Research Article An Exact Solution of the Second-Order Differential Equation with the Fractional/Generalised Boundary Conditions

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1 Advances in Mathematical Physics Volume 218, Article ID , 9 pages Research Article An Eact Solution of the Second-Order Differential Equation with the Fractional/Generalised Boundary Conditions Mariusz Ciesielski 1 and Tomasz Blaszczyk 2 1 Institute of Computer and Information Sciences, Czestochowa University of Technology, Czestochowa, Poland 2 Institute of Mathematics, Czestochowa University of Technology, Czestochowa, Poland Correspondence should be addressed to Tomasz Blaszczyk; tomasz.blaszczyk@im.pcz.pl Received 3 December 217; Accepted 26 February 218; Published 11 April 218 Academic Editor: Giorgio Kaniadakis Copyright 218 Mariusz Ciesielski and Tomasz Blaszczyk. 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. We analysed the initial/boundary value problem for the second-order homogeneous differential equation with constant coefficients in this paper. The second-order differential equation with respect to the fractional/generalised boundary conditions is studied. We presented particular solutions to the considered problem. Finally, a few illustrative eamples are shown. 1. Introduction The second-order differential equations provide an important mathematical tool for modelling the phenomena occurring in dynamicalsystems.eamplesoflinearornonlinearequations appear in almost all of the natural and engineering sciences and arise in many fields of physics. Many scientists have studied various aspects of these problems, such as physical systems described by the Duffing equation [1], noncommutative harmonic oscillators [2], oscillators in quantum physics [3], the dynamic properties of biological oscillators [4], the analysis of single and coupled low-noise microwave oscillators [5], the Mathieu oscillator [6], the relativistic oscillator [7], or the Schrodinger type oscillator [8]. Classical differential equations are defined by using the integer order derivatives. In recent years, the class of differential equations containing fractional derivatives (known as fractionaldifferentialequations)havebecomeanimportant topic. There are two approaches to obtain these types of equations. The first one is to replace the integer order derivative in classical differential equations with a fractional derivative (see, e.g., [9 14]). The second approach is a generalisation of a method known in classical and quantum mechanics, where the differential equations are obtained from conservative Lagrangian or Hamiltonian functions. These equations are known in the literature as fractional Euler-Lagrange equations, and they contain both the left and right fractional derivatives. New mechanics models for nonconservative systems, in terms of fractional derivatives, were developed by Riewe in [15, 16] and etended by Klimek [17, 18] and Agrawal [19, 2]. Since then, many authors have studied the fractional differential equationsofthevariationaltype(see[21 26]). In contrast to the above-mentioned references, where the authors analysed the integer order differential equations with classical boundary/initial conditions or fractional differential equations with the Dirichlet or natural boundary conditions, in this paper we consider the second-order differential equation with the fractional/generalised boundary conditions. 2. Statement of the Problem In this paper, we solve the second-order differential equation D 2 y () ±k 2 y () =, [a, b], k > (1) with respect to the following fractional/generalised boundary conditions: Φ(D α 1 y () a +,Dα 2 =b y () a +,Dβ 1 =b y () b, =a (2) D β 2y () b )=, =a

2 2 Advances in Mathematical Physics where α 1,α 2,β 1,β 2 [, 2], α 1 = α 2, β 1 = β 2,andthe operators D α a +,Dα b denote the left and right Riemann- Liouville derivatives, defined, respectively, by [27] D α a +y () fl 1 Γ (n α) d n y (τ) a ( τ) α n+1 dτ, for α R \ N, n = [α] +1 D n y () for α = n N D α () n d n b b y () fl Γ (n α) d n y (τ) (τ ) α n+1 dτ, for α R \ N, n = [α] +1 () n D n y (), for α=n N d n (3) and D n d n /d n. Let us consider two particular cases of (1). Case i. Here, we study the following differential equation: D 2 y () +k 2 y () =. (4) It is well known that (4) has a general solution, for k given by =, y () =C 1 sin (k) +C 2 cos (k). (5) The solution (5) contains two arbitrary independent constants of integration C 1 and C 2.Aparticularsolutioncanbe derived from the general solution by applying the set of initial or boundary conditions. The fractional differentiation of general solution (5) (using the left-side fractional operator) gives us D α a +y () =Dα a + (C 1 sin (k) +C 2 cos (k)) =C 1 D α a + sin (k) +C 2D α a + cos (k) (6) and differentiation by using the right-side operator leads to D α b y () =Dα b (C 1sin (k) +C 2 cos (k)) =C 1 D α b sin (k) +C 2D α b cos (k). (7) Now, we formulate the following properties for Riemann- Liouville derivatives of the sine and cosine functions. Property 1 (the left-sided Riemann-Liouville fractional derivatives of the sine and cosine functions). Let α and k =. Then, the following relations hold: D α a D α a ( a) α () i (k ( a)) 2i+1 () i (k ( a)) 2i (cos (ka) + sin (ka) ) for α> α N + sin (k) = Γ (2i+1 α) k n sin (k + nπ 2 ) for α=n N ( a) α () i (k ( a)) 2i () i (k ( a)) 2i+1 (cos (ka) sin (ka) ) for α> α N +cos (k) = Γ (2i+ 1 α) k n cos (k + nπ 2 ) for α=n N. (8) Property 2 (the right-sided Riemann-Liouville fractional derivatives of the sine and cosine functions). Let α and k =. Then, the following relations hold: D α b D α b (b ) α () i (k (b )) 2i () i (k (b )) 2i+1 (cos (kb) + sin (kb) ) for α> α N sin (k) = Γ (2i+1 α) () n k n sin (k + nπ 2 ) for α=n N (b ) α () i (k (b )) 2i () i (k (b )) 2i+1 (cos (kb) + sin (kb) ) for α> α N cos (k) = Γ (2i+1 α) () n k n cos (k + nπ 2 ) for α=n N. (9)

3 Advances in Mathematical Physics 3 Proof (Properties 1 and 2). We use Taylor s series epansions of sine and cosine functions [28] sin () = () i 2i+1 Γ (2i+ 2), () i 2i cos () = Γ (2i+ 1) (1) and properties of the left and right-sided fractional derivatives of power functions [27] D α a + ( a)m = D α b (b )m = Γ (m+1) Γ (m+1 α) ( a)m α, Γ (m+1) Γ (m+1 α) (b )m α, α, m> α, m>. (11) Also we apply the known fundamental trigonometric identities sin (k) = sin (k ( a) +ka) = sin (k ( a)) cos (ka) + cos (k ( a)) sin (ka) cos (k) = cos (k ( a) +ka) = cos (k ( a)) cos (ka) sin (k ( a)) sin (ka) sin (k) = sin (k (b ) kb) = sin (k (b )) cos (kb) + cos (k (b )) sin (kb) cos (k) = cos (k (b ) kb) (12) where D α a +sin (k ( a)) =D α a ( () i + Γ (2i+ 2) (k ( a))2i+1 ) = () i k 2i+1 Γ (2i+ 2) Dα a + ( a)2i+1 () i k 2i+1 = Γ (2i+ 2 α) ( a)2i+1 α = ( a) α () i (k ( a)) 2i+1 and in a similar way we obtain D α a cos (k ( a)) = () i (k ( a)) 2i + ( a) α Γ (2i+ 1 α) D α b sin (k (b )) = (b ) α () i (k (b )) 2i+1 (17) (18) (19) D α b cos (k (b )) = () i (k (b )) 2i (b ) α. (2) Γ (2i+ 1 α) Finally, putting (17) (2) into (13) (16), we obtain the formulas in Properties 1 and 2. Remark 3. Note that the infinite series included in formulas(17) (2)canbeepressedbyformulascontainingthe Mittag-Leffler function. This observation leads us to the following epressions: D α a +sin (k ( a)) Then = cos (k (b )) cos (kb) + sin (k (b )) sin (kb). D α a +sin (k) = cos (ka) Dα a +sin (k ( a)) (13) + sin (ka) D α a +cos (k ( a)) D α a +cos (k) = cos (ka) Dα a +cos (k ( a)) (14) sin (ka) D α a +sin (k ( a)) D α b sin (k) = cos (kb) Dα b sin (k (b )) (15) + sin (kb) D α b cos (k (b )) D α b cos (k) = cos (kb) Dα b cos (k (b )) + sin (kb) D α b sin (k (b )), (16) =k( a) 1 α E 2,2 α ( k 2 ( a) 2 ) D α a +cos (k ( a)) = ( a) α E 2,1 α ( k 2 ( a) 2 ) D α b sin (k (b )) =k(b ) 1 α E 2,2 α ( k 2 (b ) 2 ) D α b cos (k (b )) = (b ) α E 2,1 α ( k 2 (b ) 2 ), (21) where E p,q denotes the Mittag-Leffler function defined in [27] E p,q () =, p,q R, p >, R. (22) Γ(pi+q) i The above-mentioned notations can be useful in case when one uses the built-in function (the Mittag-Leffler function) in a mathematical software.

4 4 Advances in Mathematical Physics Case ii. The second problem has the following form: D 2 y () k 2 y () =. (23) In this case the general solution of (23), for k =,isgivenby y () =C 1 sinh (k) +C 2 cosh (k). (24) The fractional differentiation of solution (24) (using the leftside operator) gives D α a +y () =Dα a + (C 1 sinh (k) +C 2 cosh (k)) =C 1 D α a + sinh (k) +C 2D α a + cosh (k) (25) andfortheright-sidederivativewehave D α b y () =Dα b (C 1 sinh (k) +C 2 cosh (k)) =C 1 D α b sinh (k) +C 2D α b cosh (k). (26) Net, we formulate the following properties for Riemann- Liouville derivatives of the hyperbolic sine and hyperbolic cosine functions. Property 4 (the left-sided Riemann-Liouville fractional derivatives of the hyperbolic sine and hyperbolic cosine functions). Let α and k =. Then, the following relations hold: D α a + sinh (k) = D α a + cosh (k) = ( a) α (cosh (ka) k n cosh (k) for n = 1, 3, 5,... sinh (k) for n =, 2, 4,... (k ( a)) 2i+1 + sinh (ka) ( a) α (k ( a)) 2i (cosh (ka) Γ (2i+1 α) + sinh (ka) k n sinh (k) for n = 1, 3, 5,... cosh (k) for n =, 2, 4,... (k ( a)) 2i Γ (2i+ 1 α) ) for α> α N for α=n N. (k ( a)) 2i+1 ) for α> α N for α=n N (27) Property 5 (the right-sided Riemann-Liouville fractional derivatives of the hyperbolic sine and hyperbolic cosine functions). Let α and k =. Then, the following relations hold: (b ) α (k (b )) 2i+1 ( cosh (kb) D α Γ (2i+ 2 α) + sinh (kb) (k (b )) 2i Γ (2i+1 α) ) for α> α N b sinh (k) = () n k n cosh (k) for n = 1, 3, 5,... for α = n N sinh (k) for n =, 2, 4,... (b ) α (k (b )) 2i (cosh (kb) D α Γ (2i+1 α) sinh (kb) (k (b )) 2i+1 ) for α> α N b cosh (k) = () n k n sinh (k) for n = 1, 3, 5,... for α=n N. cosh (k) for n =, 2, 4,... (28) Proof (Properties 4 and 5). Here we apply Taylor s series epansions of the hyperbolic sine and cosine functions [28] sinh () = Γ (2i+ 2), 2i+1 cosh () = Γ (2i+ 1) and the trigonometric identities 2i (29) sinh (k) = sinh (k ( a) +ka) = sinh (k ( a)) cosh (ka) + cosh (k ( a)) sinh (ka) cosh (k) = cosh (k ( a) +ka) = cosh (k ( a)) cosh (ka) + sinh (k ( a)) sinh (ka)

5 Advances in Mathematical Physics 5 sinh (k) = sinh (k (b ) kb) = sinh (k (b )) cosh (kb) + cosh (k (b )) sinh (kb) cosh (k) = cosh (k (b ) kb) Remark 6. In formulas (32), the infinite series can be also epressed by using the Mittag-Leffler function, and we obtain D α a +sinh (k ( a)) =k( a) 1 α E 2,2 α (k 2 ( a) 2 ) = cosh (k (b )) cosh (kb) sinh (k (b )) sinh (kb). (3) D α a +cosh (k ( a)) = ( a) α E 2,1 α (k 2 ( a) 2 ) D α b sinh (k (b )) =k(b ) 1 α E 2,2 α (k 2 (b ) 2 ) (33) Then D α b cosh (k (b )) = (b ) α E 2,1 α (k 2 (b ) 2 ). where D α a +sinh (k) = cosh (ka) Dα a +sinh (k ( a)) + sinh (ka) D α a +cosh (k ( a)) D α a +cosh (k) = cosh (ka) Dα a +cosh (k ( a)) + sinh (ka) D α a +sinh (k ( a)) D α b sinh (k) = cosh (kb) Dα b sinh (k (b )) + sinh (kb) D α b cosh (k (b )) D α b cosh (k) = cosh (kb) Dα b cosh (k (b )) sinh (kb) D α b sinh (k (b )), (31) 3. Eamples of the Determination of Particular Solutions The boundary conditions, written in the general form (2), can be used in many combinations. Now, we show three selected eamples. Other combinations of the particular boundary conditions can be easily adopted by the reader (in a similar way). Eample 7. Equation(4)withthefollowingboundaryconditions given on both sides of the domain: b y () =a =L 1 D α 1 a + y () =b =L 2. (34) We substitute the general solution (5) into (34) and we have D α a +sinh (k ( a)) = (k ( a)) 2i+1 ( a) α D α a +cosh (k ( a)) = (k ( a)) 2i ( a) α Γ (2i+1 α) D α b sinh (k (b )) = (k (b )) 2i+1 (b ) α D α b cosh (k (b )) = (k (b )) 2i (b ) α Γ (2i+1 α). (32) Finally, putting (32) into (31), we obtain the formulas in Properties 4 and 5. C 1 b sin (k) =a +C 2 b cos (k) =a =L 1 C 1 D α 1 a + sin (k) =b +C 2D α 1 a + cos (k) =b =L 2. (35) The independent constants of integration C 1 and C 2 can be determined from the solution of the following system of equations: [ C 1 ]=[ C 2 [ b sin (k) =a D α 1 a + sin (k) =b [ L 1 L 2 ]. cos (k) b =a ] cos (k) a + =b ] D α 1 The analytical solution of (36) is of the form (36) C 1 = C 2 = L 1 D α 1 a + cos (k) =b L 2 b cos (k) =a b sin (k) =a Dα 1 a + cos (k) =b Dα 1 a + sin (k) =b Dβ 1 b cos (k) =a L 1 D α 1 a + sin (k) =b +L 2 b sin (k) =a b sin (k) =a Dα 1 a + cos (k) =b Dα 1 a + sin (k) =b Dβ 1 b cos (k) =a. (37)

6 6 Advances in Mathematical Physics Eample 8. Equation (23) with the conditions given on the left side of the domain ( =a): this case corresponds to the initial value problem y () b =L 1 D β 2y () =a b =L =a 2, β 1 =β 2. (38) When we put the general solution (24) into (38), then we obtain the following linear system of equations: C 1 b sinh (k) =a +C 2 b cosh (k) =a =L 1 C 1 D β 2 b sinh (k) =a +C 2D β 2 b cosh (k) =a =L 2 thatcanbealsowritteninthematriform (39) Eample 9. Equation (4) with the following set of boundary conditions: μ 1 b y () =a +μ 2D β 2 b y () =a =L 1 D α 1 a + y () =b =L 2, μ 1,μ 2 R, μ 1 + μ 2 >. (41) This case corresponds to the generalisation of the Robin boundary condition given at =a.alsohereweputgeneral solution (5) into (41) and we obtain μ 1 (C 1 b sin (k) =a +C 2 b cos (k) =a ) [ C 1 ]=[ C 2 [ b sinh (k) =a D β 2 b sinh (k) =b [ L 1 L 2 ]. cosh (k) b =a ] D β 2 cosh (k) b =a ] (4) or +μ 2 (C 1 D β 2 b sin (k) =a +C 2D β 2 b cos (k) =a ) =L 1 C 1 D α 1 a + sin (k) =b +C 2D α 1 a + cos (k) =b =L 2 (42) [ C 1 μ 1 sin (k) b ]=[ +μ =a 2D β 2 sin (k) b =a C 2 D α 1 sin (k) [ a + =b μ 1 cos (k) b +μ =a 2D β 2 cos (k) b =a D α 1 cos (k) a + =b ] ] [ L 1 L 2 ] (43) from which we can easily determine the constants of integration C 1 and C 2. Remark 1. One can note that the fractional boundary conditions D β b =a and D α a + =b for integral values of parameters α and β take the classical forms of boundary conditions; this means D n b =a () n y (n) (a) and D n a + =b y (n) (b). Inparticular,itshouldbenoted that the difference occurs, among others, in the boundary condition D 1 b =a y (a) andthisformshouldbe takenintoaccount. 4. Eample of Solutions Onthebasisoftheproposedmethod,tofindtheparticular solutions to the considered equations (4) and (23), we calculated constants of integration C 1 and C 2 occurring in the general solutions that satisfy the sets of the given initial or boundary conditions (various combinations). In Figures 1 3, numerous eamples of solutions have been presented. 5. Conclusions The initial/boundary value problem for the second-order homogeneous differential equations with constant coefficients has been considered. The general solutions to these equations are widely known and involve arbitrary constants. Our aim was to find the particular solutions to this problem which satisfy the generalised boundary conditions. Such boundary conditions complement the set of classic boundary conditions (including the Dirichlet, Neumann, and Robin types) by including the fractional ones. The use of the fractional boundary conditions in the considered initial/boundary value problem required the fractional differentiation of the general solutions. We derived the formulas for the left- and right-sided Riemann-Liouville fractional derivatives of the sine, cosine, hyperbolic sine, and hyperbolic cosine functions that occur in the general solutions. On this basis, the integration constants in these solutions were determined analytically. On the plots, one can observe that the obtained results for the fractional boundary conditions are located between the solutions to the considered problem with respect to the classical (integer order) boundary conditions. Such behaviouroftheparticularsolutionsgivesnewpossibilitiesin physical phenomena modelling, like the harmonic oscillator modelling, among others. In the future, we plan to apply this approach to seek solutions to other types of the initial/boundary value problems, in particular to the four-order problems. Conflicts of Interest The authors declare that they have no conflicts of interest.

7 Advances in Mathematical Physics D 2 +k 2 = k = 2.5 a = 1 b = 3 D 1 b =a =5D a + =b =1 1 D 2 +k 2 = k=5a=1b=3 D 1 b =a =2D a + =b = =,.25,.5,.75, 1} D 2 +k 2 y () = k = 2.5 a = 1 b = 3 D 1.5 b =a =5D a + =b =1 5 =1, 1.25, 1.5, 1.75, 2} D 2 +k 2 = 2 k = 2.5 a = 1 b = 3 D.5 b 15 =a =1D a + =b = =1, 1.25, 1.5, 1.75, 2} 2 3 =,.25,.5,.75, 1} Figure 1: The particular solutions of (4) for selected boundary conditions (see details in the legends)..7 D 2 k 2 =.6 k=.1a=1b=3.5 D.1 b =a =.1 D b =a =.1 =.25,.5,.75, 1, 1.25, 1.5, 1.75} D 2 y () k 2 = k=.5a=1b=3 D b y () =a =1D b =a = =,.25,.5,.75, 1, 1.25, 1.5, 1.75} D 2 k 2 = k=2a=1b=3 D.5 b =a =1D a + =b = D 2 k 2 = k=2a=1b=3 D b =a =D a + =b =1.6.4 =,.25,.5,.75, 1, 1.25, 1.5, 1.75, 2}.6.4 =,.25,.5,.75, 1, 1.25, 1.5, 1.75, 2} Figure 2: The particular solutions of (23) for selected boundary conditions (see details in the legends).

8 8 Advances in Mathematical Physics 8 6 D 2 +k 2 = k = 2.5 a = 1 b = 3.5D.5 b =a +.5D1.5 b =a = 5D a + =b = D 2 +k 2 = k=5a=1b=3.5d.5 b =a +.5D1.5 b =a = 2 D a + =b = =1, 1.25, 1.5, 1.75, 2} =,.25,.5,.75, 1} Figure 3: The particular solutions of (4) for selected boundary conditions (see details in the legends). Acknowledgments The research is supported by Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology. References [1] M. J. Brennan and I. Kovaci, Eamples of physical systems described by the Duffing equation, in The Duffing Equation: Nonlinear Oscillators and their Behaviour, I.KovacicandM.J. Brennan, Eds., John Wiley & Sons, Ltd., 211. [2] A. Parmeggiani, Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction, vol of Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, 21. [3] P. Blaise and O. Henri-Rousseau, Quantum Oscillators, John Wiley & Sons, Inc., Hoboken, NJ, USA, 211. [4] T. Pavlidis, Biological Oscillators: Their Mathematical Analysis, Academic Press, New York, NY, USA and London, UK, [5] U.L.Rohde,A.K.Poddar,andG.Bock,The Design of Modern Microwave Oscillators for Wireless Applications: Theory and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, USA, 25. [6] C. Floris, Stochastic stability of damped Mathieu oscillator parametrically ecited by a gaussian noise, Mathematical Problems in Engineering,vol.212,ArticleID375913,18pages,212. [7] L.S.Osborne, Arelativisticoscillatormodelappliedtoleptonnucleon reactions, Physics Letters B, vol. 63, no. 4, pp , [8] J. M. Cornwall and G. Tiktopoulos, Functional Schrödinger equation approach to high-energy multiparticle scattering, Physics Letters B,vol.282,no.1-2,pp.195 2,1992. [9] B.N.Achar,J.W.Hanneken,T.Enck,andT.Clarke, Dynamics of the fractional oscillator, Physica A: Statistical Mechanics and its Applications,vol.297,no.3-4,pp ,21. [1] B. N. Narahari Achar, J. W. Hanneken, and T. Clarke, Damping characteristics of a fractional oscillator, Physica A: Statistical MechanicsanditsApplications,vol.339,no.3-4,pp , 24. [11] Y. Kang and X. Zhang, Some comparison of two fractional oscillators, Physica B: Condensed Matter, vol.45,no.1,pp , 21. [12] S. Kukla and U. Siedlecka, Fractional heat conduction in multilayer spherical bodies, Applied Mathematics and Computational Mechanics,vol.15,no.4,pp.83 92,216. [13] Y. Povstenko and J. Klekot, The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption, Applied Mathematics and Computational Mechanics,vol.16,no.2,pp ,217. [14] A. A. Stanislavsky, Twist of fractional oscillations, Physica A: Statistical Mechanics and its Applications, vol.354,no.1-4,pp , 25. [15] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics,vol.53,no.2,pp ,1996. [16] F. Riewe, Mechanics with fractional derivatives, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics,vol.55, no. 3, part B, pp , [17] M. Klimek, Fractional sequential mechanics models with symmetric fractional derivative, Czechoslovak Physics,vol.51,no.12,pp ,21. [18] M. Klimek, Lagrangean and Hamiltonian fractional sequential mechanics, Czechoslovak Physics, vol. 52, no. 11, pp , 22. [19] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Mathematical Analysis and Applications,vol.272,no.1,pp ,22. [2] O. P. Agrawal, Fractional variational calculus and the transversality conditions, Physics A: Mathematical and General,vol.39,no.33,pp ,26. [21] D. Baleanu, J. H. Asad, and I. Petras, Fractional Bateman - Feshbach Tikochinsky oscillator, Communications in Theoretical Physics, vol. 61, no. 2, pp , 214. [22] T. Blaszczyk, A numerical solution of a fractional oscillator equation in a non-resisting medium with natural boundary conditions, Romanian Reports in Physics, vol. 67, no. 2, pp , 215. [23] T. Blaszczyk and M. Ciesielski, Fractional oscillator equation: analytical solution and algorithm for its approimate computation, Vibration and Control,vol.22,no.8,pp , 216. [24] M. Ciesielski and T. Blaszczyk, Numerical solution of nonhomogenous fractional oscillator equation in integral form, JournalofTheoreticalandAppliedMechanics,vol.53,no.4,pp , 215.

9 Advances in Mathematical Physics 9 [25] W. Liu, M. Wang, and T. Shen, Analysis of a class of nonlinear fractional differential models generated by impulsive effects, Boundary Value Problems,vol.175,217. [26] Y. Xu and O. P. Agrawal, Models and numerical solutions of generalized oscillator equations, Vibration and Acoustics, vol. 136, no. 5, Article ID 515, 214. [27] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 26. [28] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Elsevier, London, UK, 6th edition, 25.

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