Download ارتباط بین (P – گراف ) ها و ماتروئیدها و ابرگراف ها

Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.7, No.4(27), Winter 2011, pp 65-70
ISSN: 2008-594x
Legendre Wavelets for Solving Fractional
Differential Equations
M. Soleymanivaraki *1, Hossein Jafari 2, M. Arab.Firoozjaee1
1Department
2Islamic
of Mathematics, University of Mazandaran, Babolsar, Iran
Azad University, Ayatollah Amoli Branch, Amol, Iran
Received: 8 May 2010
Accepted: 16 November 2010
Abstract
In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre
wavelets approximations. The continues Legendre wavelets constructed on [0, 1] are utilized as
a basis in collocation method. Illustrative examples are included to demonstrate the validity and
applicability of the technique.
Keywords: Legendre Wavelet, Fractional Differential Equations, Collocation Method.
1 Introduction
Ordinary differential equations involving fractional order derivatives are used to
model a variety of systems, of which an important engineering application lies in
viscoelastic damping [1, 8, 3]. Another important application of fractional derivatives
lies in control theory [6]. Linear ODEs with fractional order derivatives or integrals are
studied using Laplace [1] and Fourier transforms [8]. Nonlinear ODEs involving
fractional order derivatives can be solved numerically using several method [9, 11].
In this paper, we present a numerical technique to solve ODEs with fractional order
derivatives using legendre wavelet. There are many different types of definitions of
fractional calculus. For example, the Riemann-Liouville integral operator [12] of order
α is defined by
 1 x f (t )
  0, t  0

dt
I  f ( x)   ( )  ( x  t )1
(1)
0
  0.

f ( x)



and its fractional derivative of order α (α ≥ 0) is normally used:
( Dl f )( x)  (
d m m 
) (I
f )( x),
dx
(  0, m  1    m),
*Corresponding author
E-mail address: [email protected]
65
(2)
H.Jafari et al, Legendre Wavelets for Solving Fractional Differential Equations
where n is an integer. For Riemann-Liouvilles definition, one has
I  xv 
(v  1)
x v 
 (v  1   )
(3)
The Riemann-Liouville integral operator plays an important role in the development of
the theory of fractional derivatives and integrals. However, it has some disadvantages
for fractional differential equations with initial and boundary conditions. Therefore, we
adopt here Caputo’s definition [3, 10], which is a modification of Rieman-Liouville
definition:
x

1
f ( m ) (t )

0 ( x  t ) m1 dt

(
m


)

( D f )( x)  
 m f ( x)


x m
(  0, m  1    n, )
  m.
(4)
where m is an integer. Caputo’s integral operator has a useful property [3, 10]:
m 1
( I D f )( x)  f ( x)  


k 0
f ( k ) (0  ) k
x ,
k!
( x  0, m  1    m.)
(5)
where n is an integer. In this study, fractional differentiation is understood in the Caputo
sense because of its applicability to real–world problems. Wavelets theory is a relatively
new and an emerging area in mathematical research. It has been applied in a wide range
of engineering disciplines; particularly, wavelets are very successfully used in signal
analysis for waveform representations and segmentations, time–frequency analysis and
fast algorithms for easy implementation [4]. Wavelets permit the accurate representation
of a variety of functions and operators. Moreover, wavelets establish a connection with
fast numerical algorithms [2].
In the present article, we are concerned with the application of Legendre wavelets to
the numerical solution of FODEs. The method consists of converting of ordinary
differential equations involving fractional order to algebraic equations and expanding
the solution by Legendre wavelets with unknown coefficients. The article is organized
as follow: In Section 2, we describe the basic formulation of wavelets and Legendre
wavelets required for our subsequent development. Section 3 is devoted to the solution
of FODEs using integral operator matrix and product operator matrix and Legendre
wavelets. In Section 4, considering numerical examples our numerical finding reported
and the accuracy of the proposed scheme is demonstrate.
2 The Properties of Legendre wavelets
2.1 Wavelets and Legendre wavelets
Wavelets constitute a family of functions constructed from the dilation and
translation of a single function called the mother wavelet. When the dilation parameter a
and the translation parameter b vary continuously we have the following family of
continuous wavelets as [7]:
66
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.7, No.4(27), Winter 2011, pp 65-70

1
 a.b (t )  a 2  (
t b
)
a
a , b  R, a  0
(6)
If we restrict the parameters a and b to discrete values as a  a0 k , b  nb0 a0 k , a0  1, b0  0
and n , and k positive integers, we have the following family of discrete wavelets:
k
 k ,n (t )  a0 2  (a0 k t  nb0 ),
Where  k ,n (t ) form a wavelet basis for L2 (R) . In particular, when a 0  2 and b0  1
then  k , n (t ) forms an orthonormal basis [7].

 nm (t )   (k , n, m, t ) have
Legendre
wavelets
four
arguments;

n  2n  1, n  1,2,3,...,2 k 1 , k can assume any positive integer, m is the order for
Legendre polynomials and t is the normalized time. They are defined on the interval
[0, 1) as follows:

1 k2
m

2 L m (2k t  n),

2

 nm (t)  


0,
for
n 1
n 1
t k
k
2
2
,
(7)
otherwise
Where m  0,1,2,..., M  1, n  1,2,...,2 k 1 .
1
is for orthonormality, the dilation parameter is a  2  k and the
2

translation parameter b  n 2  k . Here, Lm (t ) are the well–known Legendre polynomials
of order m which are defined on the interval [−1, 1], and can be determined with the
aid of the following recurrence formulae:
The coefficient
m
L0 (t )  1,
L1 (t )  t ,
Lm 1 (t )  (
2m  1
m
)tLm (t )  (
) Lm 1 (t ),
m 1
m 1
m  1,2,3,...
The set of { nm (t )] are composed an orthonormal system for L2 [0,1]. [13]
2.2 Function approximation
A function f (t ) defined over [0, 1) may be expanded by Legendre wavelet series as
 
f (t)    c nm nm (t),
(8)
n 1 m  0
where
67
H.Jafari et al, Legendre Wavelets for Solving Fractional Differential Equations
cnm  f (t ), nm (t ) .
(9)
In (9), . denotes the inner product. If the infinite series in Eq. (8) is truncated, then
Eq.(8) can be written as
2 k 1 M 1
f (t )    cnm nm (t )  C T  (t ),
(10)
n 1 m 0
where C and  (t ) are 2 k 1 M  1 matrices given by
C(t )  [c10 , c11 ,..., c1M 1 , c20 ,..., c2M 1 ,..., c2k 1 o ,..., c2k 1 M 1 ]T
(11)
(t )  [ 10 (t ), 11 (t ),..., 1M 1 (t ), 20 (t ),..., 2M 1 (t ),..., 2k 1 o (t ),..., 2k 1 M 1 (t )]T
(12)
3 Solution of fractional differential equations
Consider the initial value problem given in Eq. (13).
D  x(t )  Nx(t )  g (t )
  0,
(13)
where N is the nonlinear operator and D is fractional derivative. In order to use Legendre wavelets, we first approximate x(t ) as
x(t )  C T (t )
(15)
where C and  (t ) are defined similarly to Eqs. (11) and (12). Then, we have:
D  C T (t )  NC T (t )  g (t ),
(16)
we now collocate Eq. (16) at 2 k 1 M points xi as
D  C T  (t i )  NC T  (t i )  g (t i )
(17)
We have chosen arbitrary collocation points. By solving equations system (17), we
obtained solution for Eq.(13).
4 Illustrative examples
We applied the method presented in this paper and solved various examples:
4.1 Example 1
Now, let us consider the Bagley-Torvik equation that governs the motion of a rigid
plate immersed in a Newtonian fluid [1].
68
Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.7, No.4(27), Winter 2011, pp 65-70
3
2
2
d x(t ) d x(t )
(18)
x(0)  1, x ' (0)  1

 x(t )  1  t
2
3
dt
2
dt
Following Diethelm and Ford [5]. We applied the method presented in this paper and
solved Eq. (18) with M  3 and K  3 For this equation we find:
c1, 0 =0.5625, c1,1 =0.0360844, c1, 2 =0. c 2, 0 =0.6875, c 2 ,1 =0.0360844, c 2, 2 =0.
c3, 0 =0.8125, c 3,1 =0.0360844, c3, 2 =0. c 4, 0 =0.9375, c 4 ,1 =0.0360844, c 4, 2 =0.
Now, we get:
x(t )  c1,0 1,0 (t )  c1,1 1,1 (t )  c1, 2 1, 2 (t )  c2,0 2,0 (t )  c2,1 2,1 (t )  c2, 2 2, 2 (t ) 
c3,0 3,0 (t )  c3,1 3,1 (t )  c3, 2 3, 2 (t )  c4,0 4,0 (t )  c4,1 4,1 (t )  c4, 2 4, 2 (t )  1  t
which is the exact solution.
Example 4.2
Next, following Momani and Odibat [11], we consider the composite fractional
oscillation equation
1
2
d x(t )
dt
1
2
 x(t )  t 2 
2
t 1.5  0 ,
(2.5)
(19)
with the initial conditions:
We applied the method presented in this paper and solved Eq. (19) with M  3 and
K  2 For this equation we find:
c1, 0 =0.0589256, c1,1 =0.051031, c1, 2 =0.0131762
c 2, 0 =0.412479, c 2 ,1 =0.153093, c 2, 2 =0.0131762
Now, we get:
x(t )  c1,0 1,0 (t )  c1,1 1,1 (t )  c1, 2 1, 2 (t )  c 2,0 2,0 (t )  c 2,1 2,1 (t )  c 2, 2 2, 2 (t )  t
which is the exact solution.
5 Conclusions
In this work, we solved fractional differential equations by using Legendre wavelets
in Collocation method. Legendre wavelets are well behaved basic functions that are
orthonormal on [0, 1]. Application of the wavelets allows the creation of more effective
and faster algorithms than the ordinary ones. Illustrative examples are included to
demonstrate the validity and applicability of the technique.
69
H.Jafari et al, Legendre Wavelets for Solving Fractional Differential Equations
References
[1] Bagley, R. L. and Torvik, P. J., Fractional calculus: A different approach to the analysis of
viscoelastically damped structures, AIAA Journal 21(5), (1983), 741-748.
[2] Beylkin G, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Commun.
Pure Appl. Math. 44 (1991) 141–183.
[3] Caputo M. Linear models of dissipation whose Q is almost frequency independent. Part II. J Roy
Austral Soc 13 (1967); 529–39.
[4] Chui. C.K, Wavelets: A Mathematical Tool for Signal Analysis, SIAM, Philadelphia, PA,1997.
[5] Diethelm K.J, Ford NJ. Numerical solution of the Bagley-Torvik equation. Bit 42: (2004); 490-507.
[6] Debnath L., Recent applications of fractional calculus to science and engineering, International
Journal of Mathematics and Mathematical Sciences (54), (2003), 34133442.
[7] Gu J.S, Jiang W.S, The Haar wavelets operational matrix of integration, Int. J. Syst. Sci. 27 (1996)
623-628.
[8] Gaul L., Klein, P., and Kempfle, S., Impulse response functions of an oscillator with fractional
derivative in damping description, Mechanics R. Comm. 16(5), 1989, 297-305.
[9] Kai Diethelm and Guido Walz, Numerical solution of fractional order differential equations by
extrapolation, Numerical Algorithms 16 (1997) 231-253.
[10] Luchko Yu, Gorenflo R. An operational method for solving fractional differential equations with the
Caputo derivatives. Acta Math Vietnamica 24(2): (1999); 207-33.
[11] Momani S, Odibat Z. Analytical approach to linear fractional partial differential equations arising in
fluid mechanics. Phys Lett A 355:(2006); 271–9.
[12] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional
differential equations, to methods of their solution and some of their applications. New York:
Academic Press; 1999.
[13] Razzaghi M, Yousefi S, The Legendre wavelets operational matrix of integration, International
Journal of Systems Science, 32(2001)495–502.
70