A numerical scheme for solving multi-term fractional

Commun. Frac. Calc. 4 (1) (2013) 38 - 49

A numerical scheme for solving multi-term fractional differential equations M. Javidi ,*

N. Nyamoradi,

Department of Mathematics, Faculty of science, Razi University, Kermanshah 67149, Iran.

Received 11 Jan 2013; accepted 12 March 2013

Abstract Recently, Atanackovic and Stankovic [T.M. Atanackovic and B. Stankovic, On a numerical scheme for solving differential equations of fractional order, Mechanics Research Communications 35 (2008) 429 − 438] presented a new numerical scheme, which is computationally more efficient, to solve the fractional differential equations (FDEs). The method is based on a transformation of the original FDEs into a system of ordinary differential equations (ODEs). In this work, we show how the numerical approximation of the solution of a multi-term FDE can be calculated by reduction of the problem to a system of ODEs. We use fourth order Runge-Kutta (RK) formula for the numerical integration of the system of ODEs. Two examples are provided to illustrate the method. The results show the simplicity and the efficiency of the method. Keywords: Fractional differential equations; Fourth order Runge-Kutta formula; Numerical methods

1

Introduction

This paper concerns the numerical solution of multi-term fractional differential equations which have the general form D2 f (t) = F (t, f (t), Dα1 f (t), Dα2 f (t), · · · , Dαm f (t)),

(1)

subject to the initial conditions f (k) (0) = 0, k = 0, 1

(2)

where αi , i = 0, 1, · · · , m are real numbers such that 0 < αi < 1. The notion of fractional calculus appears in diverse fields of science and engineering. If the notion first appears during the 17th century, a precise definition has been proposed by RiemannLiouville and more recently by Caputo [1]. Fractional differential equations (FDEs) appeared in many fields of science and engineering, including viscoelastic [2-5], continuum mechanics [6], wave-diffusion equation [7], heat conduction equation and associated thermal stress [8], diffusion equation [9-10], kinetic equation [11], Fokker-planck equation [12]. Recently, Many researchers worked on the FDEs, Srivastava and Rai [13], proposed a new mathematical model, namely a multi-term fractional diffusion equation, for oxygen delivery through a capillary to tissues. Agrawal [14] developed a new formulation allows accurate computation of initial compliance and creep response of a system for a long duration. Kumar and Agrawal [15], proposed a numerical solution scheme for a class of fractional differential equations (FDEs). In this paper the total time is divided into a set of small intervals, and between ∗

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M. Javidi and N. Nyamoradi, Commun. Frac. Calc. 4 (1) (2013) 38 - 49

39

two successive intervals the unknown functions are approximated using quadratic polynomials. In [16] a method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE) proposed. Momani and Odibat [17], applied Adomian decomposition method for the solution of a time-fractional Navier-stokes equation in a tube. In [18], fractional diffusion equations of order υ ∈ (0, 2) are examined and solved under different types of boundary conditions. Recently, Many researchers worked on the multi-term FDEs. In [19], an algorithm based on a new modified homotopy perturbation method (MHPM), is presented to obtain approximate solutions of multi-term diffusion-wave equations of fractional order. In [20], two methods are used to solve this type of equations. The first is an analytical method; Adomian decomposition method (ADM). The second method is a proposed numerical method (PNM). This method is a modification of the numerical method introduced in [21, chapter 8]. In [22], an application of the new technique is applied to solve fractional differential equations. In this paper, we give a numerical method [23] for solving multi-term fractional differential equations based on transformation of the multi-term fractional differential equation into a system of ODEs. We use fourth order Runge-Kutta formula for the numerical integration of the system of ODEs.

2

Preliminaries

We begin by stating some preliminaries definitions from fractional calculus. There exist different approaches to fractional derivatives [21]. Defnition 1. A function f : R → R+ is said to be in the space Cµ , with µ ∈ R, if it can be written as f (x) = xp f1 (x) with p > µ, f1 (x) ∈ C[0, ∞) and it is said to be in the space Cµm if f (m) ∈ Cµ for S m ∈ N {0}. Defnition 2. The Riemann-Liouville fractional integral of f ∈ Cµ with order α > 0 and µ ≥ −1 is defined as: Z t 1 α J f (t) = (t − τ )α−1 f (τ )dτ, α > 0, t > 0, Γ(α) 0 (3) 0 J f (t) = f (t). m with order α > 0 and m ∈ Defnition 3. The Riemann-Liouville fractional derivative of f ∈ C−1 S N {0}, is defined as:

Dα f (t) =

dm m−α J f (t), m − 1 < α ≤ m, m ∈ N. dtm

m with order α > 0 and m ∈ N Defnition 4. The Caputo fractional derivative of f ∈ C−1 defined as: ( J m−α f (m) (t), m − 1 < α ≤ m, m ∈ N, Dα f (t) = dm f (t) α = m. dtm ,

(4) S

{0}, is

(5)

Defnition 5. A two-parameter Mittag-Leffler function is defined by the following series

Eα,β (t) =

P∞

tk k=0 Γ(αk+β) .

Consequently, we have E1,1 (t) = et and E1,1 (−t) = e−t .

(6)

40

3


Expansion formulas for the fractional derivatives

In order to solve (1) and (2) numerically, we may use different procedures (see [21] for example). Here, we shall use a method introduced by Atanackovic and Stankovic [24] to solve linear FDEs. Also the same authors [23] developed the method to solve nonlinear FDEs. In this paper we developed the same method to solve linear and nonlinear multi-term FDEs. Now we explain the method. It is well known that for an analytic function f(t) the Caputo fractional derivative Dα f (t), 0 < α < 1 defined as 1 Dα f (t) = Γ(1−α)

Rt

−α f 0 (τ )dτ, 0 (t − τ )

(7)

where f (1) (τ ) denote the first derivative of f (τ ). As stated in [24], Dα f (t) can be expanded in a power series of involving integer order derivatives as ¶ ∞ µ X tk−α α f (k) (t), D f (t) = k Γ(k + 1 − α) α

k=0

where

µ

¶

α k

=

(−1)(k−1) Γ(k − α) Γ(k + 1). Γ(1 − α)

But the above expansion is not useful for our purposes since it involves all derivatives of the function f of integer order. In this paper our intention is to use an expansion of Dα f (t) that will involve a function and a finite number of its integer order derivatives. Now from Eq. (7), by using the integral by part, we obtain (1)

(0) 1 Dα f (t) = Γ(2−α) [ ftα−1 +

Rt

1−α f (2) (τ )dτ ] 0 (t − τ )

(8)

By using the binomial formula ([25], pp. 217), we have 1−α

(t − τ )

=t

1−α

∞

∞

p=0

p=0

X X Γ(p − 1 + α) τ τ τ τ (1 − )1−α = t1−α (1 − α) (−1)p ( )p = t1−α ( )p , | | < 1. t t Γ(α − 1)p! t t

(9)

Substituting Eq. (8) into Eq. (7), we obtain R t P∞

0

(0) 1 1 Dα f (t) = Γ(2−α) [ ftα−1 + tα−1

Γ(p−1+α) τ p (2) (τ )dτ ]. p=0 Γ(α−1)p! ( t ) f

0

(10)

As a result, we can rewrite Eq. (10) as follows 0

(t) 1 1 [ ftα−1 + tα−1 Dα f (t) = Γ(2−α)

P∞

Γ(p−1+α) p=1 Γ(α−1)p!tp

Rt 0

τ p f (2) (τ )dτ ].

(11)

Using the integration by part, we get Z

t

Z 0

τ p f (2) (τ )dτ = tp f (t) − p

0

t

0

τ p−1 f (τ )dτ

0

Z 0

= tp f (t) − ptp−1 f (t) + p(p − 1) 0

(12) t

τ p−2 f (τ )dτ, p ≥ 2.


41

By substitution of Eq. (12) into Eq. (11), we obtain ∞

X Γ(p − 1 + α) 1 f (t) D f (t) = { α−1 [1 + ] Γ(2 − α) t Γ(α − 1)p! 0

α

p=1

(13) −[

α−1 f (t) + tα

∞ X p=2

Vp (t) Γ(p − 1 + α) f (t) ( α + p−1+α )]}. Γ(α − 1)(p − 1)! t t

where Vp (t) = −(p − 1)

Rt 0

τ p−2 f (τ )dτ, p = 2, 3, · · ·

(14)

with the following properties d p−2 f (t), dt Vp = −(p − 1)t

p = 2, 3, · · · .

(15)

We approximate Dα f (t) by using M terms in sums appearing in Eq. (13) as follows M

X Γ(p − 1 + α) 1 f (t) { α−1 [1 + ] Γ(2 − α) t Γ(α − 1)p! 0

Dα f (t) '

p=1

(16) −[

α−1 f (t) + tα

M X p=2

Vp (t) Γ(p − 1 + α) f (t) ( α + p−1+α )]} Γ(α − 1)(p − 1)! t t

Now by setting x1 (t) = f (t), x2 (t) = f (1) (t) and xp+1 (t) = Vp (t), p = 2, 3, · · · , we can rewrite Eq. (16) as follows M X x2 (t) Γ(p − 1 + α) 1 α { α−1 [1 + ] D x1 (t) ' Γ(2 − α) t Γ(α − 1)p! p=1

M

−[

X Γ(p − 1 + α) x1 (t) xp+1 (t) α−1 x1 (t) + ( + p−1+α )]} α t Γ(α − 1)(p − 1)! tα t

(17)

p=2

= Gα (x1 (t), x2 (t), · · · , xM +1 (t)).

4

Transform of multi-term FDE to system of ODEs Now by using Eq. (17) we can rewrite Eq. (1) as follows (2)

x1 (t) = F (t, x1 (t), Gα1 (x1 (t), x2 (t), · · · , xM +1 (t)), Gα2 (x1 (t), x2 (t),

(18)

· · · , xM +1 (t)), · · · , Gαm (x1 (t), x2 (t), · · · , xM +1 (t))), where M

X Γ(p − 1 + αi ) 1 x2 (t) Gαi (x1 (t), x2 (t), · · · , xM +1 (t)) = { α −1 [1 + ] Γ(2 − αi ) t i Γ(αi − 1)p! p=1

(19) M

−[

X Γ(p − 1 + αi ) x1 (t) xp+1 (t) αi − 1 x1 (t) + ( + p−1+α )]}. α t i Γ(α − 1)(p − 1)! tα t p=2

42


Therefore, from Eqs. (18) and (14), we can convert the multi-term FDE (1) into a system of ODEs as follows x01 (t) = x2 (t), x02 (t) = F (t, x1 (t), Gα1 (x1 (t), x2 (t), · · · , xM +1 (t)), Gα2 (x1 (t), x2 (t), · · · , xM +1 (t)), · · · , Gαm (x1 (t), x2 (t), · · · , xM +1 (t))), x03 (t) = −x1 (t),

(20)

x04 (t) = −2tx1 (t), .. . x0M +1 (t) = −(M − 1)tM −2 x1 (t),

subject to the initial conditions xk (0) = 0, k = 0, 1, · · · , M + 1. We solve the above system of ODEs by using the well known RK formula of order four with step size ∆t.

5

Numerical results

To demonstrate the effectiveness of the method this scheme, we consider two examples, one linear and one nonlinear. These examples are considered because the closed form solutions are available and they have also been solved using other numerical schemes. This allows one to compare the results obtained using this scheme with the analytical solution and the solutions obtained using other schemes. Example 1. As the first example, we consider a nonlinear multi-term FDE [26-27], which is defined as follows: aD2 f (t) + bDα2 f (t) + cDα1 f (t) + f (t)3 = g(t), 0 < α1 < α2 ≤ 1, 2b 2c t3 g(t) = 2at + t3−α2 + t3−α1 + e( ), Γ(4 − α2 ) Γ(4 − α1 ) 3 subject to the initial conditions f (0) = f 0 (0) = 0. It is easily verified that the exact solution of this problem is f (t) =

t3 . 3

(21)


43

From Eqs. (20), the multi-term FDE (21) is converted into a system of ODEs as follows x01 (t) = x2 (t), M

X Γ(p − 1 + α1 ) 1 1 x2 (t) x02 (t) = [g(t) − ex31 − c( { α1 −1 [1 + ] a Γ(2 − α1 ) t Γ(α1 − 1)p! p=1

M

−[

X Γ(p − 1 + α1 ) x1 (t) xp+1 (t) α1 − 1 ( + p−1+α1 )]} x (t) + 1 tα 1 Γ(α1 − 1)(p − 1)! tα1 t p=2

M

− b(

X Γ(p − 1 + α2 ) x2 (t) 1 { α2 −1 [1 + ] Γ(2 − α2 ) t Γ(α2 − 1)p! p=1

(22) M

−[

X Γ(p − 1 + α2 ) x1 (t) xp+1 (t) α2 − 1 x (t) + ( + p−1+α2 )]}), 1 tα 2 Γ(α2 − 1)(p − 1)! tα2 t p=2

x03 (t) = −x1 (t), x04 (t) = −2tx1 (t), .. . x0M +1 (t) = −(M − 1)tM −2 x1 (t),

subject to the initial conditions x1 (0) = x2 (0) = · · · = xM +1 (0) = 0, where x1 (t) = f (t), x2 (t) = f 0 (t). Table 1 compares the absolute errors and CPU times in the solution at t = 1 reported in [27] and obtained using this scheme for α1 = 0.00196, α2 = 0.07621, a = 1, b = 2, c = 0.5, e = 1, M = 9 and various values of ∆t. We showed the presented method with J-M in in this section. We observe that this scheme gives much lower errors and CPU times. Table 2 compares the absolute errors and CPU times in the solution reported in [27] and obtained using this scheme for α1 = 0.00196, α2 = 0.07621, a = 1, b = 2, c = 0.5, e = 1, M = 49, ∆t = 0.001 and various values of t. We observe that this scheme gives much lower errors and CPU times. Note that as the step size is reduced, the error is also reduced as expected. Table 3 compares the absolute errors times in the solution at t = 1 reported √ and CPU √ in [27] and obtained using this scheme for α1 = 55 , α2 = 22 , a = 1, b = 0.1, c = 0.2, e = 0.3, M = 9 and various values of ∆t. As can be seen from table 3, our scheme gives much lower errors and CPU times. Table 4 compares the absolute errors and CPU times in the solution reported in [27] and obtained √ √ 5 2 using our scheme for α1 = 5 , α2 = 2 , a = 1, b = 0.1, c = 0.2, e = 0.3, ∆t = 0.001, M = 49 and various values of t. As can be seen from Table 4, our scheme gives much lower errors and CPU times.

44


M ethod, CP U/∆t [27] CP U (s) J − M, M = 9 CP U (s)

0.1 0.066473 0.265 0.0062 0.0470

0.01 0.007849 0.375 1.54e − 4 0.3130

0.001 0.000797 95.063 9.41e − 5 2.9840

0.0005 0.000399 894.999 9.39e − 5 5.9380

Table 1. Comparison of absolute errors and CPU time reported in [27] and obtained using this scheme at t = 1

t/M ethod 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

J −M 1.96e − 7 4.17e − 7 8.52e − 7 1.99e − 6 4.72e − 6 1.03e − 5 2.01e − 5 3.46e − 5 5.05e − 5 5.64e − 5

[27] 0.00003934 0.00015354 0.00033056 0.00055244 0.00079775 0.00079775 0.00129004 0.00156444 0.00200641 0.00289864

Table 2. Comparison of absolute errors and CPU time reported in [27] and obtained using this scheme at ∆t = 0.001

M ethod, CP U/∆t [27] CP U (s) J − M, M = 9 CP U (s)

0.1 0.0830668 0.282 0.0098 0.0470

0.01 0.0093860 1.078 3.5819e − 4 0.2970

0.001 0.0008682 106.392 2.6876e − 4 2.9690

0.0005 0.0003888 952.907 2.6866e − 4 5.9060

Table 3. Comparison of absolute errors and CPU time reported in [27] and obtained using this scheme at t = 1

t/M ethod 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

J −M 2.21e − 7 9.16e − 7 3.69e − 6 1.17e − 5 2.99e − 5 6.54e − 5 1.26e − 4 2.19e − 4 3.47e − 4 4.91e − 4

[27] 0.000039 0.000157 0.000352 0.000613 0.000868 0.000686 0.001847 0.013453 0.053538 0.168659

Table 4. Comparison of absolute errors and CPU time reported in [27] and obtained using this scheme at ∆t = 0.001


45

−3

1

x 10

−4

α1=0.2 α1=0.4 α1=0.3 α1=0.8

t=0.5 t=1 t=1.5 t=2

−5

0.8

−6

Log (Absolute Error)

−8

−9

10

Absolute Error

−7

0.6

0.4

−10

−11

0.2

−12

−13

0

0

0.5

1

1.5

2 t

2.5

3

3.5

−14

4

Fig. 1. Absolute errors with ∆t = 0.01, M = 49, a = 1, b = 2, c = 0.5, e = 1 and α2 = 0.2

0

10

20

30

40

50 M

60

70

80

90

100

Fig. 2. Logarithm of absolute errors with ∆t = 0.01, a = 1, b = 0.1, c = 0.2, e = 0.3, α1 = √ √ 5 2 , α2 = 2 for various values of M and t 5 −3

2.5

−2

x 10

t=0.5 t=1 t=1.5 t=2

−4

α2=0.2 α2=0.4 α2=0.3 α2=0.8

2

Absolute Error

Log10(Absolute Error)

−6

−8

−10

1.5

1

−12

0.5 −14

−16

0

10

20

30

40

50 M

60

70

80

90

0 0.1

100

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

Fig. 3. Logarithm of absolute errors with ∆t = 0.01, a = 1, b = 2, c = 0.5, e = 1, α1 = 0.1, α2 = 0.2 for various values of M and t.

Fig. 4. Absolute errors with ∆t = 0.005, M = 10 in case 2.

In Fig. 1, we plot absolute errors with ∆t = 0.01, M = 49, a = 1, b = 2, c = 0.5, e = 1 and α2 = 0.2. In Fig. √ 2, we √plot the logarithm of absolute errors with ∆t = 0.01, a = 1, b = 0.1, c = 0.2, e = 0.3, α1 = 55 , α2 = 22 for various values of M and t. In Fig. 3, we plot the logarithm of absolute errors with ∆t = 0.01, a = 1, b = 2, c = 0.5, e = 1, α1 = 0.1, α2 = 0.2 for various values of M and t. Example 2. In this example, we consider a linear multi-term FDE with variable coefficient [26-27], which is defined as follows: D2 f (t) + b(t)Df (t) + c(t)α2 f (t) + e(t)Dα1 f (t) + k(t)f (t) = g(t), 0 < α1 < α2 ≤ 1, c(t) e(t) t2 g(t) = −a − b(t)t − t2−α2 − t2−α1 + k(t)(2 − ), Γ(3 − α2 ) Γ(3 − α1 ) 2

(23)

subject to the initial conditions f (0) = 2, f 0 (0) = 0.

(24)

It is easily verified that the exact solution of this problem is 2

f (t) = 2 − t2 .

(25)

46

M. Javidi and N. Nyamoradi, Commun. Frac. Calc. 4 (1) (2013) 38 - 49 −7 ∆ t=0.005 ∆ t=0.001 ∆ t=0.0005 ∆ t=0.0001

−7.5

−8

log10(Absolute Error)

−8.5

−9

−9.5

−10

−10.5

−11

−11.5

−12 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

Fig. 5. Logarithm of absolute errors with M = 20 for various values of t and ∆t in case 2.

By setting x(t) = f (t) − 2, from Eqs. (23)-(25), we have aD2 x(t) + b(t)Dx(t) + c(t)α2 x(t) + e(t)Dα1 x(t) + k(t)x(t) = h(t), 0 < α1 < α2 ≤ 1, (26) h(t) = g(t) − 2k(t), subject to the initial conditions x(0) = x0 (0) = 0. It is easily verified that the exact solution of this problem is x(t) = −

t2 . 2

Now we put x1 (t) = x(t) and x2 (t) = x0 (t). By using Eqs. (20) the multi-term FDE (26) is converted into a system of ODEs as follows x01 (t) = x2 (t), M

x02 (t) = g(t) − 2k(t) − k(t)x1 − (

X Γ(p − 1 + α1 ) e(t) x2 (t) { α1 −1 [1 + ] Γ(2 − α1 ) t Γ(α1 − 1)p! p=1

M

M

p=2

p=1

X Γ(p − 1 + α1 ) x1 (t) xp+1 (t) X Γ(p − 1 + α2 ) c(t) x2 (t) α1 − 1 ( α + p−1+α1 )]) − ( { α2 −1 [1 + ] − [ α1 x1 (t) + t Γ(α1 − 1)(p − 1)! t1 t Γ(2 − α2 ) t Γ(α2 − 1)p! M

−[

X Γ(p − 1 + α2 ) x1 (t) xp+1 (t) α2 − 1 + p−1+α2 )]}), x (t) + ( 1 α t 2 Γ(α2 − 1)(p − 1)! tα2 t p=2

x03 (t) = −x1 (t), x04 (t) = −2tx1 (t), x0M +1 (t) = −(M − 1)tM −2 x1 (t) (27)


47

subject to the initial conditions x1 (0) = x2 (0) = · · · = xM +1 (0) = 0. We consider two cases for the functions and constants which are defined in Eq. (27): case 1: a = 0.1, b(t) = t, c(t) = t + 1, e(t) = t2 , k(t) = (t + 1)2 , α1 =√0.781, α2√= 0.891. √ case 2: a = 5, b(t) = t, c(t) = t2 − t, e(t) = 3t, k(t) = t3 − t, α1 = 707 , α2 = 1313 . Now we solve the system of ODEs (27) by using RK formula of order 4. In table 5 we list absolute errors in the solution using this scheme in case 1 for various values of t and S = M + 1. For this case, we take ∆t = 0.001. Note that as the number of equations (S)is increased, the error is decreased in table 5, as expected. In table 6 we list absolute errors in the solution using this scheme with S = 10 for various values of ∆t and t in the case 1. In the rows 1-3 we note that as the step size is reduced, the error is also reduced. t/S 0.2 0.6 1 1.4 1.8 2.0

10 1.4292e − 4 8.42e − 4 0.0022 0.0042 0.0067 0.0081

20 1.1203e − 4 4.13e − 4 0.0010 0.0019 0.0030 0.0036

40 9.40e − 5 2.14e − 4 4.77e − 4 8.53e − 4 0.0013 0.0016

80 4.28e − 5 9.72e − 5 2.16e − 4 3.83e − 4 3.83e − 4 7.11e − 4

100 1.4379e − 5 4.57e − 5 1.50e − 4 2.85e − 4 4.49e − 4 5.41e − 4

Table 5. Absolute errors in the solution with S = 10 for various values of ∆t and t in the case 1

t/∆t 0.2 0.4 0.6 0.8 1.0

0.1 0.0112 0.0083 0.0066 0.0055 0.0036

0.05 0.0050 0.0041 0.0036 0.0036 0.0039

0.005 4.90e − 4 6.78e − 4 0.0010 0.0016 0.0023

0.001 1.42e − 4 3.96e − 4 8.42e − 4 0.0015 0.0022

0.0005 9.98e − 5 3.61e − 4 8.18e − 4 0.0014 0.0022

0.0001 6.54e − 5 3.34e − 4 7.97e − 4 0.0014 0.0022

Table 6. Absolute errors in the solution with S = 10 for various values of t and ∆t in the case 2 In Table 7 we list absolute errors in the solution with S = 10 for various values of ∆t and t in the case 2. As can be seen from this table, as the step size is reduced, the error is also reduced, as expected. t/∆t 0.2 0.4 0.6 0.8 1.0

0.005 0.0019 0.0039 0.0058 0.0075 0.0091

0.001 9.82e − 4 0.0020 0.0029 0.0038 0.0046

0.001 1.98e − 4 3.93e − 4 5.80e − 4 7.56e − 4 9.13e − 4

0.0005 9.93e − 5 1.96e − 4 2.90e − 4 3.76e − 4 4.55e − 4

0.0001 1.98e − 5 3.91e − 5 5.70e − 5 7.29e − 5 8.78e − 5

0.00005 9.94e − 6 1.94e − 5 2.79e − 5 3.49e − 5 4.18e − 5

Table 7. Absolute errors in the solution with S = 10 for various values of t and ∆t in the case 2 In Table 8 we list the ratio of the absolute error R = E(2∆t)/E(∆t) with S = 10 for various values of ∆t and t in case 2. As can be seen from this, R is for most part very close to 2, which indicate that the error is of the order 1.

48


t/∆t 0.2 0.4 0.6 0.8 1.0

0.005 1.9747 1.9876 1.9920 1.9945 1.9959

0.001 1.9950 1.9981 2.0004 2.0027 2.0034

0.0005 1.9976 2.0000 2.0017 2.0062 2.0075

Table 8. E(2∆t)/E(∆t) with S = 10 for various values of t and ∆t in the case 2 In Fig. 4, we plot absolute errors with ∆t = 0.005, M = 10 in the case 2. In Fig. 5, we plot the logarithm of absolute errors with M = 20 for various values of t and ∆t in case 2.

6

Conclusions

In this paper, we applied an expansion formula for fractional derivatives given by (16). It contains integer derivatives up to the finite order k and time moments of k-th derivative, given by (14). Firstly, by using Eq. (16) we convert the fractional differential equation to a system of ordinary differential equations of order integer. We use fourth order Runge-Kutta formulae for the numerical integration of the system of ODEs. Two examples, one linear and the other nonlinear , were solved to demonstrate the performance of the method. Results obtained using this scheme agrees with analytical solutions and the numerical results obtained using other schemes. It is shown that results converge as the step size is reduced. Also it is shown that results converge as the number of equations is increased. References 1 M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part 2, Geophys. J. R. Astr. Soc. 13 (1967) 529–539. 2 A. Schmidt and L. Gaul, Finite element formulation of viscoelastic constitutive equations using fractional time derivatives, Nonlinear Dynam. 29 (2002) 37–55. 3 R. Bagley and P. Torvik, Fractional calculus a different approach to the analysis of viscoelastically damped structures, AIAA J. 21 (1983) 741–748. 4 M. Nerantzaki and N.G. Babouskos, Vibrations of inhomogeneous anisotropic viscoelastic bodies described with fractional derivative models (submitted for publication). 5 A. Galucio, J.F. Deu and R. Ohayon, Finite element formulation of viscoelastic sandwich beams using fractional derivative operators, Comput. Mech. 33 (2004) 282–291. 6 A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, 1997, 223–276. 7 J.T. Katsikadelis, The fractional wave-diffusion equation in bounded inhomogeneous anisotropic media. An AEM solution, in: G.D. Manolis, D. Polyzos (Eds.), Advances in Boundary Element Methods: A Volume to Honor Professor Dimitri Beskos, Springer Science, Dordrecht, Netherlands, 2008 (Chapter 17). 8 Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Therm. Stres. 28 (2005) 83–102. 9 T.A.M. Langlands, Solution of a modified fractional diffusion equation, Physica A 367 (2006) 136–144. 10 W. Wyss, The fractional diffusion equation, J. Math. Phys. 27 (1986) 2782–2785. 11 A.I. Saichev and G.M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997) 753–764. 12 R. Metzler and T.F. Nonnenmacher, Space- and time-fractional diffusion and wave equations, fractional Fokkerplanck equations, and physical motivation, Chem. Phys. 284 (2002) 67–90. 13 V. Srivastava and K.N. Rai, A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues, Math. Comput. Model. 51 (2010) 616–624. 14 O.P. Agrawal, A numerical scheme for initial compliance and creep response of a system, Mech. Res. Commun. 36 (2009) 444–451. 15 P. Kumar and O.P. Agrawal, An approximate method for numerical solution of fractional differential equations, Sign. Proc. 86 (2006) 2602–2610. 16 J. Chen, F. Liu and V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl. 338 (2008) 1364–1377.


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