In this paper, we use three operators called K-, A-, and B-operators to define the equation of motion of an oscillator. In contrast to fractional integral and derivative operators which use fractional power kernels or their variations in their definitions, the K-, A-, and B-operators allow the kernel to be arbitrary. In the case, when the kernel is a power kernel, these operators reduce to fractional integral and derivative operators. Thus, they are more general than the fractional integral and derivative operators. Because of the general nature of the K-, A-, and B-operators, the harmonic oscillators are called the generalized harmonic oscillators. The equations of motion of a generalized harmonic oscillator are obtained using a generalized Euler–Lagrange equation presented recently. In general, the resulting equations cannot be solved in closed form. A finite difference scheme is presented to solve these equations. To verify the effectiveness of the numerical scheme, a problem is considered for which a closed form solution could be found. Numerical solution for the problem is compared with the analytical solution, which demonstrates that the numerical scheme is convergent.

References

1.
Strogatz
,
S. H.
,
2001
,
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
,
Westview Press
,
Boulder, CO
.
2.
Hirsch
,
M.
,
Smale
,
S.
, and
Devaney
,
R.
,
2004
,
Differential Equations, Dynamical Systems, and an Introduction to Chaos
,
Academic Press
,
Amsterdam
.
3.
David
,
F.
, and
Hendry
,
W.
,
1968
, “
Harmonic-Oscillator Model for Baryons
,”
Phys. Rev.
,
173
(
5
), pp.
1720
1729
.10.1103/PhysRev.173.1720
4.
Dekker
,
H.
,
1981
, “
Classical and Quantum Mechanics of the Damped Harmonic Oscillator
,”
Phys. Rep.
,
80
(
1
), pp.
1
110
.10.1016/0370-1573(81)90033-8
5.
Kells
,
G.
,
Twamley
,
J.
, and
Heffernan
,
D.
,
2008
, “
Stability Analysis of the Kicked Harmonic Oscillator's Accelerator Modes
,”
Chaos, Solitons Fractals
,
36
(
3
), pp.
772
780
.10.1016/j.chaos.2006.07.017
6.
Chalykh
,
O.
, and
Oblomkov
,
A.
,
2000
, “
Harmonic Oscillator and Darboux Transformations in Many Dimensions
,”
Phys. Lett. A
,
267
(
4
), pp.
256
264
.10.1016/S0375-9601(00)00087-6
7.
Kreyszig
,
E.
,
2011
,
Advanced Engineering Mathematics
,
Wiley
,
Hoboken, NJ
.
8.
Tipler
,
P.
, and
Mosca
,
G.
,
2008
,
Physics for Scientists and Engineers
,
W.H. Freeman & Company
,
Cranbury, NJ
.
9.
Macfarlane
,
A.
,
1989
, “
On q-Analogues of the Quantum Harmonic Oscillator and the Quantum Group SU(2)q
,”
J. Phys. A: Math. Gen.
,
22
(
21
), pp.
4581
4588
.10.1088/0305-4470/22/21/020
10.
Mickens
,
R.
,
2001
, “
Mathematical and Numerical Study of the Duffing-Harmonic Oscillator
,”
J. Sound Vib.
,
244
(
3
), pp.
563
567
.10.1006/jsvi.2000.3502
11.
Diethelm
,
K.
,
2010
,
The Analysis of Fractional Differential Equations
,
Springer
,
Berlin
.
12.
Kilbas
,
A.
,
Srivastava
,
H.
, and
Trujillo
,
J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam
.
13.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego
.
14.
Agrawal
,
O. P.
,
2010
, “
Generalized Variational Problems and Euler-Lagrange Equations
,”
Comput. Math. Appl.
,
59
(
5
), pp.
1852
1864
.10.1016/j.camwa.2009.08.029
15.
Achar
,
B.
,
Hanneken
,
J.
,
Enck
,
T.
, and
Clarke
,
T.
,
2001
, “
Dynamics of the Fractional Oscillator
,”
Physica A
,
297
(
3–4
), pp.
361
367
.10.1016/S0378-4371(01)00200-X
16.
Achar
,
B.
,
Hanneken
,
J.
, and
Clarke
,
T.
,
2002
, “
Response Characteristics of a Fractional Oscillator
,”
Physica A
,
309
(
3–4
), pp.
275
288
.10.1016/S0378-4371(02)00609-X
17.
Achar
,
B.
,
Hanneken
,
J.
, and
Clarke
,
T.
,
2004
, “
Damping Characteristics of a Fractional Oscillator
,”
Physica A
,
339
(
3–4
), pp.
311
319
.10.1016/j.physa.2004.03.030
18.
Li
,
M.
,
Lim
,
S.
, and
Chen
,
S.
,
2011
, “
Exact Solution of Impulse Response to a Class of Fractional Oscillators and Its Stability
,”
Math. Prob. Eng.
,
2011
, p.
657839
.10.1155/2011/657839
19.
Stanislavsky
,
A.
,
2005
, “
Twist of Fractional Oscillations
,”
Physica A
,
354
(
2
), pp.
101
110
.10.1016/j.physa.2005.02.033
20.
Tofighi
,
A.
,
2003
, “
The Intrinsic Damping of the Fractional Oscillator
,”
Physica A
,
329
(
1–2
), pp.
29
34
.10.1016/S0378-4371(03)00598-3
21.
Kang
,
Y.
, and
Zhang
,
X.
,
2010
, “
Some Comparison of Two Fractional Oscillators
,”
Physica B
,
405
(
1
), pp.
369
373
.10.1016/j.physb.2010.04.036
22.
Um
,
C.
,
Yeon
,
K.
, and
George
,
T.
,
2002
, “
The Quantum Damped Harmonic Oscillator
,”
Phys. Rep.
,
362
(
2–3
), pp.
63
192
.10.1016/S0370-1573(01)00077-1
23.
Riewe
,
F.
,
1996
, “
Nonconservative Lagrangian and Hamiltonian Mechanics
,”
Phys. Rev. E
,
53
(
2
), pp.
1890
1899
.10.1103/PhysRevE.53.1890
24.
Riewe
,
F.
,
1997
, “
Mechanics With Fractional Derivatives
,”
Phys. Rev. E
,
55
(
3
), pp.
3581
3592
.10.1103/PhysRevE.55.3581
25.
Klimek
,
M.
,
2001
, “
Fractional Sequential Mechanics—Models With Symmetric Fractional Derivative
,”
Czech. J. Phys.
,
51
(
12
), pp.
1348
1354
.10.1023/A:1013378221617
26.
Klimek
,
M.
,
2002
, “
Lagrangian and Hamiltonian Fractional Sequential Mechanics
,”
Czech. J. Phys.
,
52
(
11
), pp.
1247
1253
.10.1023/A:1021389004982
27.
Agrawal
,
O. P.
,
2002
, “
Formulation of Euler-Lagrange Equations for Fractional Variational Problems
,”
J. Math. Anal. Appl.
,
272
(
1
), pp.
368
379
.10.1016/S0022-247X(02)00180-4
28.
Agrawal
,
O. P.
,
2006
, “
Fractional Variational Calculus and the Transversality Conditions
,”
J. Phys. A: Math. Gen.
,
39
(
33
), pp.
10375
10384
.10.1088/0305-4470/39/33/008
29.
Blaszczyk
,
T.
,
Ciesielski
,
M.
,
Klimek
,
M.
, and
Leszczynski
,
J.
,
2011
, “
Numerical Solution of Fractional Oscillator Equation
,”
Appl. Math. Comput.
,
218
(
6
), pp.
2480
2488
.10.1016/j.amc.2011.07.062
30.
Agrawal
,
O. P.
,
2008
, “
A General Finite Element Formulation for Fractional Variational Problems
,”
J. Math. Anal. Appl.
,
337
(
1
), pp.
1
12
.10.1016/j.jmaa.2007.03.105
31.
Xu
,
Y.
, and
Agrawal
,
O. P.
,
2013
, “
Models and Numerical Schemes for Generalized Van der Pol Equations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
12
), pp.
3575
3589
.10.1016/j.cnsns.2013.04.022
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