Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 586870, 7 pages
http://dx.doi.org/10.1155/2013/586870
Research Article
Numerical Simulations for the Space-Time Variable Order
Nonlinear Fractional Wave Equation
Nasser Hassan Sweilam1 and Taghreed Abdul Rahman Assiri2
1
2
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Department of Mathematics, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia
Correspondence should be addressed to Nasser Hassan Sweilam; n sweilam@yahoo.com
Received 3 March 2013; Revised 29 April 2013; Accepted 3 May 2013
Academic Editor: Chein-Shan Liu
Copyright © 2013 N. H. Sweilam and T. A. R. Assiri. 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.
The explicit finite-difference method for solving variable order fractional space-time wave equation with a nonlinear source term
is considered. The concept of variable order fractional derivative is considered in the sense of Caputo. The stability analysis and the
truncation error of the method are discussed. To demonstrate the effectiveness of the method, some numerical test examples are
presented.
1. Introduction
It is well known that the fractional calculus definitions are
extensions of the usual calculus definitions [1–8], where the
orders need not to be positive integers. On the other hand, the
variable order calculus is a natural extension of the constant
order (integer or fractional) calculus. In this sense, the order
may function in any variable such as time and space variables
or a system of other variables [9, 10]. In general, one can say
that this extension is introduced by Samko and Ross in [11],
where Marchaud fractional derivative and Riemann-Liouville
derivative are extended to the variable order cases the order
in this case is a function in the space variable only. Many
authors have introduced different definitions of variable order
differential operators, each of these with a specific meaning
to suit desired goals. These definitions such as RiemannLiouville, Grünwald, Caputo, Riesz [3, 12–16], and some notes
as Coimbra definition [17, 18].
Coimbra in [17] used Laplace transform of Caputo’s
definition of the fractional derivative as the starting point to
suggest a novel definition for the variable order differential
operator. Because of its meaningful physical interpretation,
Coimbra’s definition is better suited for modeling physical
problems. The variable order differentials are an important
tool to study some systems such as the control of nonlinear
viscoelasticity oscillator (for more details see [17–19] and
the references cited therein), where the order changes with
respect to a parameter or more parameters.
In the following, we present the basic definition for the
variable order fractional derivatives which we will use in this
paper.
Definition 1 (see [14]). The Caputo space variable order
derivative is defined as follows:
1
𝐷𝑥∝(𝑥,𝑡) 𝑢 (𝑥, 𝑡) =
Γ (𝑛 − ∝ (𝑥, 𝑡))
(1)
𝑥
1
𝜕𝑛 𝑢 (𝜉, 𝑡)
𝑑𝜉,
×∫
∝(𝑥,𝑡)−𝑛+1
𝜕𝜉𝑛
0 (𝑥 − 𝜉)
where 0 < ∝ (𝑥, 𝑡) < 1.
The main aim of this work is to use the explicit finite difference method (EFDM) to study numerically the following
nonlinear space-time variable order wave equation:
𝜕𝛽(𝑥,𝑡) 𝑢 (𝑥, 𝜏)
𝜕∝(𝑥,𝑡) 𝑢 (𝑥, 𝜏)
=
𝐵
𝑡)
+ 𝑓 (𝑢, 𝑥, 𝑡) ,
(𝑥,
𝜕𝜏𝛽(𝑥,𝑡)
𝜕𝑥∝(𝑥,𝑡)
(2)
1 < ∝ (𝑥, 𝑡) , 𝛽 (𝑥, 𝑡) ≤ 2,
subject to initial conditions
𝑢 (𝑥, 0) = 𝜑1 (𝑥) ,
𝑢𝑡 (𝑥, 0) = 𝜑2 (𝑥) ,
(3)
2
Journal of Applied Mathematics
and the following boundary conditions
𝑢 (0, 𝑡) = Ψ1 (𝑥) ,
𝑢 (𝑎, 𝑡) = Ψ2 (𝑥) ,
(4)
where 0 ≤ 𝑥 < 𝑎, 0 ≤ 𝑡 < 𝑇, 𝐵(𝑥, 𝑡) > 0 is a constant,
Ψ1 (𝑥), Ψ2 (𝑥) are smooth functions, and 𝑓(𝑢, 𝑥, 𝑡) is a nonlinear scour term that satisfies the Lipschitz condition, that
is,
󵄨
󵄨
󵄨
󵄨󵄨
(5)
󵄨󵄨𝑓 (𝑢1 , 𝑥, 𝑡) − 𝑓 (𝑢2 , 𝑥, 𝑡)󵄨󵄨󵄨 ≤ 𝐿 󵄨󵄨󵄨𝑢1 − 𝑢2 󵄨󵄨󵄨 ,
Now, pick two positive integers 𝑁, 𝑀 and define the step size
of space and time by ℎ, 𝜏, respectively, where ℎ = 𝑎/𝑀 and
𝜏 = 𝑇/𝑁. Also we introduce the following notations:
𝑥𝑖 = 𝑖ℎ,
𝑡𝑗 = 𝑗𝜏,
𝑗
𝑗
𝑗
(𝑥 − 𝜉)
𝑗
𝛽(𝑥,𝑡)
𝐷𝑡
𝜕 𝑢 (𝜉, 𝑡)
𝑑𝜉
𝜕𝜉2
𝑗
𝑗−1
Γ (3 −
𝑗
𝛽𝑖 ) 𝑘=0
𝑗
𝑅𝑖
𝑗−𝑘
− 2𝑢𝑖
(10)
=
𝑗
𝑗
𝑗
𝑧=𝑥−𝜉
𝐵𝑖 ℎ−∝𝑖
𝑗
Γ (3 − ∝𝑖 )
Γ (3 − 𝛽𝑖 )
𝑗
,
𝑄𝑖 =
𝑗
𝜏−𝛽𝑖
𝑗
𝑗
𝑗
𝑗
,
(11)
𝐻𝑖𝑘 = ((𝑘 + 1)2−𝛽𝑖 − 𝑘2−𝛽𝑖 ) .
Then, we can rewrite (2) in the following form:
𝑗−1
𝑗−𝑘+1
𝑗−𝑘
∑ (𝑢𝑖
− 2𝑢𝑖
𝑘=0
≈
−1
+𝑢 (𝑥 − (𝑘 + 1) ℎ, 𝑡)) (ℎ2 ) )
𝑗−𝑘−1
+ 𝑢𝑖
𝑖−1
𝑗 𝑗
𝑗
𝑄𝑖 𝑅𝑖 ∑ (𝑢𝑖−𝑘+1
𝑘=0
) 𝐻𝑖𝑘
(12)
−
𝑗
2𝑢𝑖−𝑘
+
𝑗
𝑗
𝑢𝑖−𝑘−1 ) 𝐺𝑘
+
𝑗 𝑗
𝑄𝑖 𝑓𝑖 ,
that is,
𝑧1−∝ 𝑑𝑧.
𝑗+1
𝑢𝑖
(6)
𝑗
𝑗−1
= 2𝑢𝑖 − 𝑢𝑖
𝑗−1
𝑗−𝑘+1
− ∑ (𝑢𝑖
Then,
𝑘=1
𝐷𝑥∝(𝑥,𝑡) 𝑢 (𝑥, 𝑡)
𝑗 𝑗
𝑖−1
𝑗−𝑘
− 2𝑢𝑖
𝑗
𝑗−𝑘−1
+ 𝑢𝑖
𝑗
) 𝐻𝑖𝑘
𝑗
𝑗
𝑗 𝑗
+ 𝑄𝑖 𝑅𝑖 ∑ (𝑢𝑖−𝑘+1 − 2𝑢𝑖−𝑘 + 𝑢𝑖−𝑘−1 ) 𝐺𝑘 + 𝑄𝑖 𝑓𝑖 ,
𝑘=0
ℎ2−∝(𝑥,𝑡)
Γ (3 − ∝ (𝑥, 𝑡))
𝑗+1
𝑢𝑖
= (2 −
𝑗
𝐻𝑖1 ) 𝑢𝑖
(13)
𝑀−2
𝑖−1
× ∑ ( (𝑢 (𝑥 − (𝑘 − 1) ℎ, 𝑡) − 2𝑢 (𝑥 − 𝑘ℎ, 𝑡)
𝑘=0
−1
+𝑢 (𝑥 − (𝑘 + 1) ℎ, 𝑡)) (ℎ2 ) )
× ((𝑘 + 1)
)
𝑗
𝑗
𝑘=0
2−∝(𝑥,𝑡)
𝑗−𝑘−1
+ 𝑢𝑖
𝐺𝑘 = ((𝑘 + 1)2−∝𝑖 − 𝑘2−∝𝑖 ) ,
× ∑ ( (𝑢 (𝑥 − (𝑘 − 1) ℎ, 𝑡) − 2𝑢 (𝑥 − 𝑘ℎ, 𝑡)
≈
𝑗−𝑘+1
∑ (𝑢𝑖
For simplicity, let us define
𝑖−1
𝑘ℎ
𝜏−𝛽𝑖
× ((𝑘 + 1)2 −𝛽𝑖 − 𝑘2 −𝛽𝑖 ) .
1
Γ (2 − ∝ (𝑥, 𝑡))
(𝑘+1)ℎ
(9)
𝑗
𝑗
∝(𝑥,𝑡)−2+1
𝑗
𝑢 (𝑥, 𝑡)
2
𝜕2 𝑢 (𝑥 − 𝑧, 𝜏)
×
𝑑𝑧,
𝜕𝑧2
×∫
𝑗
Γ (3 − 𝛼𝑖 ) 𝑘=0
By the same way, we have
𝑖−1 (𝑘+1)ℎ
1
=
𝑧1−∝(𝑥,𝑡)
∑∫
Γ (2 − ∝ (𝑥, 𝑡)) 𝑘=0 𝑘ℎ
≈
𝑗
∑ (𝑢𝑖−𝑘+1 − 2𝑢𝑖−𝑘 + 𝑢𝑖−𝑘−1 )
𝑗
=
1
𝑗
× ((𝑘 + 1)2 −𝛼𝑖 − 𝑘2 −𝛼𝑖 ) .
1
=
Γ (2 − ∝ (𝑥, 𝑡))
0
𝑖−1
ℎ−𝛼𝑖
=
𝐷𝑥∝(𝑥,𝑡) 𝑢 (𝑥, 𝑡)
×∫
𝑗
𝐷𝑥∝(𝑥,𝑡) 𝑢 (𝑥, 𝑡)
2. Discretization for EFDM
In this section, EFDM is used to study the model problem
(2), then the space-time solutions domain will be discretized.
The discrete form for the pervious Caputo derivative can be
written as follows:
(8)
for 𝑗 = 1, . . . , 𝑀,
𝑢𝑖 ≈ 𝑢(𝑥𝑖 , 𝑡𝑗 ), 𝐵𝑖 = 𝐵(𝑥𝑖 , 𝑡𝑗 ), and 𝑓𝑖 = 𝑓(𝑢𝑖 , 𝑥𝑖 , 𝑡𝑗 ). Then,
where the constant 𝐿 > 0 is called a Lipschitz constant for 𝑓.
𝑥
for 𝑖 = 1, 2, . . . , 𝑁,
−𝑘
2−∝(𝑥,𝑡)
).
(7)
𝑗−𝑘+1
− ∑ (𝐻𝑖𝑘−2 − 2𝐻𝑖𝑘−1 + 𝐻𝑖𝑘 ) 𝑢𝑖
𝑘=2
𝑗−2
− (𝐻𝑖
𝑗 𝑗
𝑗−1
𝑗−1
− 2𝐻𝑖 ) 𝑢𝑖1 − 𝐻𝑖 𝑢𝑖0
𝑖−1
𝑗
𝑗
𝑗
𝑗
𝑗 𝑗
+ 𝑄𝑖 𝑅𝑖 ∑ (𝑢𝑖−𝑘+1 − 2𝑢𝑖−𝑘 + 𝑢𝑖−𝑘−1 ) 𝐺𝑘 + 𝑄𝑖 𝑓𝑖 .
𝑘=0
Journal of Applied Mathematics
3
The previous equation can be expressed in the following
matrix form:
𝑈𝑖0 = 01 ,
𝑈𝑖1 = 𝑈𝑖0 + 𝜏02 ,
Proof. Let us define 𝑊𝑗+1 − 𝑈𝑗+1 = 𝜀𝑗+1 .
From (15) we have
(14)
𝑗+1
𝜀𝑖
and for 𝑗 ≥ 2
𝑗+1
𝑈𝑖
𝑗
= 𝐴𝑗 𝜀𝑖
𝑀−2
𝑗
= 𝐴𝑗 𝑈𝑖
𝑀−2
𝑗−2
𝑗−𝑘+1
𝑗−2
(𝐻𝑖
−
𝑗−1
2𝐻𝑖 ) 𝑈𝑖1
𝑗
−
𝑗−1
𝐻𝑖 𝑈𝑖0
𝑗
− (𝐻𝑖
(15)
𝑘=2
𝑗
𝑗−1
𝑗−1
− 2𝐻𝑖 ) 𝜀𝑖1 − 𝐻𝑖 𝜀𝑖0 + 𝐹𝜀𝑗 ,
where
𝑗
+𝐹 ,
𝑗
𝑗
𝑇
𝑗
𝑗
𝐹𝜀𝑗 = (𝑄𝑚−1 𝑓 (𝑢𝑚−1 , 𝑥𝑚−1 , 𝑡𝑗 )
𝑗
𝑗
𝑗
where 𝐹𝑗 = (𝑄𝑖 𝑓(𝑢𝑚−1 , 𝑥𝑚−1 , 𝑡𝑗 ), . . . , 𝑄𝑖 𝑓(𝑢1 , 𝑥1 , 𝑡𝑗 )) , 𝑈𝑗 =
− 𝑄𝑚−1 𝑓 (𝑤𝑚−1 , 𝑥𝑚−1 , 𝑡𝑗 ) , . . . , 𝑄1 𝑓 (𝑢1 , 𝑥1 , 𝑡𝑗 )
(𝑢𝑀−1 , 𝑢𝑀−2 , . . . , 𝑢1 ) ,
− 𝑄1 𝑓 (𝑤1 , 𝑥1 , 𝑡𝑗 ))
𝑗
𝑗 𝑇
𝑗
𝑗
𝑇
01 = (𝜑1 (𝑥1 ) , 𝜑1 (𝑥2 ) , . . . , 𝜑1 (𝑥𝑁)) ,
𝑇
(16)
𝑗
𝑎𝑛𝑚
𝑗
𝑗
0,
𝜃={
1,
𝑗
where 𝑚 = 1,
where 𝑚 ≤ 𝑛,
where 𝑚 = 𝑛 + 1,
𝑀−2
󵄩󵄩 𝑗−𝑘+1 󵄩󵄩
󵄩󵄩
+ ∑ (𝐻𝑖𝑘−2 − 2𝐻𝑖𝑘−1 + 𝐻𝑖𝑘 ) 󵄩󵄩󵄩𝜀𝑖
󵄩󵄩∞
󵄩
𝑘=2
𝑗−2
+ (𝐻𝑖
for 𝑛 = 1, 2, . . . , 𝐾 − 1, and 𝑚 = 1, 2, . . . , 𝐾 − 1. Also, we note
that
𝐾
󵄨 󵄨
‖𝐴‖∞ = max ∑ 󵄨󵄨󵄨𝑎𝑛𝑚 󵄨󵄨󵄨 = max {2 − 𝐻𝑖𝑛 } = 2 − 𝐻𝑖0 , (18)
1≤𝑛≤𝐾
1≤𝑛≤𝐾
𝑚=1
𝑘=2
𝑗
(1)
(2)
= 1, 𝑎𝑛𝑑
>
𝑗
𝐺𝑘+1 , 𝑎𝑛𝑑
+
3. The Stability Analysis and
the Truncation Error
Let us consider 𝑊𝑗+1 and 𝑈𝑗+1 to be two different numerical
solutions of (15) with initial values given by 𝑊0 and 𝑈0 ,
respectively.
Theorem 3. The explicit method approximation defined by
(15) to the variable order space-time wave equation (2) is
unconditionally stable, that is,
󵄨
󵄨
󵄨
󵄨󵄨 𝑗+1
󵄨󵄨𝑊 − 𝑈𝑗+1 󵄨󵄨󵄨 ≤ 𝐶 󵄨󵄨󵄨𝑊0 − 𝑈0 󵄨󵄨󵄨 , for any 𝑗.
(19)
󵄨
󵄨
󵄨
󵄨
𝑗−2
𝐶 (𝐻𝑖
−
𝑗−1 󵄩 󵄩
2𝐻𝑖 ) 󵄩󵄩󵄩󵄩𝜀𝑖0 󵄩󵄩󵄩󵄩∞
𝑗−1 󵄩 󵄩
+ 𝐻𝑖 󵄩󵄩󵄩󵄩𝜀𝑖0 󵄩󵄩󵄩󵄩∞ ≤ 𝐶1
= 1,
𝐻𝑘𝑖 > 𝐻𝑖𝑘+1 , for 𝑘 = 0, 1, . . .
𝑗−1 󵄩 󵄩
𝑗−1 󵄩 󵄩
− 2𝐻𝑖 ) 󵄩󵄩󵄩󵄩𝜀𝑖1 󵄩󵄩󵄩󵄩∞ + 𝐻𝑖 󵄩󵄩󵄩󵄩𝜀𝑖0 󵄩󵄩󵄩󵄩∞ .
𝑀−2
󵄩 󵄩
+ ∑ (𝐻𝑖𝑘−2 − 2𝐻𝑖𝑘−1 + 𝐻𝑖𝑘 ) 𝐶 󵄩󵄩󵄩󵄩𝜀𝑖0 󵄩󵄩󵄩󵄩∞
Lemma 2. The coefficients 𝐺𝑘 and 𝐻𝑖𝑘 satisfy the following
conditions:
𝐻𝑖0
(22)
Now, we analyze the stability via mathematical induction [10].
From (14) we have ‖𝜀𝑖1 ‖∞ ≤ 𝐶‖𝜀𝑖0 ‖∞ , where 𝐶 is a constant.
𝑗
Now, assume that ‖𝜀𝑖 ‖∞ ≤ 𝐶‖𝜀𝑖0 ‖∞ , then from (22), we
have
󵄩 󵄩
󵄩󵄩 𝑗+1 󵄩󵄩
󵄩󵄩𝜀𝑖 󵄩󵄩 ≤ 𝐶 (2 + 𝑄𝐿) 󵄩󵄩󵄩𝜀𝑖0 󵄩󵄩󵄩
󵄩∞
󵄩 󵄩∞
󵄩
then ‖𝐴‖∞ = 1.
𝑗
𝐺0
𝑗
𝐺𝑘
𝑗 𝑗 𝑇
𝑗
󵄩 󵄩 󵄩
󵄩󵄩 𝑗+1 󵄩󵄩
󵄩
󵄩󵄩𝜀𝑖 󵄩󵄩 ≤ 󵄩󵄩󵄩𝐴𝑗 + Δ𝐹𝑗 󵄩󵄩󵄩 󵄩󵄩󵄩𝜀𝑖𝑗 󵄩󵄩󵄩
󵄩∞ 󵄩
󵄩∞ 󵄩 󵄩∞
󵄩
where 𝑚 > 𝑛 + 2,
(17)
𝑗 𝑗 𝑗 𝑇
𝑗
and Δ𝐹𝑗 = diag (𝑄𝑚−1 𝐿 𝑚−1 , . . . , 𝑄1 𝐿 1 ) .
𝑗
Noting that |𝐿 𝑖 | ≤ 𝐿, for any 𝑖, 𝑗.
𝑗
𝑗
Let 𝑄 = max{𝑄𝑚−1 , . . . , 𝑄1 }. From (20), we have
𝑗
𝑗
‖𝐴 + Δ𝐹 ‖𝑚 ≤ (2 + 𝑄𝐿), then
where 𝑚 = 𝑛 + 2,
where 𝑚 = 2,
otherwise,
(21)
≤ (𝑄𝑚−1 𝐿 𝑚−1 𝜀𝑚−1 , . . . , 𝑄1 𝐿 1 𝜀1 ) = Δ𝐹𝑗 𝜀𝑗 ,
𝑗
𝑗
) is a matrix with the following coefficients:
and 𝐴𝑗 = (𝑎𝑛𝑚
𝑄𝑛 𝑅𝑛 𝐺𝑛−1 ,
{
{
{
𝑗 𝑗
𝑗
𝑗
𝑗
{
{
𝑄𝑛 𝑅𝑛 (𝐺𝑛−𝑚 − 2𝐺𝑛−𝑚+1 + 𝜃𝐺𝑛−𝑚+2 ) ,
{
{
{
{
= {2 − 𝐻𝑗 + 𝑄𝑗 𝑅𝑗 (𝜃𝐺𝑗 − 2𝐺𝑗 ) ,
𝑛
𝑛 𝑛
{
1
0
{
{
{
𝑗 𝑗 𝑗
{
𝑅
𝐺
,
𝑄
{
{
{ 𝑛 𝑛 0
{0,
𝑇
𝑗
𝑗
02 = (𝜑2 (𝑥1 ) , 𝜑2 (𝑥2 ) , . . . , 𝜑2 (𝑥𝑁)) ,
𝑗
(20)
𝑘=2
− ∑ (𝐻𝑖𝑘−2 − 2𝐻𝑖𝑘−1 + 𝐻𝑖𝑘 ) 𝑈𝑖
−
𝑗−𝑘+1
− ∑ (𝐻𝑖𝑘−2 − 2𝐻𝑖𝑘−1 + 𝐻𝑖𝑘 ) 𝜀𝑖
(23)
󵄩󵄩 0 󵄩󵄩
󵄩󵄩𝜀𝑖 󵄩󵄩 .
󵄩 󵄩∞
Then, the theorem holds.
Lemma 4. Let
∝(𝑥𝑖 ,𝑡𝑗 )
0 𝐷𝑥
𝑢 (𝑥𝑖 , 𝑡𝑗 ) =
ℎ−∝(𝑥𝑖 ,𝑡𝑗 )
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
𝑗−1
𝑗
𝑗
(24)
𝑗
𝑗
× ∑ 𝐺𝑖 (𝑢𝑖−𝑘+1 − 2𝑢𝑖−𝑘 + 𝑢𝑖−𝑘+1 )
𝑘=0
be a smooth function; then
󵄨󵄨 ∝(𝑥𝑖 ,𝑡𝑗 )
󵄨󵄨
∝(𝑥 ,𝑡 )
󵄨󵄨𝐷
𝑢 (𝑥𝑖 , 𝑡𝑗 ) − 𝐷𝑥 𝑖 𝑗 𝑢 (𝑥𝑖 , 𝑡𝑗 )󵄨󵄨󵄨 = 𝑂 (ℎ) .
󵄨󵄨 𝑥
󵄨
(25)
4
Journal of Applied Mathematics
1.4
By the integral mean value theorem, we have
1.2
∝(𝑥𝑖 ,𝑡𝑗 )
𝑢 (𝑥𝑖 , 𝑡𝑗 )
0 𝐷𝑥
1
=
u(x, t)
0.8
1
Γ (2 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
𝑘−1
×∑∫
0.6
(𝑗+1)ℎ
𝑗=0 𝑗ℎ
0.4
=
0.2
𝑧1−∝(𝑥𝑖 ,𝑡𝑗 )
ℎ2−∝(𝑥𝑖 ,𝑡𝑗 )
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
𝜕2 𝑢 (𝑥 − 𝑧, 𝑡)
𝑑𝑧
𝜕𝑧2
,
(27)
0
−0.2
0
1
2
3
4
5
6
7
8
x
󵄨󵄨 ∝(𝑥𝑖 ,𝑡𝑗 )
󵄨󵄨
∝(𝑥 ,𝑡 )
󵄨󵄨𝐷
𝑢 (𝑥𝑖 , 𝑡𝑗 ) − 𝐷𝑥 𝑖 𝑗 𝑢 (𝑥𝑖 , 𝑡𝑗 )󵄨󵄨󵄨
󵄨󵄨 𝑥
󵄨
󵄨󵄨
ℎ2−𝛼(𝑥𝑖 ,𝑡𝑗 )
󵄨󵄨
= 󵄨󵄨󵄨󵄨
󵄨󵄨 Γ (3 − 𝛼 (𝑥𝑖 , 𝑡𝑗 ))
󵄨
Exact
Approximate
Figure 1
Proof. In terms of standard centered difference formula, we
have
∝(𝑥𝑖 ,𝑡𝑗 )
0 𝐷𝑥
=
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
𝑗
∑ 𝐺𝑖
𝑗=0
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
𝑗𝜕
2
𝑗=0
= 𝑂 (ℎ) + 𝑂 (ℎ2 ) = 𝑂 (ℎ) .
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
Now, by using Lemma 4, we can derive the truncation error of
explicit finite difference scheme (14). It has a local truncation
error of 𝑂(𝜏) (from the left side) and 𝑂(ℎ) (from the right
side).
𝑂 (ℎ2 )
ℎ2−∝(𝑥𝑖 ,𝑡𝑗 )
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
𝑘−1
𝑗𝜕
× ∑ 𝐺𝑖
2
𝑗=0
+
=
𝑢 (𝑥 − 𝑗ℎ, 𝑡)
𝜕𝑧2
ℎ2−∝(𝑥𝑖 ,𝑡𝑗 ) 𝑘2−∝(𝑥𝑖 ,𝑡𝑗 )
+
Remark 5. The pervious explicit method was shown to be
stable. This method is consistent with a local truncation
error which is 𝑂(𝜏) + 𝑂(ℎ). Therefore, according to the Lax
Equivalence Theorem [2], it converges at this rate.
𝑢 (𝑥 − 𝑗ℎ, 𝑡)
𝜕𝑧2
𝑥2−∝(𝑥𝑖 ,𝑡𝑗 )
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
𝑂 (ℎ2 )
4. Numerical Examples
Example 1. Consider the following variable-order linear fractional wave equation:
ℎ2−∝(𝑥𝑖 ,𝑡𝑗 )
Γ (3 − ∝ (𝑥𝑖 , 𝑡𝑗 ))
×
𝑘−1
𝑗𝜕
∑ 𝐺𝑖
𝑗=0
2
(28)
󵄨󵄨
󵄨󵄨
𝑘−1
󵄨󵄨
󵄨
ℎ2−𝛼(𝑥𝑖 ,𝑡𝑗 )
𝑗
2 󵄨󵄨
󵄨
= 󵄨󵄨󵄨
∑ 𝐺𝑖 ⋅ 𝑂 (ℎ) + 𝑂 (ℎ )󵄨󵄨󵄨
󵄨󵄨 Γ (3 − 𝛼 (𝑥𝑖 , 𝑡𝑗 )) 𝑗=0
󵄨󵄨
󵄨
󵄨
󵄨󵄨 2−𝛼(𝑥 ,𝑡 ) 2−𝛼(𝑥 ,𝑡 )
󵄨󵄨
𝑖 𝑗
𝑖 𝑗
󵄨󵄨 ℎ
󵄨󵄨
𝑘
= 󵄨󵄨󵄨󵄨
⋅ 𝑂 (ℎ) + 𝑂 (ℎ2 )󵄨󵄨󵄨󵄨
󵄨󵄨 Γ (3 − 𝛼 (𝑥𝑖 , 𝑡𝑗 ))
󵄨󵄨
󵄨
󵄨
𝜕2 𝑢 (𝑥 − 𝑗ℎ, 𝑡)
[
+ 𝑂 (ℎ2 )]
𝜕𝑧2
ℎ2−∝(𝑥𝑖 ,𝑡𝑗 )
𝑘−1
2
𝜕2 𝑢 (𝑥 − 𝑗ℎ, 𝑡) 𝜕 𝑢 (𝑥 − 𝜉𝑗 , 𝑡)
−
]
𝜕𝑧2
𝜕𝑧2
󵄨󵄨
󵄨󵄨
󵄨
+𝑂 (ℎ ) 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
ℎ
𝑘−1
𝑗
2
2−∝(𝑥𝑖 ,𝑡𝑗 )
× ∑ 𝐺𝑖
=
𝑘−1
× ∑ 𝐺𝑖 [
𝑗=0
𝑢 (𝑥𝑖 , 𝑡𝑗 )
×
=
where 𝜉𝑗 ∈ [𝑗ℎ, (𝑗+1)ℎ]. Combining the pervious two formulae, we have
𝑢 (𝑥 − 𝑗ℎ, 𝑡)
+ 𝑂 (ℎ2 ) .
𝜕𝑧2
(26)
𝜕𝛽(𝑥,𝑡) 𝑢 (𝑥, 𝑡)
∝ (𝑥, 𝑡) 𝜋
= − 0.5 cos (
)
𝛽(𝑥,𝑡)
2
𝜕𝜏
𝜕∝(𝑥,𝑡) 𝑢 (𝑥, 𝑡)
+ 𝑓 (𝑢, 𝑥, 𝑡) ,
×
𝜕𝑥∝(𝑥,𝑡)
(29)
5
1
1.8
0.9
1.6
0.8
1.4
0.7
1.2
0.6
1
u(x, t)
u(x, t)
Journal of Applied Mathematics
0.5
0.4
0.8
0.6
0.3
0.4
0.2
0.2
0.1
0
0
0
1
2
3
4
5
6
7
8
−0.2
0
1
2
3
4
x
5
6
7
8
x
(a)
(b)
2.5
2.5
2
2
1.5
1.5
u(x, t)
u(x, t)
Figure 2
1
0.5
1
0.5
0
0
−0.5
8
−0.5
8
6
4
x
2
0
0
0.2
0.4
0.6
0.8
1
6
4
x
t
Figure 3
2
0
1.55
1.5
1.65
1.6
1.7
𝛼
Figure 4
2
subjected to the following initial conditions:
𝑢 (𝑥, 0) = 𝜑 (𝑥) = 𝑥2 (8 − 𝑥) ,
𝑢𝑡 (𝑥, 0) = Ψ (𝑥) = 0,
(31)
where 𝑋𝑎 = 0, 𝑋𝑏 = 8, and 𝑇 = 1.
The exact solution of this problem when 𝛽 = 2 is 𝑢(𝑥, 𝑡) =
𝑥2 (8 − 𝑥)(𝑡2 + 1).
In Figure 1, a comparison between the numerical and the
exact solutions when 𝛽 = 2 at 𝑡 = 0.416 is presented.
In Figures 2(a) and 2(b), we report the approximate
solutions at 𝑡 = 0.052 and 𝑡 = 0.78, respectively.
2.5
2
1.5
u(x, t)
with ∝ (𝑥, 𝑡) = 1.5 + 0.5𝑒−(𝑥𝑡) −1 , 𝛽(𝑥, 𝑡) = 1.5 +
0.25 cos(𝑥) sin(2𝑡), and
2𝑢
− (𝑡2 + 1)
𝑓 (𝑢, 𝑥, 𝑡) = 2
𝑡 +1
6𝑥3−∝(𝑥,𝑡)
16𝑥2−∝(𝑥,𝑡)
+
),
×(
Γ (3 − ∝ (𝑥, 𝑡))
Γ (4 − ∝ (𝑥, 𝑡))
(30)
1
0.5
2
0
−0.5
8
1.5 𝛽
6
4
x
2
0
1
Figure 5
In Figures 3, 4, and 5, respectively, we report the approximate solutions in three dimensions, where the axis’s are
(𝑡, 𝑥, 𝑢), (alfa, 𝑥, 𝑢), and (beta, 𝑥, 𝑢), respectively.
Journal of Applied Mathematics
2.5
2.5
2
2
1.5
1.5
u(x, t)
u(x, t)
6
1
1
0.5
0.5
0
0
−0.5
0
1
2
3
4
5
6
7
8
9
−0.5
10
0
1
2
3
4
5
6
7
8
9
10
x
x
𝛼=𝛽=2
𝛼, 𝛽 are variable
(b)
(a)
5
4
u(x, t)
3
2
1
0
−1
0
1
2
3
4
5
6
7
8
9
10
x
(c)
Figure 6
Example 2. Consider the following variable-order nonlinear
fractional wave equation:
𝑢 (𝑥, 𝑡)
𝜕𝑡𝛽(𝑥,𝑡)
= 2 cos 𝑡
𝜕
∝(𝑥,𝑡)
𝑢 (𝑥, 𝑡)
𝜕𝑥∝(𝑥,𝑡)
+ 𝑓 (𝑢, 𝑥, 𝑡) ,
(32)
2
with ∝ (𝑥, 𝑡) = 2 − cos2 (𝑥)sin2 (𝑡), 𝛽(𝑥, 𝑡) = 1.8 + 0.5𝑒−(𝑥𝑡) −1
and 𝑢(𝑥, 0) = 𝜑(𝑥) = 1 + sin 𝑥, 𝑢𝑡 (𝑥, 0) = Ψ(𝑥) = 0, where
0 ≤ 𝑥 ≤ 10, 𝑇 = 1, and 𝑓(𝑢, 𝑥, 𝑡) = 𝑢2 − sin2 (𝑥) − cos2 (𝑡).
This problem has the following exact solution, when
∝= 2
𝑢 (𝑥, 𝑡) = sin 𝑥 + cos 𝑡.
5
4
(33)
In Figure 6(a), we report the numerical solution when ∝
(𝑥, 𝑡), 𝛽(𝑥, 𝑡) are variables at 𝑡 = 0.52 and the exact solutions
when ∝ = 𝛽 = 2.
In Figures 6(b) and 6(c), we report the approximate
solution at 𝑡 = 0.052 and 𝑡 = 0.78, respectively.
u(x, t)
𝜕
𝛽(𝑥,𝑡)
3
2
1
0
−1
10
5
x
0
0
0.2
0.4
0.6
0.8
1
t
Figure 7
Figures 7, 8, and 9 show the approximate solution in
three dimensions, where the axes are (𝑡, 𝑥, 𝑢), (alfa, 𝑥, 𝑢) and
(beta, 𝑥, 𝑢), respectively.
Journal of Applied Mathematics
7
5
u(x, t)
4
3
2
1
0
−1
10
5
x
1.4
0
1.6
𝛼
1.8
2
Figure 8
5
u(x, t)
4
3
2
1
0
−1
10
5
x
0
1.8
1.85
1.9
1.95
2
𝛽
Figure 9
5. Conclusions
In this paper, numerical studies using a simple explicit FDM
for solving the variable order space-time wave equation are
presented. The stability analysis and the truncation error of
the proposed method are proved. Some test examples are
given, and the results obtained by the method are compared
with the exact solutions in integer order cases. Several figures
are presented to simulate the solutions behaviors when the
variable orders change with respect to space and time. The
comparison certifies that FDM gives good results. Summarizing these results, we can say that the finite difference
method in its general form gives reasonable calculations, easy
to use, and can be applied for the variable order differential
equations in general form. All results were obtained by using
MATLAB version 7.6.0 (R2008a).
References
[1] I. Podlubny, Fractional Differential Equations, Academic Press,
1999.
[2] R. D. Richtmyer and K. W. Morton, Difference Methods for
Initial-Vlue Problems, Krieger, Malabar, Fla, USA, 2nd edition,
1994.
[3] R. Lin, F. Liu, V. Anh, and I. Turner, “Stability and convergence of a new explicit finite-difference approximation for the
variable-order nonlinear fractional diffusion equation,” Applied
Mathematics and Computation, vol. 212, no. 2, pp. 435–445,
2009.
[4] C. F. Lorenzo and T. T. Hartley, “A Solution to the fundamental
linear fractional order differential equation,” NASA/TP 1998208693, 1998.
[5] C. F. Lorenzo and T. T. Hartley, “The vector linear fractional initialization problem,” NASA/TP 1999-208919, 1999.
[6] N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical
studies for a multi-order fractional differential equation,”
Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007.
[7] N. H. Sweilam and M. M. Khader, “A Chebyshev pseudospectral method for solving fractional-order integrodifferential equations,” The ANZIAM Journal, vol. 51, no.
4, pp. 464–475, 2010.
[8] N. H. Sweilam, M. M. Khader, and A. M. Nagy, “Numerical
solution of two-sided space-fractional wave equation using
finite difference method,” Journal of Computational and Applied
Mathematics, vol. 235, no. 8, pp. 2832–2841, 2011.
[9] N. H. Sweilam, M. M. Khader, and H. M. Almarwm, “Numerical
studies for the variable-order nonlinear fractional wave equation,” Fractional Calculus and Applied Analysis, vol. 15, no. 4, pp.
669–683, 2012.
[10] S. Shen and F. Liu, “Error analysis of an explicit finite difference
approximation for the space fractional diffusion equation with
insulated ends,” The ANZIAM Journal, vol. 46, pp. C871–C887,
2005.
[11] S. G. Samko and B. Ross, “Integration and differentiation to a
variable fractional order,” Integral Transforms and Special Functions, vol. 1, no. 4, pp. 277–300, 1993.
[12] D. Ingman, J. Suzdalnitsky, and M. Zeifman, “Constitutive
dynamic-order model for nonlinear contact phenomena,” Journal of Applied Mechanics, Transactions ASME, vol. 67, no. 2, pp.
383–390, 2000.
[13] D. Ingman and J. Suzdalnitsky, “Control of damping oscillations
by fractional differential operator with time-dependent order,”
Computer Methods in Applied Mechanics and Engineering, vol.
193, no. 52, pp. 5585–5595, 2004.
[14] F. Liu, P. Zhuang, V. Anh, and I. Turner, “A fractional-order
implicit difference approximation for the space-time fractional
diffusion equation,” The ANZIAM Journal, vol. 47, pp. C48–C68,
2006.
[15] C. F. Lorenzo and T. T. Hartley, “Initialized fractional calculus,”
International Journal of Applied Mathematics, vol. 3, no. 3, pp.
249–265, 2000.
[16] C. F. Lorenzo and T. T. Hartley, “Variable order and distributed
order fractional operators,” Nonlinear Dynamics, vol. 29, no. 1–
4, pp. 57–98, 2002.
[17] C. F. M. Coimbra, “Mechanics with variable-order differential
operators,” Annalen der Physik, vol. 12, no. 11-12, pp. 692–703,
2003.
[18] C. F. M. Coimbra and L. E. S. Ramirez, “A variable order constitutive relation for viscoelasticity,” Annalen der Physik, vol. 16, no.
7-8, pp. 543–552, 2007.
[19] C. F. M. Coimbra, C. M. Soon, and M. H. Kobayashi, “The variable viscoelasticity oscillator,” Annalen der Physik, vol. 14, no. 6,
pp. 378–389, 2005.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014