THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL OSCILLATION EQUATION

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1 RESEARCH PAPER THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL OSCILLATION EQUATION Jun-Sheng Duan 1,2, Zhong Wang 2, Shou-Zhong Fu 2 Abstract We consider the zeros of the solution α (t) =E α ( t α ), 1 <α<2, of the fractional oscillation equation in terms of the Mittag-Leffler function, and give a wholly and clarified description for these zeros. We find that the number of zeros can be any finite number: 1, 2, 3, 4,..., not necessarily an odd number. When the number of zeros of α (t) isanevennumber, α (t) has a critical zero. All of the values of α for which α (t) has an even number of zeros constitute a countable set S. Foreachα (1, 2) \ S, α (t) has an odd number of zeros. These results are a supplement and a perfecting for the existed related documents. We also show that the eigenvalue problems are related with the zeros of the Mittag-Leffler functions. MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 44A10 Key Words and Phrases: fractional calculus; fractional differential equation; fractional oscillation equation; zero of function; Mittag-Leffler function 1. 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 [22,19,10,6,16,23,9,17,1,13]. The solutions of many fractional differential equations are related to an important class of special functions the Mittag-Leffler functions (with Mathematics c 2014 Diogenes Co., Sofia pp , DOI: /s x

2 THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL Subject Classification 33E12). The Mittag-Leffler function was introduced by Mittag-Leffler [20, 21] and was included in the famous manual on special functions [3]. For some surveys on the Mittag-Leffler functions, see e.g. [17, 8], and for their multi-index generalizations, see [11, 12]. The Mittag-Leffler function and its zeros were investigated by Wiman [25, 26], Gorenflo and Mainardi [5, 6], Mainardi [15], Mainardi and Gorenflo [18] and Pskhu [24], etc. The existing results are for 1 <α<2, there is an odd number of zeros for the Mittag-Leffler functions E α (t). In this paper we give a whole and clarified description for the zeros of the solution of the fractional oscillation equation, and therefore of the Mittag-Leffler functions E α (t), 1 <α<2. We find that the number of zeros can be any finite number: 1, 2, 3, 4,..., not necessarily an odd number. The results in this paper are supplement and perfecting for the existing related studies. The text is organized as follows. In the next Section 2, we recall the definitions of the fractional derivatives and the Mittag-Leffler functions. In Sections 3 and 4, we consider the solution of the fractional oscillation equation and the zeroes of this solution. In Section 5, we show the relationship of the zeroes of the Mittag-Leffler functions and the eigenvalue problems. Section 6 summarizes our conclusions. 2. Fractional derivatives and the Mittag-Leffler functions We recall the definitions of the fractional derivatives and the Mittag- Leffler functions. For additional details we refer the readers to references as [22, 19, 10, 6, 16, 23, 9, 17, 1, 8, 14, 7, 4]. Let f(t) be piecewise continuous on (, + ) and integrable on any finite subinterval of (, + ). Then for t >, the Riemann-Liouville fractional integral of f(t) oforderβ is defined as t J β t f(t) = (t τ) β 1 f(τ)dτ, (1) Γ(β) where β is a positive real number, and Γ( ) is Euler s gamma function. The Riemann-Liouville fractional derivative of f(t) oforderα is defined as (when it exists) D α t f(t) = dm ( dt m Jt m α f(t) ), t >,m 1 <α<m. (2) Let m N + and f (m) (t) exist and be integrable on any finite subinterval of (, + ). Then the Caputo fractional derivative of f(t) oforderα is

3 12 J.-S. Duan, Z. Wang, S.-Z. Fu defined as D α t f(t) = { f (m) (t), α = m, J m α t f (m) (t), m 1 <α<m. Another expression of the Caputo fractional derivative is [5, 23, 6, 2], [ ] m 1 Dt α f(t) = D α (t ) k t f(t) f (k) (t + 0 k! ), m 1 <α<m. (4) k=0 In the sequel, we denote the operator 0 Dt α as Dt α for short. The Mittag-Leffler function is defined by the series expansion [6,20,21, 3, 15, 18, 23, 4] E λ (z) = k=0 (3) z k,λ>0,z C, (5) Γ(λk +1) which is analytic on the whole complex plane. We confine z R and consider the real zeroes of the Mittag-Leffler functions E λ ( z). Obviously E λ ( z) > 0 for all z 0. Thus the zeroes of E λ ( z), if any, must be positive real numbers. The values of λ determine if the function E λ ( z) has real zeroes. If 0 <λ 1, then function E λ ( z) does not have real zeroes. If λ 2, then the function E λ ( z) has infinitely many real zeroes [3,5,24]. We list two special cases of the Mittag-Leffler functions, { E 1 ( z) =e z cos( z), z 0,,E 2 ( z) = cosh( (6) z), z < Solution of the fractional oscillation equation We consider the initial value problem for the following homogeneous fractional relaxation-oscillation equation [5, 6, 15, 18] D α t u(t) = u(t), t > 0, 0 n 1 <α n, (7) u(0) = c, u (k) (0) = 0, k =1, 2,...,n 1. (8) The solution can be expressed by the Mittag-Leffler function [5,6,15] We introduce the notation u(t) =ce α ( t α ). (9) α (t) =E α ( t α ), (10) which is the solution of the problem (7) (8) for the case of c =1,is called fundamental solution for the problem (7) (8), and in [5,6,15,18] is denoted by e α (t). Here we adopt a different notation in order to distinguish from the irrational number e.

4 THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL We note that for the integer-order cases, α =1, 2, 3, the solution α (t) degenerates to (see [5]) 1 (t) =e t, 2 (t) =cost, 3 (t) = 1 3 e t et/2 cos( t). (11) 2 The Laplace transform of α (t) is sα 1 α (s) =L[ α (t),s]= s α, Re(s) > 1. (12) +1 By the inversion formula of the Laplace transform we have α (t) = 1 s α 1 2πi Br s α +1 est ds, (13) where Br denotes the Bromwich path, i.e. the straight line from s = σ i to s = σ + i, whereσ>1. For α not integer, α (s) has branch points s =0ands =, sowe take the negative real axis as a cut and consider the one-valued branch satisfying π <arg s<π. The right hand side of Eq. (13) can be expressed as a Hankel contour integral plus the residues, i.e. α (t) =f α (t)+g α (t), (14) with f α (t) = 1 s α 1 2πi Ha(ɛ) s α +1 est ds, (15) where the Hankel path Ha(ɛ) denotes a loop constructed by a small circle s = ɛ with ɛ 0 and by the two sides of the negative real axis, and g α (t) = [ s e s h t α 1 ] Res s α, (16) +1 h s h where s s α 1 h are the relevant poles of s α +1. The poles are s h = e i(2h+1)π/α with unitary modulus. The relevant poles are only those situated in the main Riemann sheet, i.e. the poles s h satisfying π <arg s h <π. In the sequel of this paper, we consider the fractional oscillation case 1 <α<2, which is the most attracting case. For other cases we can refer to [5]. In Fig. 1 we display the Bromwich path, Hankel path and the poles s 0 and s 0 for α =1.5. From Eq. (15) we derive that f α (t) = 0 e rt K α (r)dr, (17)

5 14 J.-S. Duan, Z. Wang, S.-Z. Fu Figure 1. The Bromwich path, Hankel path and the poles s 0 and s 0 for α =1.5. with K α (r) = 1 π Im ([ s α 1 = 1 π s α +1 ] s=re iπ ), r α 1 sin(απ) r 2α +2r α cos(απ)+1. (18) Since the denominator in Eq. (18) is always positive, K α (r) has the same sign with sin(απ), i.e. K α (r) is negative for the considered case 1 <α<2. Therefore, f α (t) is increasing towards zero as t +. For the another part, g α (t), there are precisely two relevant poles, s 0 = e iπ/α and s 1 = s 0 = e iπ/α, for our considered case, and both are simple. Calculating the residues we obtain g α (t) = 2 (e α Re s t) 0 = 2 α et cos(π/α) cos [t sin( π ] α ). (19) This function exhibits oscillations with circular frequency ω(α) = sin(π/α) and with an amplitude which exponentially decays with the rate cos(π/α). The monotone part f α (t) has the asymptotic representation [5] f α (t) t α, t +, (20) Γ(1 α) which stands for the algebraic decay for t +. Hence we also have α (t) t α, t +. (21) Γ(1 α) From Eq. (14), α (t) exhibits oscillations induced by g α (t) alongthe monotone curve f α (t). In Fig. 2 we display the superposition of g α (t) and f α (t) to form the solution α (t), and the asymptotic behavior of α (t) for

6 THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL (a) (b) Figure 2. The function f α (t) (dotline), f α (t) (dot-side line), g α (t) (dash line) and α (t) (solid line) for α =1.55. α =1.55, where two subfigures correspond to different ranges of t and we adopt different vertical scales to clarify the asymptotic behavior. 4. The zeros of the function α (t) In this section, we indicate that the number of the zeros of α (t) can be 2, 4, 6,..., does not limit to be an odd number. For sufficiently large t, α (t) turns out to be permanently negative as shownin(21)bythenegativesignofγ(1 α). The smallest zero lies in the first positivity interval of cos[t sin(π/α)], i.e. in the interval I 0 : 0 <t<π/[2 sin(π/α)]. (22) All other zeros, if any, can only lie in the succeeding positivity intervals of cos[t sin(π/α)], i.e. I n : ( π 2 +2nπ)/ sin(π/α) <t<(π +2nπ)/ sin(π/α),n =1, 2,... (23) 2 If α (t) has zeros in some interval I m,wherem 2, then there are exactly two zeros in each left intervals I n,n<m, except for the first interval I 0, where always one zero. If α (t) does not have zeros in some interval I m, where m 1, then there do not exist zeros in each right intervals I n,n >m. α (t) has zeros on the interval I n, n 1, if and only if for some t in this interval, the inequality 2 α et cos(π/α) cos [t sin( π ] α ) f α (t) (24)

7 16 J.-S. Duan, Z. Wang, S.-Z. Fu holds. Only one zero exists for α sufficiently close to 1, and as α 1 + the zero approaches infinity [5,6,15,18]. By increasing α, more and more zeros arise. We conclude the following case exists: if there is exactly one value of t in I n, n 1, satisfying (24), then α (t) has exactly one zero on this interval, and in this case, α (t) has an even number of zeros in all. When the number N of the zeros is an even number, we denote the corresponding value of α by α[n ], and the greatest zero is called a critical zero. We demonstrate this procedure numerically for α increasing from 1.4 to In Fig. 3, we plot the curves of 1.4 (t), (t), 1.45 (t), 1.57 (t), 1.6 (t) and (t). When α = , α (t) has a critical zero, and has two zeros in all. Here α = denotes α is a number between and Throughout this paper we adopt such convention. When α = , α (t) has four zeros, and when α = , α (t) hassix zeros. N α[n ] Least and greatest zeros , , , , , , , , , , Table 1. The values of α foranevennumberofzeros. In Table 1 we list the values of α when the number N of zeros of α (t) is an even number and no more than 20. In this case the least and greatest zeros are calculated, where the greatest zeros are the critical zeros. If α satisfies α[2m] <α<α[2m+2], the function α (t) has2m+1 zeros, where m =1, 2, 3,... When N =2m, the critical zero is located in the interval I m,andthe critical zero and the value of α satisfy g α (t) = f α (t), g α (t) = f α (t). (25)

8 THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL (a) (b) (c) (d) (e) (f) Figure 3. The functions α (t) (a)α =1.4, (b) α =1.422, (c) α =1.45, (d) α =1.57, (e) α =1.6, and (f) α =1.649.

9 18 J.-S. Duan, Z. Wang, S.-Z. Fu Figure 4. The surface of α (t) as a function of t and α. Figure 5. The (t, α) coordinate surface and the surface of α (t) abovethecoordinatesurface. For α 2, there is an increasing number of zeros up to infinity since 2 (t) =g 2 (t) =cost has infinitely many zeros t =(n +1/2)π, n =0, 1, 2,... In order to wholly look into the relationship of the zeros of the function α (t) and the values of α we plot the surface of α (t) as a function of t and α in Fig. 4. In Fig. 5 we plot the (t, α) coordinate surface and the surface of α (t) above the coordinate surface, where the intersecting lines

10 THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL Figure 6. The intersecting lines of the surface of α (t) and the (t, α) coordinatesurface. of the two surfaces comprise zeros of the function α (t) for all α (1, 2]. In Fig. 6 these intersecting lines are shown. 5. Zeros of the Mittag-Leffler functions and the eigenvalue problems Since α (t) =E α ( t α ), the curve of E α ( t) isastretchofthatof α (t) along t axis. The Mittag-Leffler function E α ( t) does not have zeros when 0 <α 1, has infinitely many zeros when α 2, and has finite many zeros when is a zero of α (t). We consider the eigenvalue problem for the fractional differential equation, i.e. the boundary value problem 1 <α<2. It is obvious that t is a zero of E α ( t) if and only if t 1/α D α t u(t)+λu(t) =0, 1 <α 2, (26) u (0) = 0, u(1) = 0. (27) As α = 2, this BVP has the eigenvalues λ n = (nπ π 2 )2 and the eigenfunctions u(t) =A cos( λ n t)=acos[(nπ π 2 )t], n =1, 2,... The solution of the fractional differential equation (26) satisfying the boundary condition u (0) = 0 is u(t) =AE α ( λt α ). (28) From the boundary condition u(1) = 0, we obtain that λ n is an eigenvalue of the BVP, (26) and (27), if and only if E α ( λ n )=0, i.e. λ n is a zero of the Mittag-Leffler function E α ( t), and the eigenfunctions are u(t) =AE α ( λ n t α ), A 0. (29)

11 20 J.-S. Duan, Z. Wang, S.-Z. Fu We have checked that if α = 2 the results of the second-ordered differential equation are obtained. 6. Conclusions We consider the zeros of the solution α (t) =E α ( t α ), 1 <α<2, of the fractional oscillation equation in terms of the Mittag-Leffler function. Thenumberofzeroscanbeanyfinitenumber:1,2,3,4,... When the number of zeros of α (t) iseven, α (t) has a critical zero. All of the values of α for which α (t) has an even number of zeros constitute a countable set S. For each α (1, 2) \ S, α (t) has an odd number of zeros. We also show that the eigenvalue problems are related with the zeros of the Mittag-Leffler functions. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos ; ) and the Innovation Program of Shanghai Municipal Education Commission (No. 14ZZ161). References [1] D. Băleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods. Ser. on Complexity, Nonlinearity and Chaos, World Scientific, Boston (2012). [2] J.S. Duan, T. Chaolu, R. Rach, Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach- Adomian-Meyers modified decomposition method. Appl. Math. Comput. 218 (2012), [3] A. Erdélyi (Ed.), Higher Transcendental Functions, Vol. 3, Chap. 18. McGraw-Hill, New York (1955). [4] R. Gorenflo, J. Loutchko, Y. Luchko, Computation of the Mittag-Leffler function E α,β (z) and its derivative. Fract. Calc. Appl. Anal. 5, No4 (2002), ; fcaa/. [5] R. Gorenflo, F. Mainardi, Fractional oscillations and Mittag-Leffler functions. In: Proc. Conf. Recent Advances in Appl. Math., Kuwait Univ.,1996 (1996), [6] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer- Verlag, Wien/New York (1997), [7] S.R. Grace, R.P. Agarwal, P.J.Y. Wong, A. Zafer, On the oscillation of fractional differential equations. Fract. Calc. Appl. Anal.

12 THE ZEROS OF THE SOLUTIONS OF THE FRACTIONAL , No 2 (2012), ; DOI: /s ; at [8] H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math (2011), [9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). [10] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Res. Notes in Math. Ser., Vol Longman Scientific & Technical and John Wiley & Sons, Inc., Harlow and New York (1994). [11] V.S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 118 (2000), ; doi: /s (00) [12] V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus. Comput. Math. Appl. 59 No 5 (2010), ; doi: /j.camwa [13] J. Klafter, S.C. Lim, R. Metzler, Fractional Dynamics: Recent Advances. World Scientific, Singapore (2011). [14] Y. Luchko, Operational method in fractional calculus. Fract. Calc. Appl. Anal. 2, No 4 (1999), ; fcaa/. [15] F. Mainardi, Fractional relaxation-oscillation and fractional diffusionwave phenomena. Chaos Solitons Fractals 7 (1996), [16] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics. In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien/New York (1997), [17] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College, London & World Sci., Singapore (2010). [18] F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118 (2000), [19] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993). [20] G. Mittag-Leffler, Sur la nouvelle fonction E α (x). C.R. Acad. Sci. Paris 137 (1903), [21] G. Mittag-Leffler, Sur la representation analytique d une branche uniforme d une fonction monogene (cinquieme note). Acta Math. 29 (1905), [22] K.B. Oldham, J. Spanier, The Fractional Calculus. AcademicPress, New York (1974). [23] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).

13 22 J.-S. Duan, Z. Wang, S.-Z. Fu [24] A.V. Pskhu, On the real zeros of functions of Mittag-Leffler type. Mathematical Notes 77 (2005), [25] A. Wiman, Uber den fundamentalsatz in der teorie der funktionen E α (x). Acta Math. 29 (1905), [26] A. Wiman, Uber die nullstellen der funktionen E α (x). Acta Math. 29 (1905), School of Sciences Shanghai Institute of Technology Shanghai , P.R. CHINA Received: January 28, duanjs@sit.edu.cn; duanjssdu@sina.com 2 School of Mathematics and Information Sciences Zhaoqing University Zhaoqing, Guang Dong , P.R. CHINA Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 17, No 1 (2014), pp ; DOI: /s x