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Archives of Control Sciences
Volume 26(LXII), 2016
No. 3, pages 429–435
Explicit finite-difference scheme for the numerical
solution of the model equation of nonlinear hereditary
oscillator with variable-order fractional derivatives
ROMAN I. PAROVIK
The paper deals with the model of variable-order nonlinear hereditary oscillator based on
a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate
the stability and convergence of the difference scheme. It is argued that the approximation,
stability and convergence are of the first order, while the scheme is stable and converges to the
exact solution.
Key words: nonlinear hereditary oscillator, finite-difference scheme, Cauchy problem,
fractional derivatives, numerical experiment.
1. Introduction
The development of hereditary processes, i.e. processes with memory, has been re-
flected in a variety of applications in the last decade. V. V. Uchaikin [9] in his "Method
of Fractional Derivatives" devotes a whole chapter to such processes, including a de-
scription of hereditary oscillator proposed by Vito Volterra [10]. From the mathemat-
ical standpoint, hereditarity, or a memory effect of oscillator, can be demonstrated by
inserting integral operator with kernel, which is a memory function, into its model equa-
tions. If this kernel is represented by a power series form, the hereditary model equation
can be naturally transformed into differential equations with variable-order fractional
derivatives [4]. The theory of fractional calculus is quite well developed, and its main
provisions can be found in reference books [1, 8].
In this paper we consider the model of nonlinear hereditary oscillator with variable-
order derivatives. To do this, we construct an explicit finite-difference scheme for the
numerical solution of the corresponding Cauchy problem [2, 5], which will be explored
further.
The Author is with Physics and Mathematics Department, Vitus Bering Kamchatka State University,
Petropavlovsk-Kamchatsky, Russia. The Author is also with Laboratory of Physical Processes Modeling,
IKIR FEB RAS, Kamchatka Region, Paratunka, Russia. E-mail: romanparovik@gmail.com.
Received 02.05.2016.
10.1515/acsc-2016-0023
430 R. I. PAROVIK
2. Problem
Consider the following Cauchy problem.
∂β(t)
0tx(τ) + λ∂γ(t)
0tx(τ) + ωβ(t)sin(x(t)) = f(t),
x(0) = x0,˙x(0) = y(0),(1)
where
∂β(t)
0tx(τ) =
t
∫
0
¨x(τ)dτ
Γ(2−β(τ))(t−τ)β(τ)−1,∂γ(t)
0tx(τ) =
t
∫
0
˙x(τ)dτ
Γ(1−γ(τ))(t−τ)γ(τ)
are the operators of variable-order fractional derivatives 1 <β(t)<2 and 0 <γ(t)<1,
Γ(x)is the Euler gamma function, λ,ω,x0and y0are the given parameters, f(t)is the
external stimulus, t∈[0,T]is the process time; the dots over the decision function x(t)
mean the classical integer-value derivatives.
Note that problem (1) when β=2 and γ=1 transforms into the problem for clas-
sical nonlinear oscillator with friction and external force. Note also that the fractional
parameters βand γrepresent any confined functions.
3. Solution method
The solution to the Cauchy problem (1) in the general case cannot be ob-tained in
an explicit form. Therefore, we will seek the solution to this problem using the theory
of finite-difference schemes [7]. Let us construct an explicit finite-difference scheme.
We divide the segment [0,T]into Nequal parts with a constant step τ. Then x(tj) = xj,
tj=jτis the grid solution approximating the solution x(t)of the differential Cauchy
problem (1). The operators of the fractional variable-order derivatives are approximated
as follows [3].
∂β(t)
0tx(τ) =
j−1
∑
k=0
τ−βk
Γ(3−βk)[(k+1)2−βj−k2−βj](xj−k+1−2xj−k+xj−k−1)+O(τ2),
∂γ(t)
0tx(τ) =
j−1
∑
k=0
τ−γk
Γ(2−γk)[(k+1)1−γj−k1−γj](xj−k+1−xj−k)+O(τ).
(2)
Substituting relation (2) into equation (1), after some transformations, we come to
the following explicit finite-difference scheme.
EXPLICIT FINITE-DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION
OF THE MODEL EQUATION OF NONLINEAR HEREDITARY OSCILLATOR
WITH VARIABLE-ORDER FRACTIONAL DERIVATIVES 431
x1=τy0+x0,j=0,
xj+1=Ajxj−Bxj−1−B
j−1
∑
k=1
τ−βk
Γ(3−βk)pj
k(xj−k+1−2xj−k+xj−k−1)−
−C
j−1
∑
k=1
τ−γk
Γ(2−γk)qj
k(xj−k+1−xj−k)−µsin(xj) + ξfj,
A=2A0+B0
A0+B0,B=A0
A0+B0,C=λ
A0+B0,
µ=ωβj
A0+B0,ξ=1
A0+B0,A0=τ−β0
Γ(3−β0),B0=λτ−γ0
Γ(2−γ0),
pj
k= (k+1)2−βj−k2−βj,qj
k= (k+1)1−γj−k1−γj,j=1,...,N−1.
(3)
Note that scheme (3) has in its internal points the second order of approximation
from the formulas (2); however, due to the approximation in the boundary points, the
order is reduced to unity. This can be eliminated by approximating the values in the
boundary points in a special way, for example, inserting a dummy node [6]. For the
purposes of this paper we do not need to improve scheme (3). We just investigate its
stability and convergence by means of a numerical experiment.
Consider the following example. It can be shown that the Cauchy problem with
homogeneous initial conditions
∂β(t)
0tx(τ) + λ∂γ(t)
0tx(τ) = f(t)
f(t) = ωβ(t)sin(t2)+2
t
∫
0
dτ
Γ(3−β(τ))(t−τ)β(τ)−1+2
t
∫
0
τdτ
Γ(2−γ(τ))(t−τ)γ(τ),
x(0) = ˙x(0) = 0,
(4)
has an exact solution x(t) = t2. A.A. Samarskii [7] provides definitions of stability on
the right side of the equation and with initial data. The essence of these definitions can
be summarized as follows. The scheme is stable if a small perturbation introduced to the
right side or the initial data leads to a small change in the solution within the accuracy
of a constant.
432 R. I. PAROVIK
Let us carry out a numerical experiment. To do this we choose the following val-
ues of the control parameters of the Cauchy problem (4): N=1000, λ=1, ω=2,
β(t) = 2−0.006cos(3πt),γ(t) = 1−0.003cos(3πt),ε=104. We find the perturbed
and the unperturbed solutions to problem (4) according to scheme (3) and calculate their
maximum absolute value error. The results of the experiment are shown in Tabs 1 and 2.
Table 24: Stability with respect to the right side.
NMaximum error
10 1.05*10−5
50 1.2*10−5
250 1.3*10−5
500 1.3*10−5
1000 1.2*10−5
2000 1.2*10−5
2500 1.3*10−5
From Tab. 1 we can conclude that for the chosen values of the control parameters
and perturbation ε, explicit finite-difference scheme (3) is stable with respect to the right
side, since the maximum error does not exceed perturbation ε.
Table 25: Stability with respect to the initial data.
NMaximum error
10 1.633*10−4
50 1.634*10−4
250 1.633*10−4
500 1.633*10−4
1000 1.632*10−4
2000 1.636*10−4
2500 1.635*10−4
From Tab. 2 it can be concluded that the maximum error values do not practically
change with increasing the number of computational grid points Nand are commensu-
rate with perturbation ε. Therefore, in this case scheme (3) is stable with respect to the
initial data. Let us demonstrate the convergence of scheme (3) for the Cauchy problem
through a numerical experiment.
EXPLICIT FINITE-DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION
OF THE MODEL EQUATION OF NONLINEAR HEREDITARY OSCILLATOR
WITH VARIABLE-ORDER FRACTIONAL DERIVATIVES 433
We choose the following values of the control parameters: N=1000, λ=100, ω=2,
t∈(0,1)and β(t) = 1.8−0.001cos(3πt),γ(t) = 0.8−0.002cos(3πt).We need to find
the maximum absolute value error between the numerical and exact solutions depending
on step as well as calculate the experimental convergence order of the numerical solution
to the exact one. The results of the experiment are shown in Tab. 3.
Table 26: The convergence of scheme (3) to the exact solution.
NτMaximum error α
10 0.1 0.1172 0.93
20 0.05 0.0573 0.954
40 0.025 0.0219 1.035
80 0.0125 0.00075 1.11
From Tab. 3 it can be concluded that when reducing step τof the computational grid,
the maximum error decreases, while the values of the experimental convergence order
α=ln(maximum error)/ln(step)are close to unity. Therefore, we can infer that scheme
(3) converges to the exact solution with the first order (Fig.1).
4. Conclusion
We have studied the model of variable-order nonlinear hereditary oscillator based
on a numerical finite-difference scheme. The stability and convergence of the difference
scheme have been evaluated by numerical experiments. The results have shown that the
approximation, stability and convergence are of the first order, while the scheme is stable
and converges to the exact solution. Certainly, if necessary, scheme (3) can be improved
through proper approximation of the initial conditions. Also, using the double counting
method we can increase its accuracy. The next step in studying the hereditary nonlinear
model of an oscillating system will be the construction and analysis of phase trajectories,
as it was carried out in [3] for linear hereditary oscillators.
References
[1] A. M. NAK HUSHEV: Fractional calculus and its applications. Fizmatlit, Moscow.
2003.
[2] R.I. PAROVIK: Finite-difference schemes for fractal oscillator with variable frac-
tional orders. Bulletin of Kamchatka Regional Association "Educational Scientific
434 R. I. PAROVIK
Figure 1: The convergence of scheme (3) to the exact solution.
Center" (KRAUNC), Physical and Mathematical Sciences,2(11), (2015), 88-95.
DOI: 10.18454/2079-6641-2015-11-2-88-95.
[3] R.I. PAROVIK: Mathematical modeling of hereditary linear oscillators. Vitus
Bering Kamchatka State University, Petropavlovsk-Kamchatsky. 2015.
[4] R.I. PAROVIK: Mathematical modeling of the hereditary oscillator. Computer Re-
search and Modeling,7(5), (2015), 1001-1021.
[5] R.I. PAROVIK: On the numerical solution of fractal oscillator equation with time-
dependant fractional variable derivative. Bulletin of Kamchatka Regional Associa-
tion "Educational Scientific Center" (KRAUNC), Physical and Mathematical Sci-
ences,1(8), (2014), 60-65. DOI: 10.18454/2079-6641-2014-8-1-60-65.
[6] I.B. PETROV and A.I. LOBA NOV: Lectures on computational mathematics. BI-
NOM, Moscow. 2006.
[7] A.A. SAMARSKII: The theory of difference schemes. Nauka, Mos-cow. 1977.
EXPLICIT FINITE-DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION
OF THE MODEL EQUATION OF NONLINEAR HEREDITARY OSCILLATOR
WITH VARIABLE-ORDER FRACTIONAL DERIVATIVES 435
[8] S.G. SAMKO, A.A. KILBA S and O.I. MARICHEV: Integrals and derivatives of
fractional order and some of their applications. Nauka i Tekhnika, Minsk. 1987.
[9] V.V. UCHAIKIN: Method of fractional derivatives. Artishok, Ulya-novsk. 2008.
[10] V. VOLTERRA: Theory of functionals and of integral and integro-differential equa-
tions. Nauka, Moscow. 1982.
