Research Journal of Applied Sciences, Engineering and Technology 7(4): 801-806, 2014
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2014
Submitted: April 15, 2013
Accepted: May 08, 2013
Published: January 27, 2014
A Numerical Solution for One-dimensional Parabolic Equation Using Pseudo-spectral
Integration Matrix and FDM
Saeid Gholami
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Abstract: This study presents a numerical method for the solution of one type of PDEs equation. In this study, apply
the pseudo-spectral successive integration method to approximate the solution of the one-dimensional parabolic
equation. This method is based on El-Gendi pseudo-spectral method. Also the Finite Difference Method (FDM) is
used as a minor method. The present numerical results are in satisfactory agreement with exact solution.
Keywords: El-Gendi method, Gauss-Lobatto points. pseudo-spectral successive integration, parabolic equation
The authors of El-Gendi et al. (1992) presentedan
operation matrix for the successive integration. In fact,
this matrix is generalization of the El-Gendi matrix.
Elbarbary (2007) presented a modification of the ElGendi successive integration matrix in (El-Gendi et al.,
1992) which yields more accurate results than those
computed by El-Gendi matrix in solving problems.
Finally, Elgindy (2009) developed a new explicit
expression of the higher order pseudo-spectral
integration matrices. Applications to initial value
problems, boundary value problems and linear integral
and integro-differential equations are presented.
The purpose of this study is to apply the Pseudospectral successive integration method to solving the
one-dimensional parabolic equation. The Pseudospectral successive integration method in based on ElGendi (1969) Also; the finite difference method is used.
INTRODUCTION
Clenshaw and Curtis (1960) presented a new
numerical method for integration of a function based on
the Chebyshev polynomials. El-Gendi (1969)
developed a new numerical scheme based on the
Clenshaw and Curtis quadrature scheme which
described a new method for the numerical solution of
linear integral equations of Fredholm type and of
Voltera type. This method has been extended to the
linear integro-differential equations and ordinary
differential equations. This method is presented an
operation matrix for integration.
Delves and Mohamed (1985) used the El-Gendi
method to solving the integral equations. They have
shown that the El-Gendi (1975) method represented a
modification of the Nystrom scheme when applied to
solving second kind of Fredholm integral equations.
Also, in Jeffreys and Jeffreys (1956) suggested a new
method to solving the differential equations based on
the successive integration of the Chebyshev expansions.
This method is accomplished by starting with
Chebyshev approximations for the highest order
derivative and generating approximations to the lower
order derivatives through successive integration of the
highest order derivative. Hatziavramidis and Ku (1985)
have been presented a Chebyshev expansion method for
the solution of boundary-value problems of O.D.E type
by using the pseudo-spectral successive integration
method. The method is easier to implement than
spectral methods employing the Galerkin and Tau
approximations and yields results of comparable
accuracy to these methods, with reduce computing
requirement. Also Nasr et al. (1990) and Nasr and ElHawary (1991) respectively, used the El-Gendi method
and successive integration method to solving the
Falkner-skan equation which uses a boundary value
technique and the Orr-sommerfeld equation for both
plane poiseulle flow and the Blasius velocity profile.
El-Gendi’s method: Clenshaw and Curtis (1960)
presented the numerical integration of a ‘well-behaved’
function 𝑓𝑓(𝑥𝑥)in −1 ≤ 𝑥𝑥 ≤ 1, by Chebyshev expansion
of 𝑓𝑓(𝑥𝑥)as follows:
𝑁𝑁
𝑓𝑓(𝑥𝑥) = ∑′′ 𝑟𝑟=0 𝑎𝑎𝑟𝑟 𝑇𝑇𝑟𝑟 (𝑥𝑥),
(1)
where:
2
and,
𝑁𝑁
𝑎𝑎𝑟𝑟 = ∑′′ 𝑗𝑗 =0 𝑓𝑓�𝑥𝑥𝑗𝑗 � 𝑇𝑇𝑟𝑟 �𝑥𝑥𝑗𝑗 �
𝑁𝑁
𝑗𝑗𝑗𝑗
𝑥𝑥𝑗𝑗 = − cos , 𝑗𝑗 = 0,1, … , 𝑁𝑁
𝑁𝑁
(2)
(3)
and integrating this series term by term. A summation
symbol with double prims denotes a sum with first and
last terms halved. The operation matrix B to
𝑥𝑥
approximate
the
indefiniteintegral
∫−1 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑,
presented by El-Gendi (1969) as follows:
801
Res. J. App. Sci. Eng. Technol., 7(4): 801-806, 2014
𝑁𝑁
𝑥𝑥
∫−1 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑 = �
𝑗𝑗 =0
∑𝑁𝑁+1
𝑟𝑟=0 𝑐𝑐𝑟𝑟 𝑇𝑇𝑟𝑟 (𝑥𝑥)
with:
𝑥𝑥
𝑎𝑎𝑟𝑟 ∫−1 𝑇𝑇𝑗𝑗 (𝑡𝑡) 𝑑𝑑𝑑𝑑 =
where,
(−1)𝑗𝑗 +1 𝑎𝑎 𝑗𝑗
𝑁𝑁
2𝑘𝑘
⎨
⎪
⎪
⎪
⎩
𝑐𝑐𝑁𝑁−1 =
𝑐𝑐𝑁𝑁 =
1
𝑎𝑎 𝑁𝑁 −2 − 𝑎𝑎 𝑁𝑁
2
2(𝑁𝑁−1)
𝑎𝑎 𝑁𝑁 −1
2𝑁𝑁
,
𝑐𝑐𝑁𝑁+1 =
,
,
(𝑃𝑃𝑁𝑁 𝑓𝑓)(𝑥𝑥) = ∑𝑁𝑁
𝑟𝑟=0 𝑎𝑎𝑟𝑟 𝑇𝑇𝑟𝑟 (𝑥𝑥),
(5)
𝑎𝑎 𝑁𝑁
(6)
𝐼𝐼𝑛𝑛 (𝑓𝑓) =
𝑁𝑁
�
𝑟𝑟=0
where,
𝐵𝐵
= �𝑏𝑏𝑖𝑖,𝑗𝑗
𝑏𝑏𝑖𝑖,𝑗𝑗 (𝑛𝑛) =
�,
(𝑥𝑥 𝑖𝑖 −𝑥𝑥 𝑗𝑗 )𝑛𝑛 −1
(𝑛𝑛−1)!
𝑏𝑏𝑖𝑖,𝑗𝑗 ,𝑖𝑖, 𝑗𝑗 = 0(1)𝑁𝑁
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑛𝑛
𝑇𝑇𝑟𝑟 (𝑥𝑥) = �
𝑚𝑚 =0
where,
𝑛𝑛
(−1)𝑚𝑚 �𝑚𝑚
�
2𝑛𝑛 𝜒𝜒 𝑚𝑚
(𝑛𝑛)
𝑇𝑇𝑟𝑟+𝑛𝑛−2𝑚𝑚 (𝑥𝑥),
𝑟𝑟 > 𝑛𝑛
𝜒𝜒𝑚𝑚 = ∏𝑛𝑛 𝑗𝑗 =0 (𝑟𝑟 + 𝑛𝑛 − 𝑚𝑚 − 𝑗𝑗)
𝑗𝑗 ≠𝑛𝑛−𝑚𝑚
Proof (Khalifa et al., 2003).
(9)
Theorem 2: Elbarbary (2007) The successive
integration of Chebyshev polynomials is expressed in
terms of Chebyshev polynomials as follows:
𝑥𝑥
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑛𝑛 −1
𝑛𝑛 −2
2
1
∫−1 ∫−1 ∫−1 … ∫−1 ∫−1 𝑇𝑇𝑟𝑟 (𝑡𝑡0 ) 𝑑𝑑𝑡𝑡0 𝑑𝑑𝑡𝑡1 … 𝑑𝑑𝑡𝑡𝑛𝑛−2 𝑑𝑑𝑡𝑡𝑛𝑛−1 =
𝑛𝑛−𝛾𝛾𝑟𝑟
�
where,
𝑚𝑚 =0
𝛽𝛽𝑟𝑟
𝑛𝑛
(−1)𝑚𝑚 �𝑚𝑚
�
2𝑛𝑛 𝜒𝜒 𝑚𝑚
𝜉𝜉𝑛𝑛,𝑚𝑚 ,𝑟𝑟 (𝑥𝑥)
𝑛𝑛−1
𝜉𝜉𝑛𝑛,𝑚𝑚 ,𝑟𝑟 (𝑥𝑥) = 𝑇𝑇𝑟𝑟+𝑛𝑛−2𝑚𝑚 (𝑥𝑥) − �
𝑖𝑖
𝜂𝜂𝑖𝑖 = �
Pseudo spectral integration matrix: We assume that
(𝑃𝑃𝑁𝑁 𝑓𝑓)(𝑥𝑥) is Nth order Chebyshev interpolating
polynomial of the function 𝑓𝑓(𝑥𝑥) in the points
�𝑥𝑥𝑘𝑘 , 𝑓𝑓(𝑥𝑥𝑘𝑘 )� where,
𝑁𝑁
𝑥𝑥
𝑎𝑎𝑟𝑟 ∫−1 ∫−1𝑛𝑛 −1 ∫−1𝑛𝑛 −2 … ∫−12 ∫−11 𝑇𝑇𝑟𝑟 (𝑡𝑡0 ) 𝑑𝑑𝑡𝑡0 𝑑𝑑𝑡𝑡1 … 𝑑𝑑𝑡𝑡𝑛𝑛−2 𝑑𝑑𝑡𝑡𝑛𝑛−1 ,
(8)
This study is organized as follows: In section 2, the
pseudo-spectral integration method which is based on
the El-Gendi (1975) method is presented. In section 3,
the main problem is solved by using the present
pseudo-spectral integration method and finite difference
method. In section 4 for a given example with
analytical solution, employed presented method and
numerical results are presented. A brief conclusion is in
section 5.
(𝑃𝑃𝑁𝑁 𝑓𝑓)(𝑥𝑥) = �𝑗𝑗 =0 𝑓𝑓𝑗𝑗 𝜑𝜑𝑗𝑗 (𝑥𝑥)
(13)
(14)
Theorem 1: Khalifa et al. (2003) the exact relation
between Chebyshev functions and its derivatives is
expressed as:
[𝐼𝐼𝑛𝑛 (𝑓𝑓)] =
𝑥𝑥
𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑡𝑡
�∫−1 ∫−1𝑛𝑛 −1 ∫−1𝑛𝑛 −2 … ∫−12 ∫−11 𝑓𝑓(𝑡𝑡0 ) 𝑑𝑑𝑡𝑡0 𝑑𝑑𝑡𝑡1 … 𝑑𝑑𝑡𝑡𝑛𝑛−2 𝑑𝑑𝑡𝑡𝑛𝑛−1 � =
𝐵𝐵(𝑛𝑛) [𝑓𝑓]
(7)
(𝑛𝑛)
�𝑗𝑗 =0 𝛼𝛼𝑗𝑗 𝑓𝑓�𝑥𝑥𝑗𝑗 � 𝑇𝑇𝑟𝑟 �𝑥𝑥𝑗𝑗 �,
𝑁𝑁
The successive integration of 𝑓𝑓(𝑥𝑥) at the points 𝑥𝑥𝑘𝑘
can be estimated by successive integration of (𝑃𝑃𝑁𝑁 𝑓𝑓)(𝑥𝑥).
Thus we have:
where, B is a square matrix of order N+1 and [𝑓𝑓] =
[𝑓𝑓(𝑥𝑥0 ), 𝑓𝑓(𝑥𝑥1 ), … , 𝑓𝑓(𝑥𝑥𝑁𝑁 )]𝑡𝑡 where 𝑥𝑥𝑗𝑗 are Gauss-Lobatto
points (3).
Finally, in El-Gendi et al. (1992) to approximate
the successive integration of function 𝑓𝑓(𝑥𝑥) we have:
(𝑛𝑛)
(12)
𝑁𝑁
2𝛼𝛼 𝑟𝑟
𝑎𝑎𝑟𝑟 =
Then, after certain re-rearrangement we define the
matrix B as follows:
𝑥𝑥
(11)
where,
4(𝑁𝑁+1)
�∫−1 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑� = 𝐵𝐵[𝑓𝑓],
𝑁𝑁
�𝑟𝑟=0 𝛼𝛼𝑟𝑟 𝑇𝑇𝑟𝑟 (𝑥𝑥) 𝑇𝑇𝑟𝑟 �𝑥𝑥𝑗𝑗 �
𝑁𝑁
where, 𝜑𝜑𝑗𝑗 (𝑥𝑥) = δ𝑗𝑗 ,𝑘𝑘 (δ𝑗𝑗 ,𝑘𝑘 is Kronecker delta) and
1
𝛼𝛼0 = 𝛼𝛼𝑁𝑁 = , 𝛼𝛼𝑗𝑗 = 1 for 𝑗𝑗 = 1(1)𝑁𝑁 − 1. Since
2
(𝑃𝑃𝑁𝑁 𝑓𝑓)(𝑥𝑥) is a unique Nth interpolating polynomial, it
can be expressed in terms of a series expansion of the
classical Chebyshev polynomials, hence we have:
1
⎧ 𝑐𝑐0 = ∑′′ 𝑗𝑗 =0 𝑗𝑗 2 −1 − 4 𝑎𝑎1 ,
𝑗𝑗 ≠1
⎪
⎪
⎪𝑐𝑐𝑘𝑘 = 𝑎𝑎 𝑘𝑘−1 −𝑎𝑎 𝑘𝑘+1 , 𝑘𝑘 = 1,2, … , 𝑁𝑁 − 2
2𝛼𝛼 𝑗𝑗
𝜑𝜑𝑗𝑗 (𝑥𝑥) =
(4)
𝑥𝑥 𝑗𝑗
𝑗𝑗 =0 (𝑖𝑖−𝑗𝑗 )!𝑗𝑗 !
𝑖𝑖=0
(𝑖𝑖)
𝜂𝜂𝑖𝑖 𝑇𝑇𝑟𝑟+𝑛𝑛−2𝑚𝑚 (−1),
, 𝜒𝜒𝑚𝑚 = ∏𝑛𝑛 𝑗𝑗 =0 (𝑟𝑟 + 𝑛𝑛 − 𝑚𝑚 − 𝑗𝑗),
𝑗𝑗 ≠𝑛𝑛−𝑚𝑚
𝑛𝑛
2 𝑖𝑖 = 0
𝛽𝛽𝑖𝑖 = �
, 𝛾𝛾 = �𝑛𝑛 − 𝑖𝑖 + 1
1 i > 0 𝑖𝑖
0
𝑖𝑖 = 0
1 ≤ 𝑖𝑖 ≤ 𝑛𝑛.
𝑖𝑖 > 𝑛𝑛
Proof (Elbarbary, 2007).
Thus, from Theorem 2 and (13), (14) we have:
(10)
802
Res. J. App. Sci. Eng. Technol., 7(4): 801-806, 2014
𝐼𝐼𝑛𝑛 (𝑓𝑓) =
�
And boundary conditions:
𝑁𝑁
𝑗𝑗 =0
�
2𝛼𝛼 𝑗𝑗
𝑁𝑁
𝑛𝑛 −𝛾𝛾𝑟𝑟
𝑁𝑁
�𝑟𝑟=0 𝛼𝛼𝑟𝑟 𝑇𝑇𝑟𝑟 �𝑥𝑥𝑗𝑗 � �
𝑚𝑚 =0
𝑛𝑛
(−1)𝑚𝑚 �𝑚𝑚
�
𝛽𝛽𝑟𝑟
2𝑛𝑛 𝜒𝜒 𝑚𝑚
𝜉𝜉𝑛𝑛,𝑚𝑚 ,𝑟𝑟 (𝑥𝑥)� 𝑓𝑓�𝑥𝑥𝑗𝑗 �
The matrix form of the successive integration of
the function 𝑓𝑓(𝑥𝑥) at the Gauss-Lobatto points 𝑥𝑥𝑘𝑘 is as
follows:
[𝐼𝐼𝑛𝑛 (𝑓𝑓)] =
��
𝑁𝑁
𝑗𝑗 =0
Θ(𝑛𝑛) [𝑓𝑓]
�
2𝛼𝛼 𝑗𝑗
𝑁𝑁
𝑛𝑛−𝛾𝛾𝑟𝑟
𝑁𝑁
�𝑟𝑟=0 𝛼𝛼𝑟𝑟 𝑇𝑇𝑟𝑟 �𝑥𝑥𝑗𝑗 � �
𝑚𝑚 =0
𝛽𝛽𝑟𝑟
𝑛𝑛
(−1)𝑚𝑚 �𝑚𝑚
�
2𝑛𝑛 𝜒𝜒 𝑚𝑚
𝑣𝑣(−1, 𝑡𝑡) = 𝐺𝐺0 (𝑡𝑡), 𝑣𝑣(1, 𝑡𝑡) = 𝐺𝐺1 (𝑡𝑡), 0 ≤ 𝑡𝑡 ≤ 𝑇𝑇,
Also, with the over-specification at a point the
special domain:
𝑣𝑣(𝑋𝑋0 , t) = 𝐾𝐾(𝑡𝑡), 0 ≤ 𝑡𝑡 ≤ 𝑇𝑇,
Now, we apply pseudo-spectral successive method and
FDM to solving the Eq. (18) with its conditions.
We apply FDM on t-dimension, assume:
𝜉𝜉𝑛𝑛,𝑚𝑚 ,𝑟𝑟 (𝑥𝑥)� 𝑓𝑓�𝑥𝑥𝑗𝑗 �� =
(15)
The elements of the matrix Θ(𝑛𝑛) are:
(𝑛𝑛)
𝜃𝜃𝑘𝑘,𝑗𝑗 =
2𝛼𝛼 𝑗𝑗
𝑁𝑁
𝑁𝑁
𝑛𝑛 −𝛾𝛾𝑟𝑟
�𝑟𝑟=0 𝛼𝛼𝑟𝑟 𝑇𝑇𝑟𝑟 �𝑥𝑥𝑗𝑗 � �
𝑚𝑚 =0
𝛽𝛽𝑟𝑟
𝑛𝑛
(−1)𝑚𝑚 �𝑚𝑚
�
2𝑛𝑛 𝜒𝜒 𝑚𝑚
𝜉𝜉𝑛𝑛,𝑚𝑚 ,𝑟𝑟 (𝑥𝑥𝑘𝑘 ),
𝑡𝑡𝑗𝑗 = 𝑗𝑗ℎ,
(16)
Hence, we have:
The matrix Θ(𝑛𝑛) which presented by ELbarbary
(2007) is the pseudo-spectral integration matrix.
𝜕𝜕𝜕𝜕 (𝑋𝑋,𝑡𝑡)
ONE-DIMENSIONAL PARABOLIC EQUATION
𝜕𝜕𝜕𝜕 (𝑋𝑋,𝑡𝑡)
We consider the one-dimensional parabolic
equation (Dehghan and Tatari, 2006) of the form:
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
=
𝜕𝜕 2 𝑢𝑢
𝜕𝜕𝑥𝑥 2
+ 𝑝𝑝(𝑡𝑡)𝑢𝑢(𝑥𝑥, 𝑡𝑡) + 𝐹𝐹(𝑥𝑥, 𝑡𝑡), 0 ≤ 𝑥𝑥 ≤ 1, 0 ≤ 𝑡𝑡 ≤ 𝑇𝑇
1
= 2ℎ [−3𝑣𝑣(𝑋𝑋, 𝑡𝑡0 ) + 4𝑣𝑣(𝑋𝑋, 𝑡𝑡1 ) − 𝑣𝑣(𝑋𝑋, 𝑡𝑡2 )]
𝜕𝜕𝜕𝜕
𝑡𝑡=𝑡𝑡 𝑗𝑗
= 2ℎ �𝑣𝑣�𝑋𝑋, 𝑡𝑡𝑗𝑗 +1 � − 𝑣𝑣�𝑋𝑋, 𝑡𝑡𝑗𝑗 −1 ��, 𝑗𝑗 = 1(1)𝑛𝑛 − 1 (20)
𝑡𝑡=𝑡𝑡 𝑛𝑛
= 2ℎ [3𝑣𝑣(𝑋𝑋, 𝑡𝑡𝑛𝑛 ) − 4𝑣𝑣(𝑋𝑋, 𝑡𝑡𝑛𝑛 −1 ) + 𝑣𝑣(𝑋𝑋, 𝑡𝑡𝑛𝑛−2 )] (21)
(19)
1
1
−[3 + 2ℎ 𝑝𝑝(𝑡𝑡0 )]𝑣𝑣(𝑋𝑋, 𝑡𝑡0 ) + 4𝑣𝑣(𝑋𝑋, 𝑡𝑡1 ) − 𝑣𝑣(𝑋𝑋, 𝑡𝑡2 ) =
8ℎ
𝜕𝜕 2 𝑣𝑣(𝑋𝑋,𝑡𝑡 0 )
𝜕𝜕𝑋𝑋 2
+ 2ℎ 𝑄𝑄(𝑋𝑋, 𝑡𝑡0 ),
(22)
+ 2ℎ 𝑄𝑄(𝑋𝑋, 𝑡𝑡𝑗𝑗 ),
(23)
𝑣𝑣�𝑋𝑋, 𝑡𝑡𝑗𝑗 +1 � − 2ℎ 𝑝𝑝�𝑡𝑡𝑗𝑗 �𝑣𝑣�𝑋𝑋, 𝑡𝑡𝑗𝑗 � − 𝑣𝑣�𝑋𝑋, 𝑡𝑡𝑗𝑗 −1 � =
And boundary conditions:
8ℎ
𝑢𝑢(0, 𝑡𝑡) = 𝑔𝑔0 (𝑡𝑡), 𝑢𝑢(1, 𝑡𝑡) = 𝑔𝑔1 (𝑡𝑡), 0 ≤ 𝑡𝑡 ≤ 𝑇𝑇,
𝜕𝜕 2 𝑣𝑣(𝑋𝑋,𝑡𝑡 𝑗𝑗 )
𝜕𝜕𝑋𝑋 2
[3 − 2ℎ 𝑝𝑝(𝑡𝑡𝑛𝑛 )]𝑣𝑣(𝑋𝑋, 𝑡𝑡𝑛𝑛 ) − 4𝑣𝑣(𝑋𝑋, 𝑡𝑡𝑛𝑛−1 ) +
With the over-specification at a point in the special
domain:
𝑣𝑣(𝑋𝑋, 𝑡𝑡𝑛𝑛−2 ) = 8ℎ
𝑢𝑢(𝑥𝑥0 , t) = 𝐸𝐸(𝑡𝑡), 0 ≤ 𝑡𝑡 ≤ 𝑇𝑇
𝜕𝜕 2 𝑣𝑣(𝑋𝑋,𝑡𝑡 𝑛𝑛 )
𝜕𝜕𝑋𝑋 2
+ 2ℎ 𝑄𝑄(𝑋𝑋, 𝑡𝑡𝑛𝑛 )
(24)
From the pseudo-spectral successive integration at
𝑖𝑖𝑖𝑖
the Gauss-Lobatto points (𝑥𝑥𝑖𝑖 = − cos ) we have:
where, 𝑔𝑔0 , 𝑔𝑔1 , 𝐹𝐹 and 𝐸𝐸 are known functions and the
functions 𝑢𝑢 and 𝑝𝑝 are unknowns. In this study, we apply
both the pseudo spectral integration method and Finite
Difference Method (FDM) (Strikverda, 2004) to
solving this equation. (In this study, we suppose that the
function 𝑝𝑝 is obtained by analytical method)
First, to translating 0 ≤ 𝑥𝑥 ≤ 1 to −1 ≤ 𝑋𝑋 ≤ 1 we
𝑋𝑋+1
. Hence the Eq. (17) changed to the form:
used 𝑥𝑥 =
𝜕𝜕 2 𝑣𝑣(𝑋𝑋,𝑡𝑡 𝑗𝑗 )
𝜕𝜕𝑋𝑋 2
𝜕𝜕𝜕𝜕 (𝑋𝑋,𝑡𝑡 𝑗𝑗 )
𝜕𝜕𝜕𝜕
2
𝑋𝑋=𝑋𝑋 𝑖𝑖
𝑋𝑋=𝑋𝑋 𝑖𝑖
= 𝜑𝜑�𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 �,
(1)
𝑁𝑁
= ∑𝑁𝑁
𝑘𝑘=0 𝜗𝜗𝑖𝑖,𝑘𝑘 𝜑𝜑�𝑋𝑋𝑘𝑘 , 𝑡𝑡𝑗𝑗 � + 𝑐𝑐1 ,
(25)
(26)
(2)
𝑣𝑣�𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 � = ∑𝑁𝑁
𝑘𝑘=0 𝜗𝜗𝑖𝑖,𝑘𝑘 𝜑𝜑�𝑋𝑋𝑘𝑘 , 𝑡𝑡𝑗𝑗 � + 𝑐𝑐1 (𝑋𝑋𝑖𝑖 + 1) + 𝑐𝑐2
(27)
𝜕𝜕 2 𝑣𝑣
= 4 2 + 𝑝𝑝(𝑡𝑡)𝑣𝑣(𝑋𝑋, 𝑡𝑡) + 𝑄𝑄(𝑋𝑋, 𝑡𝑡), −1 ≤ 𝑋𝑋 ≤
𝜕𝜕𝑋𝑋
1,0 ≤ 𝑡𝑡 ≤ 𝑇𝑇,
(18)
(𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖 = 0(1)𝑁𝑁, 𝑗𝑗 = 0(1)𝑛𝑛 ).
With initial condition:
𝑣𝑣(𝑋𝑋, 0) = 𝑓𝑓(𝑋𝑋), −1 ≤ 𝑋𝑋 ≤ 1
𝑗𝑗 = 0(1)𝑛𝑛
Substituting (19)-(21) into (18) gives:
𝑢𝑢(𝑥𝑥, 0) = 𝑓𝑓(𝑥𝑥), 0 ≤ 𝑥𝑥 ≤ 1
𝜕𝜕𝜕𝜕
,
𝑡𝑡=𝑡𝑡 0
𝜕𝜕𝜕𝜕
(17)
𝑇𝑇
𝑛𝑛
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 (𝑋𝑋,𝑡𝑡)
With initial condition:
𝜕𝜕𝜕𝜕
ℎ=
803
The constants 𝑐𝑐1 and 𝑐𝑐2 are obtained to satisfy the
boundary conditions. Thus:
Res. J. App. Sci. Eng. Technol., 7(4): 801-806, 2014
1
(2)
𝑐𝑐1 = − �∑𝑁𝑁
𝑘𝑘=0 𝜗𝜗𝑁𝑁,𝑘𝑘 𝜑𝜑�𝑋𝑋𝑘𝑘 , 𝑡𝑡𝑗𝑗 � − 𝐺𝐺1 �𝑡𝑡𝑗𝑗 � + 𝐺𝐺0 �𝑡𝑡𝑗𝑗 ��, for 𝑗𝑗 =
2
0(1)𝑛𝑛
𝑐𝑐2 = 𝐺𝐺0 �𝑡𝑡𝑗𝑗 �, for 𝑗𝑗 = 0(1)𝑛𝑛
and:
Substituting 𝑐𝑐1 and 𝑐𝑐2 into (27) gives us the
approximation solution at point �𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 � as follows:
𝑣𝑣�𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 � =
1
(2)
(2)
𝑁𝑁
∑𝑁𝑁
𝑘𝑘=0 𝜗𝜗𝑖𝑖,𝑘𝑘 𝜑𝜑�𝑋𝑋𝑘𝑘 , 𝑡𝑡𝑗𝑗 � − (𝑋𝑋𝑖𝑖 + 1) �∑𝑘𝑘=0 𝜗𝜗𝑁𝑁,𝑘𝑘 𝜑𝜑�𝑋𝑋𝑘𝑘 , 𝑡𝑡𝑗𝑗 �� +
1
2
(𝑋𝑋𝑖𝑖 + 1)𝐺𝐺1 �𝑡𝑡𝑗𝑗 � −
1
2
2
(𝑋𝑋𝑖𝑖 − 1)𝐺𝐺0 �𝑡𝑡𝑗𝑗 �
𝜑𝜑𝑁𝑁,0 =
𝜑𝜑𝑁𝑁,𝑗𝑗 =
(28)
But, to complete our work, we need to find all the
unknowns 𝜑𝜑𝑖𝑖,𝑗𝑗 . Hence for this purpose we substitute
(28) into (22)-(24) and solve the system of linear
equations to present more details we define:
1
(2)
(2)
(2)
(2)
(2)
(2)
𝐴𝐴𝑖𝑖 = �𝜗𝜗𝑖𝑖,0 , 𝜗𝜗𝑖𝑖,1 , … , 𝜗𝜗𝑖𝑖,𝑛𝑛 � − (𝑋𝑋𝑖𝑖 + 1)�𝜗𝜗𝑁𝑁,0 , 𝜗𝜗𝑁𝑁,1 , … , 𝜗𝜗𝑁𝑁,𝑛𝑛 �
2
𝑡𝑡
Φ𝑗𝑗 = �𝜑𝜑0,𝑗𝑗 , 𝜑𝜑1,𝑗𝑗 , … , 𝜑𝜑𝑁𝑁,𝑗𝑗 �
1
1
Υ𝑖𝑖,𝑗𝑗 = (𝑋𝑋𝑖𝑖 + 1)𝐺𝐺1 �𝑡𝑡𝑗𝑗 � − (𝑋𝑋𝑖𝑖 − 1)𝐺𝐺0 �𝑡𝑡𝑗𝑗 �
2
2
𝜑𝜑𝑁𝑁,𝑛𝑛 =
(30)
•
�3 + 2ℎ 𝑝𝑝(𝑡𝑡0 )� Υ𝑖𝑖,0 + 4 Υ𝑖𝑖,1 − Υ𝑖𝑖,2 − 2ℎ 𝑄𝑄(𝑋𝑋, 𝑡𝑡0 ) = 0
(31)
•
�2ℎ 𝑝𝑝�𝑡𝑡𝑗𝑗 �� Υ𝑖𝑖,𝑗𝑗 − Υ𝑖𝑖,𝑗𝑗 −1 − 2ℎ 𝑄𝑄�𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 � = 0
(36)
and:
(32)
Φ∗ = [Φ0 , Φ1 , Φ2 , … , Φ𝑛𝑛 ]𝑡𝑡
�3 + 2ℎ𝑝𝑝(𝑡𝑡0 )�Υ𝑖𝑖,0 − 4Υ𝑖𝑖,1 + Υ𝑖𝑖,2 + 2ℎ𝑄𝑄(𝑋𝑋𝑖𝑖 , 𝑡𝑡0 )
⎡
⎤
−Υ𝑖𝑖,2 + �2ℎ𝑝𝑝(𝑡𝑡1 )�Υ𝑖𝑖,1 + Υ𝑖𝑖,0 + 2ℎ𝑄𝑄(𝑋𝑋𝑖𝑖 , 𝑡𝑡1 )
⎢
⎥
⎢
⎥
)�Υ𝑖𝑖,2 + Υ𝑖𝑖,1 + 2ℎ𝑄𝑄(𝑋𝑋𝑖𝑖 , 𝑡𝑡2 )
−Υ
+
�2ℎ𝑝𝑝(𝑡𝑡
𝑖𝑖,3
2
𝐵𝐵 = ⎢
⎥
⋮
⎢
⎥
−Υ𝑖𝑖,𝑛𝑛 + �2ℎ𝑝𝑝(𝑡𝑡𝑛𝑛−1 )�Υ𝑖𝑖,𝑛𝑛−1 + Υ𝑖𝑖,𝑛𝑛−2 + 2ℎ𝑄𝑄(𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑛𝑛 )
⎢
⎥
⎣−�3 − 2ℎ𝑝𝑝(𝑡𝑡𝑛𝑛 )�Υ𝑖𝑖,𝑛𝑛 + 4Υ𝑖𝑖,𝑛𝑛−1 + Υ𝑖𝑖,𝑛𝑛−2 + 2ℎ𝑄𝑄(𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑛𝑛 )⎦
(33)
�3 − 2ℎ 𝑝𝑝(𝑡𝑡𝑛𝑛 )� 𝐴𝐴𝑖𝑖 Φ𝑛𝑛 − 4 𝐴𝐴𝑖𝑖 Φ𝑛𝑛−1 + 𝐴𝐴𝑖𝑖 Φ𝑛𝑛 −2 − 8ℎ 𝜑𝜑𝑖𝑖,𝑛𝑛 +
�3 − 2ℎ 𝑝𝑝(𝑡𝑡𝑛𝑛 )� Υ𝑖𝑖,𝑛𝑛 − 4 Υ𝑖𝑖,𝑛𝑛−1 + Υ𝑖𝑖,𝑛𝑛−2 − 2ℎ 𝑄𝑄(𝑋𝑋, 𝑡𝑡𝑛𝑛 ) = 0
Solving the systems (32)-(34) leads to obtaining
the all unknowns 𝜑𝜑𝑖𝑖,𝑗𝑗 . Notice that if 𝑖𝑖 = 0, 𝑁𝑁 then
𝐴𝐴𝑖𝑖 = 0, thus we can obtained the first and last rows of
the matrix Φ as follows:
1
�−Υ0,𝑗𝑗 +1 + 2ℎ𝑝𝑝�𝑡𝑡𝑗𝑗 �Υ0,𝑗𝑗 + Υ0,𝑗𝑗 −1 + 2ℎ𝑄𝑄�𝑋𝑋0 , 𝑡𝑡𝑗𝑗 ��
𝑗𝑗 = 1(1)𝑛𝑛 − 1
(38)
Table 1: Absolute error for n = 4, N = 4 at points �𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 �
X/ t
𝑋𝑋1
𝑋𝑋2
𝑋𝑋3
5.2×10−3
1.1×10−2
4.8×10−3
𝑡𝑡0
8.5×10−4
8.3×10−4
1.6×10−3
𝑡𝑡1
−3
−3
3.1×10
3.8×10
2.6×10−4
𝑡𝑡2
9.4×10−4
8.0×10−4
4.8×10−4
𝑡𝑡3
2.1×10−2
1.3×10−2
1.1×10−3
𝑡𝑡4
The absolute error at all points �𝑋𝑋0 , 𝑡𝑡𝑗𝑗 � and �𝑋𝑋4 , 𝑡𝑡𝑗𝑗 � for any 𝑗𝑗 = 0(1)4
are equals to zero
1
��3 + 2ℎ 𝑝𝑝(𝑡𝑡0 )� Υ0,0 − 4 Υ0,1 + Υ0,2
8ℎ
+ 2ℎ 𝑄𝑄(𝑋𝑋0 , 𝑡𝑡0 )�
8ℎ
(37)
Finally, by solving (35), all unknowns 𝜑𝜑𝑖𝑖,𝑗𝑗 to be
determined. Hence we can approximate the solutions of
the main problem at all points �𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 � by substituting
𝜑𝜑𝑖𝑖,𝑗𝑗 into (28).
(34)
𝜑𝜑0,𝑗𝑗 =
(35)
𝐴𝐴 =
−�3 + 2ℎ𝑝𝑝(𝑡𝑡0 )�𝐴𝐴𝑖𝑖 4𝐴𝐴𝑖𝑖 −𝐴𝐴𝑖𝑖 0 ⋯
0
⎡
⎤
⋯
0
⎢ −𝐴𝐴𝑖𝑖 −�2ℎ𝑝𝑝(𝑡𝑡1 )�𝐴𝐴𝑖𝑖 𝐴𝐴𝑖𝑖 0
⎥
⎢
⎥
⋮
⋱
⋮
⎢
0 −𝐴𝐴𝑖𝑖 −�2ℎ𝑝𝑝(𝑡𝑡𝑛𝑛−1 )�𝐴𝐴𝑖𝑖 𝐴𝐴𝑖𝑖 ⎥
0
⎢
⎥
⋯
0
𝐴𝐴𝑖𝑖 −4𝐴𝐴𝑖𝑖 �3 − 2ℎ𝑝𝑝(𝑡𝑡𝑛𝑛 )�𝐴𝐴𝑖𝑖 ⎦
⎣
For 𝑖𝑖 = 0(1)𝑁𝑁 and 𝑗𝑗 = 𝑛𝑛
𝜑𝜑0,0 =
1
�−�3 − 2ℎ𝑝𝑝(𝑡𝑡𝑛𝑛 )�Υ𝑁𝑁,𝑛𝑛 + 4 Υ𝑁𝑁,𝑛𝑛−1 − Υ𝑁𝑁,𝑛𝑛−2
8ℎ
+ 2ℎ𝑄𝑄(𝑋𝑋𝑁𝑁 , 𝑡𝑡𝑛𝑛 )�,
where, A is 3-D matrix:
For𝑖𝑖 = 0(1)𝑁𝑁 and 𝑗𝑗 = 1(1)𝑛𝑛 − 1
𝐴𝐴𝑖𝑖 Φ𝑗𝑗 +1 − �2ℎ 𝑝𝑝�𝑡𝑡𝑗𝑗 �� 𝐴𝐴𝑖𝑖 Φ𝑗𝑗 − 𝐴𝐴𝑖𝑖 Φ𝑗𝑗 −1 − 8ℎ 𝜑𝜑𝑖𝑖,𝑗𝑗 + Υ𝑖𝑖,𝑗𝑗 +1 −
1
�−Υ𝑁𝑁,𝑗𝑗 +1 + 2ℎ𝑝𝑝�𝑡𝑡𝑗𝑗 �Υ𝑁𝑁,𝑗𝑗 + Υ𝑁𝑁,𝑗𝑗 −1
8ℎ
+ 2ℎ𝑄𝑄�𝑋𝑋𝑁𝑁 , 𝑡𝑡𝑗𝑗 �� , 𝑗𝑗 = 1(1)𝑛𝑛 − 1
𝐴𝐴Φ∗ − 8ℎ𝜑𝜑𝑖𝑖,𝑗𝑗 = 𝐵𝐵
For 𝑖𝑖 = 0(1)𝑁𝑁 and 𝑗𝑗 = 0
−�3 + 2ℎ 𝑝𝑝(𝑡𝑡0 )� 𝐴𝐴𝑖𝑖 Φ0 + 4 𝐴𝐴𝑖𝑖 Φ1 − 𝐴𝐴𝑖𝑖 Φ2 − 8ℎ 𝜑𝜑𝑖𝑖,0 −
1
��3 + 2ℎ 𝑝𝑝(𝑡𝑡0 )� Υ𝑁𝑁,0 − 4 Υ𝑁𝑁,1 + Υ𝑁𝑁,2
8ℎ
+ 2ℎ 𝑄𝑄(𝑋𝑋𝑁𝑁 , 𝑡𝑡0 )�
Also, to computing another 𝜑𝜑𝑖𝑖,𝑗𝑗 for 𝑖𝑖 = 1(1)𝑁𝑁 − 1
and 𝑗𝑗 = 0(1)𝑛𝑛 we always have the system of form:
(29)
Finally, if (29)-(31) substitute into (22)-(24),
respectively then we have 3 systems of linear equations
as:
•
1
�−�3 − 2ℎ 𝑝𝑝(𝑡𝑡𝑛𝑛 )� Υ0,𝑛𝑛 + 4 Υ0,𝑛𝑛 −1
8ℎ
− Υ0,𝑛𝑛−2 + 2ℎ 𝑄𝑄(𝑋𝑋0 , 𝑡𝑡𝑛𝑛 )�
𝜑𝜑0,𝑛𝑛 =
804
Res. J. App. Sci. Eng. Technol., 7(4): 801-806, 2014
Table 2: Absolute error for n = 8, N = 8 at points �𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 �
X/ t
𝑋𝑋1
𝑋𝑋2
𝑋𝑋3
𝑋𝑋4
3.9×10−4
1.4×10−3
2.5×10−3
2.8×10−3
𝑡𝑡0
−4
−4
−5
3.4×10
6.9×10
8.8×10−4
𝑡𝑡1
8.7×10
1.7×10−4
2.0×10−4
𝑡𝑡2
5.7×10−5
9.4×10−5
1.4×10−4
4.7×10−4
7.1×10−4
6.6×10−4
𝑡𝑡3
1.8×10−4
6.2×10−4
9.7×10−4
9.5×10−4
𝑡𝑡4
2.1×10−4
7.0×10−4
1.1×10−3
1.1×10−3
𝑡𝑡5
1.8×10−4
6.1×10−4
9.8×10−4
9.8×10−4
𝑡𝑡6
1.9×10−4
6.7×10−4
1.1×10−3
1.2×10−3
𝑡𝑡7
1.7×10−4
𝑡𝑡8
3.5×10−5
6.3×10−5
1.8×10−5
The absolute error at all points �𝑋𝑋0 , 𝑡𝑡𝑗𝑗 � and �𝑋𝑋8 , 𝑡𝑡𝑗𝑗 � for any 𝑗𝑗 = 0(1)8 are equals to zero.
𝑋𝑋5
2.1×10−3
7.7×10−4
4.2×10−5
3.7×10−4
5.9×10−4
7.4×10−4
6.4×10−4
8.1×10−4
2.7×10−4
Table 3: Absolute error for n = 10, N = 10 at points �𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 �
X/t
𝑋𝑋1
𝑋𝑋2
𝑋𝑋3
𝑋𝑋4
𝑋𝑋5
𝑋𝑋6
1.6×10−4
6.2×10−4
1.3×10−3
1.7×10−3
1.9×10−3
1.6×10−3
𝑡𝑡0
2.1×10−4
4.4×10−4
6.6×10−4
7.6×10−4
6.8×10−4
𝑡𝑡1
5.3×10−5
5.1×10−6
8.4×10−6
1.4×10−4
1.8×10−4
𝑡𝑡2
1.4×10−5
7.2×10−5
1.4×10−4
2.5×10−4
2.9×10−4
2.4×10−4
1.3×10−4
𝑡𝑡3
4.1×10−5
−4
−4
−4
−4
−5
2.3×10
4.1×10
5.1×10
4.7×10
3.2×10−4
𝑡𝑡4
6.4×10
2.8×10−4
5.2×10−4
6.6×10−4
6.3×10−4
4.5×10−4
𝑡𝑡5
7.9×10−5
3.0×10−4
5.5×10−4
7.1×10−4
6.8×10−4
5.0×10−4
𝑡𝑡6
8.3×10−5
3.1×10−4
5.7×10−4
7.5×10−4
7.3×10−4
5.5×10−4
𝑡𝑡7
8.5×10−5
2.5×10−4
4.7×10−4
6.2×10−4
6.0×10−4
4.5×10−4
𝑡𝑡8
7.0×10−5
2.8×10−4
5.3×10−4
7.1×10−4
7.1×10−4
5.4×10−4
𝑡𝑡9
7.6×10−5
5.7×10−6
3.2×10−6
1.2×10−4
1.9×10−4
2.3×10−4
𝑡𝑡10
3.8×10−5
The absolute error at all points �𝑋𝑋0 , 𝑡𝑡𝑗𝑗 � and �𝑋𝑋10 , 𝑡𝑡𝑗𝑗 � for any j = 0(1)10 are equals to zero
0.025
For n = 4 and N = 4
Max error
Max error
0.020
0.015
0.010
0.005
0
0
0.1 0.2
0.3 0.4 0.5
t
0.6 0.7 0.8
0.9 1.0
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
×10-3
0
0.2
𝑋𝑋6
1.1×10−3
4.4×10−4
8.3×10−5
1.2×10−4
2.2×10−4
3.1×10−4
2.7×10−4
3.6×10−4
2.2×10−4
𝑋𝑋7
1.0×10−3
4.8×10−4
1.5×10−4
3.8×10−5
1.6×10−4
2.4×10−4
2.7×10−4
3.1×10−4
2.5×10−4
3.2×10−4
2.0×10−4
𝑋𝑋7
2.8×10−4
1.2×10−4
3.5×10−5
1.5×10−5
4.2×10−5
6.4×10−5
5.6×10−5
8.5×10−5
7.7×10−5
𝑋𝑋8
4.8×10−4
2.4×10−5
9.1×10−5
3.7×10−6
5.0×10−5
8.8×10−5
1.0×10−4
1.3×10−4
1.0×10−4
1.4×10−4
1.2×10−4
𝑋𝑋9
1.2×10−4
6.4×10−5
2.7×10−5
5.1×10−6
8.1×10−6
1.8×10−5
2.2×10−5
2.8×10−5
2.2×10−5
3.3×10−5
3.5×10−5
For n = 10 and N = 10
0.4
0.6
0.8
1.0
t
Fig. 1: The maximum error for n = 4 and N = 4
3.0 ×10
-3
Fig. 3: The maximum error for n = 10 and N = 10
For n = 8 and N = 8
𝐹𝐹(𝑥𝑥, 𝑡𝑡) = (𝜋𝜋 2 − (𝑡𝑡 + 1)2 )𝑒𝑒𝑒𝑒 𝑝𝑝(−𝑡𝑡 2 ) (cos(𝜋𝜋𝜋𝜋)
+ sin(𝜋𝜋𝜋𝜋))
Max error
2.5
2.0
𝐸𝐸(𝑡𝑡) = √2 exp(−𝑡𝑡 2 ) , 𝑥𝑥0 = 0.25
1.5
where, 𝑝𝑝(𝑡𝑡) = 1 + 𝑡𝑡 2 and the exact solution is:
1.0
0.5
0
0
0.2
0.4
0.6
0.8
𝑢𝑢(𝑥𝑥, 𝑡𝑡) = exp(−𝑡𝑡 2 ) (cos(𝜋𝜋𝜋𝜋) + sin(𝜋𝜋𝜋𝜋))
1.0
By presented approach in section 3 we solved this
problem. The present numerical results are in the above
Table 1 to 3 and Fig. 1 to 3.
t
Fig. 2: The maximum error for n = 8 and N = 8
NUMERICAL RESULTS
CONCLUSION
Consider the Eq. (17) in (14) with below details:
𝑓𝑓(𝑥𝑥) = cos(𝜋𝜋𝜋𝜋) + sin(𝜋𝜋𝜋𝜋)
𝑔𝑔0 (𝑡𝑡) = exp(−𝑡𝑡 2 ) , 𝑔𝑔1 (𝑡𝑡) = −exp(−𝑡𝑡 2 )
805
This study applies the pseudo-spectral successive
integration method to approximate the solutions of the
one-dimensional parabolic equation at the points
�𝑋𝑋𝑖𝑖 , 𝑡𝑡𝑗𝑗 �. So far this method, don’t applied to solve the
Res. J. App. Sci. Eng. Technol., 7(4): 801-806, 2014
partial differential equations. This study, demonstrate
that the pseudo-spectral successive integration method
can solve the partial differential equations.
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806