International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 2, Jun 2013, 23-32 ©TJPRC Pvt Ltd.
A CHEBYCHEV COLLOCATIONMETHOD FOR SOLVING TROESCH'S PROBLEM MOHAMED EL-GAMEL & MONA SAMEEH Department of Mathematical Sciences, Faculty of Engineering, Mansoura University, Egypt
ABSTRACT This paper presents a study of the performance of the collocation method using Chebychev basis functions for solving Troesch’s problem. Some numerical experiments are made. Numerical resultsare included to confirm the efficiency and accuracy of the method and a comparison with the modified homotopy perturbation technique(MHP), the variational iteration method, B-spline method and the sinc-collocation method are made. It is shown that the chebychev collocation method yields better results
KEYWORDS: Troesch's, Chebychev Polynomial, Nonlinear INTRODUCTION In this paper, we consider a nonlinear two-point boundary value problem, Troesch’s problem [1, 2, 3], defined by
u " = µ sinh( µ u ) u (0) = 0,
u (1) = 1
(1) (2)
where µ is a positive constant. This problem arises in an investigation of the confinement of a plasma column by radiation pressure [4] and also in the theory of gas porous electrodes [5, 6]. This problem has been studied extensively. Troesch found its numerical solution in [7] using the shooting method, in [8] using the decomposition technique, in [9, 10] using the variational iteration method, in [11] using a combination of the multipoint shooting method with the continuation and perturbation technique, in [12] using the quasilinearization method, in [13] using the method of transformation groups, in [14] the invariant imbedding method, in [15] using the inverse shooting method, in [16] using the modified homotopy perturbation method, in [17] using the differential transform method and in [18] using sinc-collocation method. The purpose of this paper is to introduce a novel approach based on Chebychev polynomial for the numerical solution of the class of nonlinear boundary value problems given in (1)-(2). Chebyshev polynomials have become increasingly important in numerical analysis. Most commonly used techniques with Chebyshev polynomials have been examined in [19, 20, 21, 22, 23, 24]and references therein. The remaining structure of this article is organized as follows: a brief introduction to the Chebchev polynomial is presented in Section 2. In Section 3, the chebychev approach for the solution of Troesch’s problem is described. The results are compared with the exact solutions and some existing numerical solutions in Section 4. Finally, in Section 5, a conclusion is given that briefly summarizes the results. SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS There are many advantages of using chebyshev polynomials as expansion function presented in that are good representation of smooth functions, provided that the function expanded by is infinitely differential, the coefficient of
24
Mohamed El-Gamel & Mona Sameeh
chebyshev expansion, approach zero faster than any inverse power in n asn goes to infinity then we need few no of iteration and less time to get the solution in any application. The well known Chebyshev polynomials are defined on the interval [−1, 1] and the following definitions are necessary for this step [25] Definition 1 Chebyshev polynomial of degree n is defined as
n = 0, 1,K
Tn ( x) = cos(n arccos(x)),
x ∈ [−1, 1]
Or, in a more instructive form,
Tn ( x ) = cos nθ ,
x = cos θ ,
θ ∈ [0, π ]
One sees at once that, on [-1, 1], 1.
Tn takes its maximal value with alternating signs (n+1) times: T n ( x k ) = ( − 1) k ,
Tn = 1, kπ x k = cos , N 2.
k = 0 , 1, K , n
(3)
Tn has n distinct zeros: Tn (t k ) = ( −1) k ,
Lemma 1: Chebyshev polynomials
( 2 k − 1)π t k = cos , 2n
k = 1,2,K , n
Tn satisfy the recurrence relation T0 ( x) = 1, T1 ( x) = x Tn+1 ( x) = 2 xTn ( x ) − Tn−1 ( x),
n f1
Tn is indeed an algebraic polynomial of degree n with the leading coefficient 2n−1 . Also,
In particular, orthogonality 1
∫T
m
( x )T n ( x )
−1
dx 1− x2
= 0,
n ≠ m (4)
∫
1 −1
T n2 ( x )
dx 1− x
2
π , = 2 π ,
n f 0 n = 0
In this paper we use orthonormal Chebyshev polynomials, noting property (4). Theorem 1 On the interval [−1, 1], among all polynomials of degree
n with
leading coefficient an
= 1 , the Chebyshev
25
A Chebychev Collocation Method for Solving Troesch's Problem
1 Tn deviates least from zero, i.e., 2n−1
polynomial
Let us assume that the function u (x) and its derivatives have truncated Chebyshev series expansion of the form N
u( x) =
∑a
r
T r ( x ),
−1 ≤ x ≤ 1
(5)
r=0
and N
u (k ) ( x) =
∑a
(k ) r
T r ( x ),
k = 1, 2 .
(6 )
r =0
where Tr (x ) denotes Chebyshev polynomials of the first kind of degree r , coefficients and
ar are unknown Chebyshev
N is chosen any positive integer such that N ≥ 2 . Then the function u (x) and its derivatives can be
written in the matrix forms
U =T A and
U ( k ) = T A( k ) or using the relation between the Chebyshev coefficient matrices A and
A(k ) [24],
A( k ) = 2k M k A U ( k ) = 2k T M k A where
T0 ( x0 ) T1 ( x0 ) T (x ) T (x ) 1 1 T = 0 1 M M T0 ( xN ) T1 ( xN )
U (k )
and
L TN ( x0 ) L TN ( x1 ) , O M L TN ( xN )
u ( k ) ( x0 ) (k ) u ( x1 ) = , M u (k ) ( x ) N
τ
1 A = a0 , a1 ,K, aN 2
u ( x0 ) u ( x1 ) U = M u( x ) N
26
Mohamed El-Gamel & Mona Sameeh
0 0 0 M = M 0 0
1/ 2
0
3/2
0
5/2
L
0
2
0
4
0
L
0
0
3
0
5
L
M
M
M
M
M
O
0
0
0
0
0
L
0
0
0
0
0
L
m1 m 2 m3 M N 0 ( N +1 ) Χ ( N +1 )
(7 )
where
m1 =
N , 2
m2 = 0,
m1 = 0,
m3 = N
m2 = N ,
m3 = 0
if
N is odd
if
N is even
The method can be developed for the problem defined in the domain [a, b] . Definition On the interval [a, b], the Chebyshev polynomial is given by
Tn* ( x ) = Tn ( y ),
Notice that its leading coefficient is equal to 2
n−1
2 a+b x− b−a 2
y=
2 b = a
n
To obtain the solution in terms of shifted Chebyshev polynomials
Tr* ( x) in the form
N
u ( x ) = ∑ a r* Tr* ( x ),
a≤x≤b
r =0
Where
2 a + b Tr* ( x) = Tr x− 2 b−a It is followed the previous procedure using the collocation points defined by
xi =
b − a a + b iπ + cos 2 b − a N
,
i = 0, 1,K , N
and the relation k
4 k * A*( k ) = M A, b−a where
k = 0, 1, 2.
(8)
27
A Chebychev Collocation Method for Solving Troesch's Problem
1 A* = a0* , a1* ,K, a*N 2 It is easily obtained
τ
T = T * because of the properties of Chebyshev polynomials.
THE DESCRIPTION OF CHEBYCHEV SCHEME To obtain a Chebyshev polynomial solution of equation (1) under general boundary conditions (2), first, we expand
) sinh ( µ u ) around u
) ) sinh ( µ u ) cosh ( µ u ) ) ) ) ) ) sinh ( µ u ) = sinh ( µ u ) + µ cosh ( µ u )(u − u ) + µ 2 (u − u ) 2 + µ 3 (u − u ) 3 2! 3! ) sinh ( µ u ) ) (u − u ) 4 + K + µ4 4! Particularly, if
) u = 0 , then sinh ( µ u ) = µ u + µ 3
u3 u5 + µ5 +K 3! 5!
Rewrite equation (1) as υ
u
"
− µ
2
2 j + 1
µ
∑
u −
j = 1
( 2 j + 1 )!
u
2 j + 1
= 0
(9 )
We assume that the approximate solution for u (x ) is represented by the formula N
u( x) =
∑a
* r
T r* ( x ),
0 ≤ x ≤1
(10 )
r =0
*
where Tr
( x) = Tr (2 x − 1) presents the shifted Chebyshev polynomials of the first kind of degree r and a r for
r = 0, 1, 2,K, N are the undetermined Chebyshev coefficients and the Chebyshev collocation points in [0, 1] are xi =
1 2
kπ 1 + cos , N
k = 0 , 1, K , N
(11 )
And its derivatives have truncated Chebyshev series expansion of the form N
u"( x) =
∑a
*( 2 ) * r r
T ( x ),
0 ≤ x ≤1
(12 )
r =0
Then the solution expressed by (10) and its derivatives can be written in the matrix forms
[u( x)] = T * A
[u ( x)] = 4 T M "
We need the following lemma
2
*
2
A
28
Mohamed El-Gamel & Mona Sameeh
Lemma 2:[26] The following relation holds
0 uυ ( x0 ) u ( x0 ) υ u ( x1 ) u ( x1 ) 0 = M M M υ u (x ) 0 0 N =
(U )υ
=
(T A ) (T A)
−1
0
u ( x0 ) u ( x1 ) M u(x ) N
U υ −1
*
υ −1
0 K O M K u ( x N ) K
*
where
T * ( x0 ) 0 * T ( x1 ) 0 T* = M M 0 0
K 0 , O M K T * ( xN ) K
A 0 K 0 0 A K 0 A = M M O M 0 0 K A
0
Proof. See [27] for details. Substituting
x = xi in (9) and applying the collocation to it, we eventually obtain the following theorem
Theorem 1 If the assumed approximate solution of the problem (9) is (10), then the discrete Chebychev system is given by υ
u " ( xi ) − µ 2u ( xi ) − ∑ j =1
µ 2 j +1
u 2 j +1 ( x i ) = 0 ( 2 j + 1)!
(13 )
The fundamental matrix equation for (13) is
WA = 0
(14 )
where υ
W = 4 2T *M
2
− µ 2T
*
−
∑ j =1
To get the expansion coefficient ar ,
µ
2 j +1
(T ( 2 j + 1 )!
*
A
)
2 j
T
r = 0, 1,K, N .in the approximate solution (10) firstly we have to obtain
the boundary conditions from equation (2)
T * (0) A = 0 T * (1) A = 1 then use them to solve the linear system (14), which can be written in the matrix form
~ WA = F
(15 )
29
A Chebychev Collocation Method for Solving Troesch's Problem
where
0 0 F =M 0 1 Now we have a nonlinear system of N + 1 equations in
N + 1 unknown coefficients. We can obtain the
coefficients in the approximate solution by solving this nonlinear system. NUMERICAL EXAMPLES The closed form solution to this problem in terms of the Jacobian elliptic function has been given [3] as
u( x) =
where
u ' (0) 1 sinh −1 sc µ x 1 − u ' ( 0 ) µ 4 2 2
[
] 2
(16 )
u ' (0) , the derivative of u at 0, is given by the expression u ' (0) = 2 1 − m , with m being the solution
of the transcendental equation
sinh ( µ2 ) = sc (µ m ) 1− m where the Jacobian elliptic function sc
(µ m )[28, 2]is defined by sc (µ m ) = sin φ
cos φ
where
φ and µ are related
by the integral φ
µ=∫ 0
1 1 − m sin2 θ
dθ
We have four tables to present results In Table 1 and Table 2 the numerical solution obtained by Chebychev collocation method is compared with the exact solution derived fromequation (16) and with the numerical solution obtained by the modified homotopy perturbation technique (MHP)[16], variation method [29] and sinc-collocation
method [18] for the case a µ = 0.5 nd µ = 0 .1
respectively In Table 3, the numerical solution obtained by the Chebychev collocation method for
µ = 5 is compared with
the numerical approximation of the exact solutions given by a Fortran code called TWPBVP and the numerical solution obtained by B-spline collocation method [30] and the numerical solution obtained by sinc-collocation method [18]. Table4, exhibits the numerical solution obtained by the Chebychev method for µ = 3 ,
µ = 5 , and µ = 7 .
DISCUSSIONS In this study, we compared the performance of the collocation method using Chebychev bases and other methods for solving Troesch’s problem. Numerical experiments are presented. The
chebychev collocation method is a simple
30
Mohamed El-Gamel & Mona Sameeh
method with high accuracy for solving nonlinear problems. So it may be easily applied by researchers and engineers familiar with the Chebychev function.
REFERENCES 1.
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions,Dover, New York, 1972.
2.
Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi,Higher Transcendental Functions, vol. 2, Mc-Graw-Hill, New York, 1953.
3.
S. Roberts, J. Shipman, On the closed form solution of Troesch's problem, J. Comput. Phys., 21 (1976) 291-304.
4.
E. Weibel, On the confinement of a plasma by magnetostaticn fields, Phys. Fluids, 2 (1959) 52-56.
5.
D. Gidaspow, B. Baker, A model for discharge of storage batteries, J. Electrochem. Soc., 120 (1973) 1005-1010.
6.
V. Markin, A. Chernenko, Y. Chizmadehev, Y. Chirkov, Aspects of the theory of gas porous electrodes, in: V.S. Bagotskii,Y.B. Vasilev11(Eds.), Fuel Cells: Their Electrochemical Kinetics, Consultants Bureau, New York, 1966, pp. 21-33.
7.
B.A. Troesch, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys., 21 (1976) 279.
8.
E. Deeba, S..Khuri, S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys., 159 (2000) 125-138.
9.
S.A. Khuri, A numerical algorithm for solving Troesch’s problem, Int. J. Comput. Math., 80 (2003) 493-498.
10. S. Momani, S. Abuasad, Z. Odibat, Variational iteration method for solving nonlinear boundary value problems, Appl. Math.Comput.,183 (2006) 1351-1358. 11. S. Roberts, J. Shipman, Solution of Troesch’s two-point boundary value problem by a combination of techniques, J. Comput. Phys., 10(1972) 232-241. 12. Miele, A. Agarwal, J. Tietze, Solution of two-point boundary-value problems with Jacobian matrix characterized by large positive eigenvalues,J. Comput. Phys., 15 (1974) 117-133. 13. J. Chiou, T. Na, On the solution of Troesch’s nonlinear two-point boundary value problem using an initial value method, J. Comput.Phys., 19 (1975) 311-316. 14. M. Scott, On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms, in:A.K. Aziz (Ed.), Numerical Solutions of Boundary-Value Problems for Ordinary Differential Equations, Academic Press, New York, 1975, pp.89-146. 15. J. Snyman, Continuous and discontinuous numerical solutions to the Troesch's problem, J. Comput. Appl. Math., 5 (1979) 171-175. 16. X. Feng, L. Mei, G. He, An efficient algorithm for solving Troesch’s problem, Appl. Math. Comput.,189 (2007) 500-507. 17. S. Chang, I. Chang, A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl. Math. Comput.195 (2008) 799-808.
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A Chebychev Collocation Method for Solving Troesch's Problem
18. M. El-Gamel, Numerical solution of Troesch’s problem by sinc-collocation method, Applied Mathematics,4 (2013) 501-508. 19. J. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, New York, 2000. 20. Clenshaw, H. Norton, The solution of nonlinear ordinary differential equations in Chebyshev series, Comput. J. 6 (1963) 88-92. 21. L. Fox, I.B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968. 22. H. Norton, The iterative solution of non-linear ordinary differential equations in Chebyshev series, Comput. J. 7 (1964) 76-5. 23. J. Mason, D.C. Handscomb, Chebyshev Polynomials, CRC Press, Boca Raton, 2003. 24. M. Sezer, M. Kaynak, Chebyshev polynomial solutions of linear differential equations, Int. Math. Educ. Sci. Technol. 27 (1996) 607-618. 25. T. Rivlin, An Introduction to the Approximation of Functions, Dover Publications, Inc., New York, 1969. 26. M. El-Gamel, M. Sameeh, An efficient technique for finding the eigen values of fourth-order sturm-liouville problems, Applied Mathematics,3 (2012),920-925. 27. Akyz-Dascio˘glu, H. Cerd¨ik-Yaslan The solution of high-order nonlinear ordinary differential equations by Chebyshev Series, Appl. Math. Comput. , 217 (2011) 5658-5666. 28. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. 29. Chang, A variational iteration method for solving Troesch’s problem, J. Comput. Appl. Math., 234 (2010) 30433047. 30. S. Khuri, A. Sayfy, Troeschs problem: A B-spline collocation approach, Math. Computer Model., 54 (2011) 19071918.
APPENDICES Table 1: Numerical Solutions of Troesch's Problem for the Case µ = 0.5
x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Exact solution 0.0951769 0.1906338 0.2866534 0.3835229 0.4815373 0.5810019 0.6822351 0.7855717 0.8913669
Chebyshev,N=12 0.0959443 0.1921287 0.2887944 0.3861848 0.4845471 0.5841332 0.6852011 0.7880165 0.8928542
MHP [16] 0.0959395 0.1921193 0.2887806 0.3861675 0.4845274 0.5841127 0.6851822 0.7880018 0.8928462
Variational [29] 0.1000416 0.2003336 0.3011275 0.4026773 0.5052411 0.6090820 0.7144698 0.8216826 0.9310084
Sinc-collocation 18] 0.0959443 0.1921352 0.2887944 0.3861848 0.4845471 0.5841332 0.6852011 0.7880165 0.8928542
32
Mohamed El-Gamel & Mona Sameeh
Table 2: Numerical Solutions of Troesch's Problem for the Case µ = 0.5
0.1
Exact Solution 0.08179699
0.2
0.16453087
0.17017132
0.16962076
0.20133869
0.17017132
0.3
0.24916736
0.25739390
0.25659292
0.30454102
0.25739390
0.4 0.5 0.6 0.7 0.8 0.9
0.33673220 0.42834716 0.52527402 0.62897114 0.74116837 0.86397002
0.34722285 0.44059983 0.53853439 0.64212860 0.7526080 0.87136251
0.34621073 0.43944227 0.53733006 0.64101046 0.75173354 0.87088353
0.41084132 0.52137347 0.63736635 0.76017896 0.89134491 1.03263022
0.34722285 0.44059983 0.53853439 0.64212860 0.75260809 0.87136251
x
Chebyshev,N=12
MHP [16]
Variational [29]
Sinc-Collocation [18]
0.08466125
0.08438170
0.10016683
0.08466125
Table 3: Numerical Solutions of Troeschs Problem for the Case µ = 5
x
Chebyshev,N=20
Fortran Code [30]
B-Spline [30]
Sinc-Collocation [18]
0.2 0.4 0.6 0.8 0.9 1.0
0.000 0.0107564 0.0331869 0.25822012 0.4550754 1.0000
0.000 0.01075342 0.03320051 0.25821664 0.45506034 1.000
0.000 0.01002027 0.03099793 0.24170496 0.42461830 1.000
0.000 0.00762552 0.03817903 0.23252435 0.44624551 1.000
Table 4: Exhibits the Numerical Solution Obtained by the Chebychev Method for µ = 3, µ = 5, and µ = 7
x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
µ = 3, N=12 0.0259412677 0.0542480446 0.0875170903 0.1287374022 0.1820596894 0.2527856362 0.3487395756 0.4831491473 0.6801445214
µ = 5, N=20 0.0047653280 0.0107564033 0.0195007950 0.0331869841 0.0554395002 0.0920696133 0.1531325264 0.2582012725 0.4550754696
µ =6, N=20 0.001886741 0.004533844 0.008899440 0.016267845 0.029976299 0.054981722 0.100281045 0.186328334 0.363367524
µ = 7, N=20 0.0002497057 0.0019548870 0.0066477254 0.0036724777 0.0171597851 0.0380891860 0.0582640325 0.1394145638 0.285199339