar et al. [19] studied the response characteristics of the fractional oscillator by extending the classic integral equation to fractional case and applying the Laplace transform and Mittag-Leffler function. Li et al. [20] considered the impulse response and the stability behavior of a class of fractional oscillators with multi-term Li [21] established the relationship between fractional oscillator processes and the corresponding fractional Brownian motion processes.
Shen et al. [22,23] analyzed the dynamical behavior and resonance for linear and Duffing-type nonlinear fractional oscillators in Caputo sense, respectively, using the averaging method.
Furthermore, the theory of fractional dynamic system has been founded and developed in [29][30][31][32][33][34]. In [29], Chaos synchronization of the Chua system with Caputo fractional derivative was considered by using numeric method. In [30], the nonlinear dynamic behaviors of oscillators described by fractional derivative were studied, the numerical scheme was developed, and the bifurcation and chaos of the oscillator in forced vibration were shown. In [31], stability of the vibration system with the Caputo fractional derivative was investigated based on stability switch. In [32], the linearization and stability theorems of the nonlinear fractional system were presented. In [33][34][35][36][37], the local fractional calculus and its fractal geometrical explanation and applications were considered.
Next, we recall the definitions of the related fractional derivatives. For additional details we refer the readers to references as [1][2][3][4][5][6][7][8][9]. Let f .t / be piecewise continuous on .t 0 ; C1/ and integrable on any finite subinterval of .t 0 ; C1/. Then for t > t 0 , the Riemann-Liouville fractional integral of f .t / of orderˇis defined as whereˇis a positive real number, and . / is Euler's gamma function. The Riemann-Liouville fractional derivative of f .t / of order˛is defined as (when it exists) For a linear fractional system, the Laplace transform is frequently used for analytical research [9,[37][38][39]. Unlike the Riemann-Liouville fractional derivative, the Laplace transform of the Caputo fractional derivatives only involves the initial values of the integer-order derivatives [9]. We model the fractional vibration equation with the Caputo fractional derivatives associated with the initial conditions in the traditional form. In the sequel, we denote the operator 0 Dt as Dt for short. The following formula of the Laplace transform will be used, where is the Laplace transform of the function f .t /.
In this paper we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order˛satisfying 0 <˛< 1 or 1 <˛< 2. We give a detailed analysis for the fundamental solution y.t / through the complex inversion integral of the Laplace transform.
The text is organized as follows. In the next section, we recall the classical vibration equation and its resolution by the Laplace transform. In Section 3, we consider the fractional vibration equation for the two cases of the order 0 <˛< 1 and 1 <˛< 2 simultaneously. Section 4 summarizes our conclusions.

Classical vibration equation
First we consider the classic vibration equation By using the Laplace transform we have The transform function is solved to be Introducing function y.t /, such that we have where we have used the initial-value theorem y.0/ D lim s!1 sY .s/ D 1: Applying the inverse Laplace transform to Eq. (7) we obtain Thus y.t / and 1 c y 0 .t / are the fundamental solution and the impulse-response solution, respectively, for Eq. (5) . In order to compare with the fractional case, we prefer to express the fundamental solution y.t / by using the complex inversion integral formula where Br denotes the Bromwich path, i.e. the straight line from s D i 1 to s D C i 1, where is chosen so that all the singularities of the integrand lie to the left of the line.
In the integer-order case, Y .s/ in Eq. (8) is a meromorphic function having two poles at most depending on the cases of the roots of the characteristic equation Hence we have where s i are the poles of Y .s/. There are the following three cases.
The characteristic equation has two different negative real roots s 1;2 D b˙pb 2 4c 2 , which are two simple poles of Y .s/. Calculating the residues we have Case 2. Critically damped case b 2 4c D 0.
The characteristic equation has two identical negative real roots s 3 D b 2 , which is a double pole of Y .s/. Calculating the residues we have Case 3. Underdamped case b 2 4c < 0.
The characteristic equation has a couple of conjugate complex roots with negative real parts, s 4;5 D b 2ṗ 4c b 2 2 i; which are two simple poles of Y .s/. Calculating the residues we have

Fractional vibration equation
In this section, we consider the fractional vibration equation where˛satisfies 0 <˛< 1 or 1 <˛< 2. We consider the two fractional cases simultaneously, and express their results separately as Case i and Case ii, if necessary.

General form of solutions
Applying the Laplace transform to Eq. (16) leads to where we use the round-off notation OE˛ D ( 0; 0 <˛< 1; 1; 1 <˛< 2; to combine the two fractional cases. Solving the transform function we have We express the fundamental solution as y.t /, i.e. it satisfies Since y.0/ D lim s!1 sY .s/ D 1, we have the following two formulas and Therefore, the inverse Laplace transform of Eq. (18) yields the solutions for the two cases, We note that the impulse-response solution is 1 c y 0 .t /. The fundamental solution y.t / satisfies the corresponding homogeneous equation x 00 .t / C b Dt x.t / C c x.t / D 0 subject to the initial conditions x 0 D 1 and x 1 D 0, and plays an important role for construction of the solution. We focus our attention to the fundamental solution y.t/ in the sequel.

Representation of fundamental solution
We investigate the fundamental solution y.t / by the complex inversion formula where Br denotes the Bromwich path, i.e. the straight line from s D i 1 to s D C i 1, where is chosen so that all the singularities of the integrand lie to the left of the line.
For the fractional cases, the original point s D 0 is a branch point of the integrand. We make a branch cut along the negative real axis and consider the problem on the principal Riemann surface. Due to Cauchy's theorem and the residue theorem we can rewrite the right hand side of Eq. (24) as the sum of residues plus a Hankel contour integral, i.e. where where s i are the relevant singularities of Y .s/, and Ha denotes the Hankel path, a loop which starts from 1 along the lower side of the negative real axis, encircles the origin counter clockwise, and ends at 1 along the upper side of the negative real axis. We display the Bromwich path and the Hankel path in Fig. 1.

Calculation of residues for f 1 .t/
To calculate f 1 .t/, we need to find out the relevant singularities of Y .s/. First we consider the roots of the characteristic equation w.s/ WD s 2 C bs˛C c D 0; for the two cases.
In [24], it was proved in detail that the characteristic equation (28) has a couple of conjugate complex roots with negative real parts for arbitrary real coefficients b; c > 0.
Eq. (28) may be rewritten as then the roots of the characteristic equation (28) are s D 1 r 1 e iÂ 1 : From the above discussion, for both cases 0 <˛< 1 and 1 <˛< 2, the characteristic equation (28) always has a couple of conjugate complex roots with the negative real parts: where r D pˇ2 C 2 and Â D C arctan. =ˇ/. Since the numerator and the denominator of the right hand side of Eq. (19) cannot equal zero simultaneously, so the roots s 6 and s 7 of the characteristic equation are the simple poles of Y .s/e st . See Fig. 1 for a graphical representation of the locations of the two simple poles.
For specified b; c and˛, we can calculate the two roots s 6 and s 7 of the characteristic equation by a software such as MATHEMATICA. In Table 1, we list the values ofˇ; ; r; Â for b D c D 1 and different˛.
Utilizing the results in (32) and the data in Table 1, we plot the curves of f 1 .t / in Fig. 2

Simplification of integrals for f 2 .t/
We look into the complex contour integral in Eq. (27). Since sY .s/ ! 0 as s ! 0; the integration on the small circle satisfies On the upper side and lower side of the negative real axis, we express s D ue˙i ; and simplify f 2 .t / as K˛.u/e ut du; (34) where ImOE denote the imaginary part and Note that i.e. the denominator of Eq.

The fundamental solution and asymptotic behavior
From the above three subsections, the fundamental solution y.t / can be expressed in the form where C D 2 2r 2 C .2 C˛/br˛cos.2Â ˛Â / C˛b 2 r 2.˛ 1/ 4r 2 C 4˛br˛cos.2Â ˛Â / C˛2b 2 r 2.˛ 1/ ; Here, f 1 .t/ represents a decaying oscillation along the t axis, where the amplitude decays exponentially. For f 2 .t /, we consider its asymptotic behavior using the Hankel integral representation in Eq. Reserving the principal term in the asymptotic series we obtain Hence the monotone part f 2 .t / has the asymptotic representation which exhibits an algebraic decay in the form of power of negative exponent. Therefore, the fundamental solution y.t/ is dominated by f 2 .t / as t ! C1, and we have The derivative of the fundamental solution has the similar expression and asymptotic behavior, If 0 <˛< 1, y 0 .t / is always negative for large enough t . If 1 <˛< 2, y 0 .t / is invariably positive for large enough t. We conclude the behaviour of y.t / as follows.
Case ii. 1 <˛< 2: y.t / is ultimately negative, and ultimately increases monotonically and approaches zero.   In Fig. 7 and Fig. 8, we plot the curves of the fundamental solution y.t / for b D c D 1 and for different values of˛. In order to clarify the variation near the t-axis, in Fig. 9 we display the curves in Fig. 8 with different vertical scales.

The zeros and the maximum extreme point
In [41] and [42], the zeros of solutions and eigenvalue problems for a class of simple fractional differential equations were considered. The two issues were found out to be closely related. Due to y.0/ D 1, we deduce that  We calculate the maximum zero of y 0 .t / such that y 00 .t / ¤ 0 to be T D 21:255755 : : : . On the interval .T; C1/, y.t/ increases monotonically and approaches zero; see Figs. 10 and 11. In Table 2, we list the number of zeros, the maximum zero and the maximum extreme point of y.t / for b D c D 1 and ten different values of˛. It shows that the number of zeros and the value of the maximum zero increase with increasing fractional order˛for both cases 0 <˛< 1 and 1 <˛< 2. We note that the maximum extreme points firstly decrease and then increase with increasing fractional order˛for the case of 0 <˛< 1.

Conclusions
We have investigated the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order˛satisfying 0 <˛< 1 or 1 <˛< 2. A detailed analysis for the fundamental solution y.t / is carried out through the Laplace transform and its complex inversion integral formula. Unlike the integer-order case, the Laplace phase function has a branch point at the original point. The fundamental solution y.t / is expressed as a sum of the oscillative part f 1 .t / and the monotone part f 2 .t/. We deduce that f 1 .t / represents a decaying oscillation whose amplitude decays exponentially, while f 2 .t / exhibits an algebraic decay in the form of power of negative exponent.
We have concluded that y.t / is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 <˛< 1, while y.t / is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 <˛< 2. We have also considered the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y.t / for specified values of the coefficients and fractional order.