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The sinc–Galerkin method for solving Troesch’s problem
Mathematical and Computer Modelling 56 (2012) 218–228
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
The sinc–Galerkin method for solving Troesch’s problem M. Zarebnia a,∗ , M. Sajjadian b a
Department of Mathematics, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran
b
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
article
abstract
info
Article history: Received 16 August 2011 Received in revised form 25 November 2011 Accepted 28 November 2011
In this paper, the sinc–Galerkin method is applied for solving Troesch’s problem. The properties of the sinc procedure are utilized to reduce the computation of Troesch’s equation to nonlinear equations with unknown coefficients. The exponential convergence √
rate of the method, O(e−k N ), is developed. We use some numerical examples to illustrate the accuracy and implementation of the method. The results are compared with homotopy perturbation method (HPM), Laplace method, perturbation method and spline method by using tables. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Numerical method Nonlinear Troesch’s problem Sinc function Galerkin
1. Introduction Consider a two-point boundary value problem, Troesch’s problem, as follows: u′′ = γ sinh(γ u), u(0) = 0,
0 ≤ x ≤ 1,
u(1) = 1,
(1)
where γ is a positive constant. Troesch’s problem is discussed by Weibel and arises in the investigation of the confinement of a plasma column by radiation pressure [1] and also in the theory of gas porous electrodes [2,3]. The closed form solution to this problem in terms of the Jacobian elliptic function has been given [4] as u(x) =
2
γ
sinh
−1
u′ (0) 2
1 sc γ x|1 − u′ (0)2 4
,
(2)
√
where 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
γ 2
√
1−m
= sc(γ |m),
(3)
where the Jacobian elliptic function sc(γ |m) is defined by sc(γ |m) =
γ =
0
∗
φ
1
1 − m sin2 θ
sin φ , cos φ
where φ, γ are related by the integral
dθ .
Corresponding author. Tel.: +98 4515514702; fax: +98 4515514701. E-mail addresses: [email protected] (M. Zarebnia), [email protected] (M. Sajjadian).
0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.11.071
(4)
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219
From (2), it was noticed that a pole of u(t ) occurs at a pole of sc γ x|1 − 41 u′ (0)2 . It was also noticed that the pole occurs at x≈
1
2γ
16
ln
.
1−m
It also has an equivalent definition given in terms of a lattice. Troesch’s problem has been solved by another method. Xinlong Fen et al. [4] have introduced an efficient algorithm for solving this problem; their method is based on the modified homotopy perturbation technique. They have compared this method with the variational iteration method and the Adomian decomposition method. The variational iteration method is applied for solving Troesch’s problem by Chang [5]. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. In another article, Chang [6] has used simple shooting method. The paper is organized into five sections. Section 2 outlines some of the main properties of sinc functions and sinc method that are necessary for the formulation of the discrete Troesch’s problem. In Section 3, we illustrate how the sinc–Galerkin method may be used to replace Eq. (1) by an explicit system of nonlinear algebraic equations. In Section 4, the convergence analysis of the method has been discussed. It is shown that the sinc procedure converges to the solution at an exponential rate. Finally, we report our numerical results and demonstrate the efficiency and accuracy of the proposed numerical scheme by considering some numerical examples in Section 5. 2. Survey of some properties of the sinc function In this section, we will review sinc function properties, sinc quadrature rule, and the sinc method. These are discussed thoroughly in [7,8]. For solving Troesch’s equation (1), these properties will be used extensively in Section 3. The sinc function. The sinc function is defined on the whole real line, −∞ < x < ∞, by
sinc(x) =
sin(π x)
1,
πx
,
x ̸= 0;
(5)
x = 0.
For any h > 0, the translated sinc functions with evenly spaced nodes are given as S (j, h)(z ) = sinc
z − jh h
,
j = 0, ±1, ±2, . . . .
(6)
The sinc functions are cardinal for the interpolating points zk = kh in the sense that (0)
S (j, h)(kh) = δjk =
1, 0,
k = j; k ̸= j.
(7)
If f is defined on the real line, then for h > 0 the series C (f , h)(z ) =
∞
f (jh)sinc
z − jh
j=−∞
h
,
(8)
is called the Whittaker cardinal expansion of f whenever this series converges. They are based on the infinite strip Ds in the complex plane
Ds = w = u + iv : |v| < d ≤
π 2
.
(9)
To construct approximation on the interval (0, 1), we consider the conformal map
φ(z ) = ln
z 1−z
.
(10)
The map φ carries the eye-shaped region
z π DE = z = x + iy : arg .
This is shown in Fig. 1.
(11)
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
Fig. 1. The domains DE and DS .
For the sinc method, the basis functions on the interval (0, 1) for z ∈ DE are derived from the composite translated sinc functions, Sj (z ) = S (j, h) ◦ φ(z ) = sinc
φ(z ) − jh
h
.
(12)
The function z = φ −1 (w) =
ew
(13)
1 + ew
is an inverse mapping of w = φ(z ). We define the range of φ −1 on the real line as
Γ = {ψ(u) = φ −1 (u) ∈ DE : −∞ < u < ∞}. (14) ∞ The sinc grid points zk ∈ (0, 1) in DE will be denoted by xk because they are real. For the evenly spaced nodes {kh}k=−∞ on the real line, the image which corresponds to these nodes is denoted by xk = φ −1 (kh) =
ekh 1 + ekh
,
k = ±1, ±2, . . . .
(15)
Sinc interpolation and quadrature rules. For further explanation of the procedure, the important class of functions is denoted by B(DE ). The properties of functions in B(DE ) and detailed discussions are given in [7,8]. We recall the following definitions and theorems for our purpose. Definition 1. Let B(DE ) denote the family of functions F which are analytic in DE and satisfy
ψ(u+L)
|F (z )dz | → 0,
u → ±∞,
where L = {iv : |v| < d ≤ π2 }, and on the boundary of DE (denoted by ∂ DE ) satisfy
N (F ) ≡
∂ DE
|F (z )dz | < ∞.
Theorem 1. If φ ′ ∈ B(DE ), then for all x ∈ Γ
∞ N (F φ ′ ) F (xk )S (k, h)oφ(x) ≤ F (x) − 2π d sinh(π d/h) k=−∞ ≤
2N (F φ ′ ) −π d/h e . πd
Moreover, if |F (x)| ≤ C1 e−α|φ(x)| , x ∈ Γ , for some positive constants C1 and α , and also h = F (x) =
N
1
F (xk )S (k, h)oφ(x) + O exp −(π dα N ) 2
.
√ π d/α N, then (16)
k=−N
In this paper, we assume that the solution to Troesch’s problem satisfies the hypotheses of Theorem 1, which mandates that this solution vanishes at the endpoints of the interval Γ = [a, b]. Nonhomogeneous boundary conditions require a
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221
slight modification of the techniques of this paper. For example, Troesch’s problem with boundary conditions u( a) = ua ,
u(b) = ub
where at least one of ua or ub is non-zero, we could proceed by using the space Mα (D) [7, p. 180] or substituting ua + ρ(x)ub
u(x) = y(x) +
1 + ρ(x)
ρ(x) = eφ(x) ,
,
in Troesch’s problem (1) and thus derive a new boundary value problem with homogeneous boundary conditions. Theorem 2. Let F ∈ B(DE ) and φ be a conformal map with constants α and C2 so that
F (z ) −α|φ(z )| , φ ′ ( z ) ≤ C2 e
z ∈ Γ,
then the sinc trapezoidal quadrature rule is
Γ
F (z )dz = h
N F (zj ) + O(exp(−α Nh)) + O(exp(−π d/h)). ′ (z ) φ j j=−N
(17)
Hence, by selecting
h=
πd αN
12 ,
(18) 1
the exponential order of the sinc trapezoidal quadrature rule in Eq. (17) is O(exp(−(π dα N ) 2 )). By applying Theorem 2, we conclude the following corollary that is a special case of (17). Corollary. Let F ∈ B(DE ), and let h be selected by Eq. (18), then
Γ
F (zk ) 1 F (z )S (k, h) ◦ φ(z )dz = h ′ + O exp −(π dα N ) 2 . φ (z )
(19)
k
Theorem 3. Let φ be a conformal one-to-one map of the simply connected domain DE onto Ds . Then
δjk(0) = [S (j, h)oφ(x)]|x=xk = δjk(1) = h
d dφ
1, 0,
[S (j, h)oφ(x)]|x=xk
k = j; k ̸= j,
(20)
0, = (−1)(k−j) , (k − j)
k = j; k ̸= j,
(21)
and
δjk(2) = h2
d2 dφ 2
[S (j, h)oφ(x)]|x=xk =
−π 2 ,
3 (k−j) −2(−1)
(k − j)2
k = j; (22)
,
k ̸= j.
3. The sinc–Galerkin method Consider Troesch’s problem that is a nonlinear boundary value problem as: u′′ (x) − γ sinh(γ u(x)) = 0,
u(0) = 0,
u(1) = 1.
(23)
Let u(x) be the solution of boundary value problem (23) and u(x) ∈ B(DE ). We approximate the solution of Eq. (23) by the sinc–Galerkin method. For our purpose, first we convert the nonhomogeneous boundary conditions (23) to homogeneous conditions by considering the transformation y(x) = u(x) − x. This change of variable yields the following boundary value problem y′′ (x) − γ sinh(γ (y(x) + x)) = 0,
(24)
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
with boundary conditions y(0) = 0,
y(1) = 0.
(25)
In this case, consider sinc approximation by the formula N
y(x) ≈ yN (x) =
wj S (j, h)oφ(x).
(26)
j=−N
Our purpose is applying the Galerkin method based on sinc functions. Therefore, consider inner product for arbitrary functions f and g in the following form
⟨f , g ⟩ =
Γ
w(x)f (x)g (x)dx,
(27)
where
w(x) =
1 1
[φ ′ (x)] 2
.
Considering Eqs. (24)–(27), we have the sinc–Galerkin method as follows
⟨y′′ (x), S (k, h)oφ(x)⟩− < γ sinh(γ (y(x) + x)), S (k, h)oφ(x) >= 0.
(28)
Multiplying both sides of Eq. (28) in h and considering Eqs. (27) and (28), we obtain:
h Γ
S (k, h)oφ(x) ′′ yN (x)dx − hγ 1 [φ ′ (x)] 2
S (k, h)oφ(x)
sinh(γ (yN (x) + x))dx = 0.
1
[φ ′ (x)] 2
Γ
(29)
We apply part by part integration for the first term of Eq. (29) and then we get
h Γ
S (k, h)oφ(x) ′′ yN dx = h 1 [φ ′ (x)] 2
S (k, h)oφ(x)
Γ
[φ ′ (x)] 2
′′ yN (x)dx.
1
(30)
By using Eqs. (19) and (26), we can write
S (k, h)oφ(x)
Γ
[φ ′ (x)] 2
h
′′
1
yN (x)dx = h
2
N
S (k, h)oφ(xj )
′′
1
[φ ′ (xj )] 2
j=−N
.
1
wj .
φ ′ (xj )
(31)
Having used Eqs. (20)–(22), we have
[S (j, h)oφ(x)]|x=xk = φ ′
d dφ
′ = φ [S (j, h)oφ(x)] dφ
[S (j, h)oφ(x)] |x=xk ′′
[S (j, h)oφ(x)]|x=xk = φ ′ (xk )h−1 δjk(1) ,
′
d
(32)
= φ ′′ (xk )h−1 δjk(1) + [φ ′ (xk )]2 h−2 δjk(2) .
(33)
x =x k
From the above relations and Eq. (31), we obtain h Γ
(2) S (k, h)oφ(x) ′′ 1 yN (x)dx = δk,j [φ ′ (zj )] 2 + hδk(1,j) 1 ′ [φ (x)] 2 j=−N N
+
(0) h2 k,j
δ
−1
[φ ′ (zj ) 2 ]′′ φ ′ (zj )
′
φ ′′ (zj )
+ 2 [φ (zj )] ′
3
[φ ′ (zj )] 2
−1 2
wj .
(34)
Now, consider the second term of Eq. (29). By applying Eq. (19), we can obtain hγ
S (k, h)oφ(x)
Γ
[φ ′ (x)]
1 2
sinh γ (yN (x) + x)dx = γ h2
= γh
sinh γ (yN (xk ) + xk )
×
1
k) [φ ′ (xk )] N sinh(γ j=−N wj S (j, h)oφ(xk ) + xk ) 2 1 2
3
[φ ′ (xk )] 2
φ ′ (x
.
(35)
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By using the properties of the sinc function and applying Eq. (20), we rewrite Eq. (35) as hγ
S (k, h)oφ(x)
1
[φ ′ (x)] 2
Γ
sinh γ (yN (x) + x) = γ h2
sinh γ (wk + xk ) 3
[φ ′ (xk )] 2
.
(36)
Substituting (34) and (36) in (29), we get the Galerkin results as follows: N
1 δk(2,j) [φ ′ (zj )] 2
+
(1) hδk,j
sinh γ (wk + xk ) 3
[φ ′ (xk )] 2
3
= 0,
+ 2([φ (zj )] ′
[φ ′ (zj )] 2
j=−N
− γ h2
φ ′′ (zj )
−1 ′ 2
)
+
(0) h2 k,j
δ
−1
[φ ′ (zj ) 2 ]′′ φ ′ (zj )
wj
k = −N , −N + 1, . . . , N .
(37) (2)
Eq. (37) is a nonlinear system with (2N + 1) equations and (2N + 1) unknowns wj , j = −N , . . . , N. We set I (2) = [δkj ], (2)
where δkj denotes the (k, j)th element of the matrix I (2) . Also, we define the diagonal matrix D(g ) as follows: g (x−N ) 0
0 0
.. .
0 0
··· ···
g (x−N +1 )
.. .
D(g ) =
··· ··· .. .
0
0 0
0 0
.. .
.. . , 0
g (xN −1 ) 0
g (xN )
(2)
where I and D(g ) are square matrices of order (2N + 1) × (2N + 1). In order to determine (2N + 1) unknowns wj , j = −N , . . . , N in nonlinear system of Eq. (37), we rewrite this system in matrix form as: AW − γ h2 D
1
F (W ) = 0,
3
[φ ′ ] 2
(38)
where
A=
I
(2)
+h D
1
2
′′
1
×D
√ ′ φ
3
φ′ 2
φ
1
′2
,
(39)
F (W ) = sinh(γ (w−N + x−N )), sinh(γ (w−N +1 + x−N +1 )), . . . , sinh(γ (wN + xN )) W = w−N , w−N +1 , . . . , wN
T
T
,
.
Solving the nonlinear system (38) by Newton’s method we obtain an approximate solution yj , j = −N , . . . , N, then we can obtain an approximation to the solution of Troesch’s problem (1): uN (x) =
N
wj S (j, h)oφ(x) + x.
(40)
j=−N
4. Convergence analysis In this section, the convergence of the sinc–Galerkin method for Troesch’s problem (1) is discussed. For this purpose, we apply the sinc–Galerkin method for the linear boundary value problem
(Ly)(x) = y′′ (x) + µy′ (x) + ν y(x) = σ (x), y(a) = 0, y(b) = 0.
a < x < b, (41)
Consider the approximate solution (26). As may be expected in view of Section 3, by using the inner product (27), we can obtain a similar result. We select yN (x) as in (26) and then we evaluate the integrals
⟨LyN , S (k, h)oφ⟩ = 0,
k = −N , . . . , N .
After integrating by parts to eliminate derivatives in y, and dropping the intermediate boundary terms, which can be shown to vanish under assumptions which we shall state below. Then the discrete sinc–Galerkin system for the determination of the unknown coefficients {wj }Nj=−N is given by
I (2) D(φ ′ w) + hI (1) D
φ ′′ w φ′
+ 2w′ − µw + +h2 D
w ′′ − (µw)′ + νw φ′
where 1 is a (2N + 1)-vector each of whose components are 1 and w(x) = (φ ′ (x))
W = h2 D
−1 2
.
σw 1, φ
(42)
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
The error in approximating the exact solution of boundary value problem (41) by (26) is given in the following theorem [9]. Theorem 4. Assume that the coefficients µ, ν and σ are analytic in the region DE and that problem (41) has a unique solution y(x) which is analytic in DE . Assume also that
′′ ′ y (z ) y (z ) ′ dz < ∞, ′ dz < ∞, ∂ D φ (z ) ∂ D φ (z ) ν(z )y(z ) σ (z ) φ ′ (z ) dz < ∞, ′ dz < ∞, ∂D ∂ D φ (z )
and for positive α ,
|y(x)| ≤ K exp(−α|φ(x)|),
x ∈ Γ.
1 2
If h = (π d/α N ) and the coefficients {wj }Nj=−N in (26) are determined from (42) with w(x) = 1
3
|y(x) − yN (x)| ≤ C3 N 2 e−(2π dαN ) 2 ,
1 1
(φ ′ (x)) 2
, then
x ∈ Γ.
Now, we discuss convergence of the sinc–Galerkin method for Troesch’s equation and obtain a bound on the error y(x) − yN (x) in the maximum norm, where y(x) is the exact solution and yN (x) is the approximate solution (26). Lemma 1. Let y be the exact solution of the nonlinear equation Ly − γ G(y) = 0, where (Ly)(x) = y′′ (x) is a second-order linear operator and G(y) = sinh(γ y(x) + x) is a nonlinear operator. Let y ∈
−1
B(DE ), G′ (y) = ∂∂Gy , and G′′ (y) = ∂∂ yG2 be well defined and bounded on the ball B(y0 , r ). Also, let L − γ G′ (y) 2
−1 L − γ G (y ) γ G(y0 ) − Ly0 be bounded on B(y0 , r ), and −1 ∥γ G(y0 ) − Ly0 ∥∞ ≤ H0 , ∥ L − γ G′ (y) ∥∞ ≤ H1 ,
′
and
0
∥γ G′′ (y)∥∞ ≤ H2 ,
y ∈ B(y0 , r ),
y ∈ B(y0 , r ).
(43)
If h˜ = H12 H2 H0 < 2,
(44)
and r > H0 H1
2k −1 ∞ h˜ k=0
2
,
then the sequence
yn+1 = yn + L − γ G′ (yn )
−1 γ G(yn ) − Lyn
(45)
is well defined. Also yn+1 ∈ B(DE ) for every positive integer n and the sequence yn converges to y∗ . Furthermore, n
∥yn − y∗ ∥∞ ≤ H1 H0
(h˜ /2)2 −1 . n 1 − (h˜ /2)2
(46)
Proof. By applying Kantorovich’s Theorem [10], we can conclude the existence of the sequence {yn }n≥0 and the bound (46). √
From Lemma 1, we can show that the sinc–Galerkin method converges at the rate of O(e−k
N
), k > 0.
Theorem 5. Let us consider all assumptions in Lemma 1 and let, the discrete equivalent of γ G′ (y) and γ G′′ (y) be well defined and bounded on the ball B(y0 , r ), and also the discrete equivalent of (L − γ G′ (y))−1 be bounded on B(y0 , r ). Let the sequence νNn be the discrete equivalent of (45). Then, (a) {νNn }n≥0 converges to νN∗ and νNn − νN∗ has a bound as defined in (46).
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225
Table 1 Numerical solutions of Troesch’s problem for the case γ = 0.5. x
Exact [11]
SGM
HPM [11]
LM [12]
PM [12]
SM [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.095176902 0.190633869 0.286653403 0.383522929 0.481537385 0.581001975 0.682235133 0.785571787 0.891366988
0.095944347 0.192128740 0.288794409 0.386184841 0.484547165 0.584133254 0.685201142 0.788016528 0.892854218
0.095948026 0.192135797 0.288804238 0.386196642 0.4845599 0.584145785 0.685212297 0.788025104 0.892859085
0.0959444 0.1921288 0.2887944 0.3861849 0.4845472 0.5841333 0.6852012 0.7880166 0.8928542
0.095995 0.192229 0.288944 0.386380 0.484782 0.584395 0.685468 0.788254 0.893011
0.095944 0.192128 0.288793 0.386380 0.484546 0.584132 0.685200 0.788015 0.892853
(b) There exists a constant C3 independent of N such that
3
1
sup |νN∗ − y∗ | ≤ C4 N 2 exp −(π dα N ) 2
x∈Γ
.
(47)
Proof. (a) Let {νNn }n≥0 be the discrete sequence by the sinc–Galerkin method that defined by the discrete equivalent of (45). Similarly, by using Lemma 1, the sequence {νNn }n≥0 exists and converges to νN∗ and moreover we have n
∥ν − νN ∥∞ n N
∗
(h˜ /2)2 −1 , ≤ H1 H0 n 1 − (h˜ /2)2
(48)
where H0 , H1 and h˜ are defined in (43) and (44). (b) Let the sequence yn defined by (45). By using Lemma 1, we know that the sequence yn exists and converges to y∗ and also n
(h˜ /2)2 −1 . n 1 − (h˜ /2)2 −1 By considering bounded inverse of L − γ G′ (y) on the ball B(y0 , r ) and Theorem 4, we have 3 3 sup |νNn − yn | ≤ C5 N 2 exp −(π dα N ) 2 . ∥yn − y∗ ∥∞ ≤ H1 H0
x∈Γ
(49)
(50)
Now, we consider the following inequality
|νn∗ − y∗ | ≤ |νN∗ − νNn | + |νNn − yn | + |yn − y∗ |.
(51)
By considering assumptions in Lemma 1 and h˜ < 2, the following inequality can be made for n large enough, n
H1 H0
(h˜ /2)2 −1 3 1 ≤ C6 N 2 exp{−(π dα N ) 2 }. n 1 − (h˜ /2)2
(52)
Having applied the relations (48)–(50) and (52), and also by having considered the maximum norm bounds for νN∗ −νNn , νNn − yn and yn − y∗ , we obtain 3
1
sup |νN∗ − y∗ | ≤ C4 N 2 exp{−(π dα N ) 2 }.
(53)
x∈Γ
5. Numerical examples In order to show numerical results of solving Troesch’s equation based on the sinc–Galerkin method and discuss the accuracy of the method, we consider the absolute errors of this method and other methods that are tabulated and compared in tables. In these examples, we suppose d = π2 and α = Error = |u(xj ) − uN (xj )|,
1 2
1
which yields h = π ( N1 ) 2 . The error function is given by
j = −N , . . . , N ,
where u and uN represent the exact and approximate solutions, respectively. These examples are solved for different values of γ . We solved Troesch’s problem (1) by using the sinc–Galerkin method(SGM). Solving the nonlinear system (38), we obtain the approximate solution uN (x). We calculated the approximate solution at the uniform grid points xi = ihu , i = 1, . . . , 9, , hu = 0.1. The results are tabulated in Table 1. Also, Table 1, shows the numerical solution of Troesch’s equation using Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods for γ = 0.5.
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228 Table 2 Errors of Troesch’s problem for the case γ = 0.5. x
SGM.E
HPM.E [11]
LM.E [12]
PM.E [12]
SM.E [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.000767445 0.001494871 0.002141006 0.002661912 0.003009780 0.003131279 0.002966009 0.002444741 0.001487230
0.000771124 0.001501928 0.002150835 0.002673713 0.003022515 0.00314381 0.002977164 0.002453317 0.001492098
7.7 × 10−4 1.5 × 10−3 2.1 × 10−3 2.7 × 10−3 3.0 × 10−3 3.1 × 10−3 3.0 × 10−3 2.4 × 10−3 1.5 × 10−3
8.2 × 10−4 1.6 × 10−3 2.3 × 10−3 2.9 × 10−3 3.2 × 10−3 3.4 × 10−3 3.2 × 10−3 2.7 × 10−3 1.6 × 10−3
7.7 × 10−4 1.5 × 10−3 2.1 × 10−3 2.7 × 10−3 3.0 × 10−3 3.1 × 10−3 3.0 × 10−3 2.4 × 10−3 1.5 × 10−3
Table 3 Numerical solutions of Troesch’s problem for the case γ = 1. x
Exact [11]
SGM
HPM [11]
LM [12]
PM [12]
SM [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.081796997 0.164530871 0.249167361 0.336732209 0.428347161 0.52527403 0.628971143 0.741168378 0.863970021
0.084661250 0.170171338 0.257393933 0.347222839 0.440599836 0.538534416 0.642128589 0.752608114 0.871362527
0.084934415 0.170697546 0.258133224 0.348116627 0.44157274 0.539498234 0.642987984 0.753267551 0.871733059
0.0846631 0.1701750 0.2573995 0.3472304 0.4406094 0.5385460 0.6421421 0.7526227 0.8713749
0.085417 0.171669 0.259603 0.350053 0.444010 0.542315 0.645984 0.756064 0.873671
0.084655 0.170160 0.257377 0.347201 0.440575 0.538508 0.642103 0.752586 0.871349
Table 4 Errors of Troesch’s problem for the case γ = 1. x
SGM.E
HPM.E [11]
LM.E [12]
PM.E [12]
SM.E [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.002864253 0.005640467 0.008226572 0.010490630 0.012252675 0.013260386 0.013157446 0.011439736 0.007392506
0.003137419 0.006166675 0.008965863 0.011384418 0.013225579 0.014224205 0.01401684 0.012099173 0.007763039
2.9 × 10−3 5.9 × 10−3 8.2 × 10−3 1.0 × 10−2 1.2 × 10−2 1.3 × 10−2 1.3 × 10−2 1.1 × 10−2 7.4 × 10−3
3.6 × 10−3 7.1 × 10−2 1.0 × 10−2 1.3 × 10−2 1.6 × 10−2 1.7 × 10−2 1.7 × 10−2 1.5 × 10−2 9.7 × 10−3
2.8 × 10−3 5.6 × 10−3 8.2 × 10−3 1.0 × 10−2 1.2 × 10−2 1.3 × 10−2 1.3 × 10−2 1.1 × 10−2 7.4 × 10−3
Table 5 Numerical solution of Troesch’s problem for the case γ = 0.5 and different values of N. x
Exact [12]
SGM N =5
SGM N = 10
SGM N = 20
BSM N =5
BSM N = 10
BSM N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.00000000 0.19063387 0.38352292 0.58100197 0.78557179 1.00000000
0.00000000 0.19211095 0.38602762 0.58425183 0.78806744 1.00000000
0.00000000 0.19214588 0.38616848 0.58414841 0.78801133 1.00000000
0.00000000 0.19212882 0.38618437 0.58413371 0.78801652 1.00000000
0.000000 0.192116 0.386162 0.584104 0.787996 1.000000
0.000000 0.192125 0.386179 0.584126 0.788011 1.000000
0.000000 0.192128 0.386183 0.584132 0.788015 1.000000
In Table 2, we compare the absolute errors of the sinc–Galerkin(SGM) with Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods for γ = 0.5. In Table 3, the sinc–Galerkin solutions are calculated for grid points xi = ihu , i = 1, . . . , 9, hu = 0.1 and γ = 1. The resulting solution is also contrasted with the other numerical techniques, namely with Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods. In Table 4, the absolute errors obtained by the sinc–Galerkin method at the mesh points x = 0, 0.1, 0.2, . . . , 0.9, 1.0 for the case γ = 1 are compared with the absolute errors of the Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods. We solved Troesch’s problem for different values N = 5, 10, 20 by presented method. Solving the nonlinear system (38), we obtain the approximate solution uN (x), the numerical results on the mesh points x = 0, 0.2, . . . , 0.8, 1 for the case γ = 0.5 are tabulated in Table 5. Also the results with the B-Spline Method (BSM) [12] have been shown for the same equation and parameters as in Table 5. Table 6, exhibits the absolute errors of the B-Spline(BSM) [12] and the sinc–Galerkin method for different values of N and γ = 0.5.
M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
227
Table 6 Errors of Troesch’s problem for the case γ = 0.5 and different values of N. x
SGM.E N =5
SGM.E N = 10
SGM.E N = 20
BSM.E N =5
BSM.E N = 10
BSM.E N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.00000000 0.00147708 0.00250470 0.00324986 0.00249565 0.00000000
0.00000000 0.00151201 0.00264556 0.00314644 0.00243954 0.00000000
0.00000000 0.00149495 0.00266145 0.00313174 0.00244473 0.00000000
0.000000 0.001482 0.002639 0.003102 0.002424 0.000000
0.000000 0.001491 0.002656 0.003124 0.002439 0.000000
0.000000 0.001494 0.002660 0.003130 0.002443 0.000000
Table 7 Numerical solution of Troesch’s problem for the case γ = 1 and different values of N. x
Exact [12]
SGM N =5
SGM N = 10
SGM N = 20
BSM N =5
BSM N = 10
BSM N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.0000000000 0.1645308709 0.3367322092 0.5252740296 0.7411683782 1.0000000000
0.0000000000 0.1701116707 0.3466496482 0.5390616505 0.7527987047 1.0000000000
0.0000000000 0.1702320815 0.3471631282 0.5385947256 0.7525814007 1.0000000000
0.0000000000 0.1701715950 0.3472212611 0.5385360476 0.7526080999 1.0000000000
0.000000 0.169980 0.346874 0.538106 0.752251 1.000000
0.000000 0.170124 0.347137 0.538429 0.752520 1.000000
0.000000 0.170160 0.347201 0.538508 0.752586 1.000000
Table 8 Errors of Troesch’s problem for the case γ = 1 and different values of N. x x
SGM.E N =5
SGM.E N = 10
SGM.E N = 20
BSM.E N =5
BSM.E N = 10
BSM.E N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.0000000000 0.0055807998 0.0099174390 0.0137876209 0.0116303265 0.0000000000
0.0000000000 0.0057012106 0.0104309190 0.0133206960 0.0114130225 0.0000000000
0.0000000000 0.0056407241 0.0104890519 0.0132620180 0.0114397217 0.0000000000
0.000000 0.005449 0.010142 0.012832 0.011083 0.000000
0.000000 0.005593 0.010405 0.013155 0.011352 0.000000
0.000000 0.005629 0.010469 0.013234 0.011418 0.000000
Fig. 2. Sinc–Galerkin solutions of Troesch’s problem for γ ≤ 5.
Table 7 shows comparison of the exact solutions, numerical solutions of the sinc–Galerkin method and the solutions of the B-spline method for different values of N in mesh points x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 for γ = 1. Table 8, indicates the absolute errors of Troesch’s problem obtained by the B-spline and the sinc–Galerkin method for N = 5, 10, 20, γ = 1 and x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. Fig. 2 displays solutions of the sinc–Galerkin method for γ = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5. 6. Conclusions The sinc–Galerkin method is used to solve the nonlinear two point boundary value problem with application to Troesch’s equation. The results obtained here were compared with the exact solution, HPM, spline, perturbation, B-spline, Laplace methods. The method is computationally attractive and applications are demonstrated through illustrative examples.
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Acknowledgments We thank the referees for their valuable comments that helped us in revising this paper. References [1] E.S. Weibel, On the confinement of a plasma by magnetostatic fields, Physics of Fluids 2 (1) (1959) 52–56. [2] V.S. Markin, A.A. Chernenko, Y.A. Chizmadehev, Y.G. Chirkov, Aspects of the theory of gas porous electrodes, in: V.S. Bagotskii, Y.B. Vasilev (Eds.), Fuel Cells: Their Electrochemical Kinetics, Consultants Bureau, New York, 1966, pp. 21–33. [3] D. Gidaspow, B.S. Baker, A model for discharge of storage batteries, Journal of The Electrochemical Society 120 (1973) 1005–1010. [4] Xinlong Feng, Liquan Mei, Guoliang He, An efficient algorithm for solving Troesch’s problem, Applied Mathematics and Computation 189 (2007) 500–507. [5] Shih-Hsiang Chang, A variational iteration method for solving Troesch’s problem, Journal of Computational and Applied Mathematics 234 (2010) 3043–3047. [6] Shih-Hsiang Chang, Numerical solution of Troesch’s problem by simple shooting method, Applied Mathematics and Computation 216 (2010) 3303–3306. [7] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York, 1993. [8] J. Lund, K. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, PA, 1992. [9] J. Lund, Symmetrization of the sinc-Galerkin method for boundary value problems, Mathematics of Computation 47 (1986) 571–588. [10] L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces, The Macmillan Company, New York, 1964. [11] S.H. Mirmoradia, I. Hosseinpoura, S. Ghanbarpourb, A. Bararia, Application of an approximate analytical method to nonlinear Troesch’s problem, Applied Mathematical Sciences 3 (32) (2009) 1579–1585. [12] S.A. Khuri, A. Sayfy, Troesch’s problem: A B-spline collocation approach, Mathematical and Computer Modelling 54 (9–10) (2011) 1907–1918.
Contents lists available at SciVerse ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
The sinc–Galerkin method for solving Troesch’s problem M. Zarebnia a,∗ , M. Sajjadian b a
Department of Mathematics, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran
b
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
article
abstract
info
Article history: Received 16 August 2011 Received in revised form 25 November 2011 Accepted 28 November 2011
In this paper, the sinc–Galerkin method is applied for solving Troesch’s problem. The properties of the sinc procedure are utilized to reduce the computation of Troesch’s equation to nonlinear equations with unknown coefficients. The exponential convergence √
rate of the method, O(e−k N ), is developed. We use some numerical examples to illustrate the accuracy and implementation of the method. The results are compared with homotopy perturbation method (HPM), Laplace method, perturbation method and spline method by using tables. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Numerical method Nonlinear Troesch’s problem Sinc function Galerkin
1. Introduction Consider a two-point boundary value problem, Troesch’s problem, as follows: u′′ = γ sinh(γ u), u(0) = 0,
0 ≤ x ≤ 1,
u(1) = 1,
(1)
where γ is a positive constant. Troesch’s problem is discussed by Weibel and arises in the investigation of the confinement of a plasma column by radiation pressure [1] and also in the theory of gas porous electrodes [2,3]. The closed form solution to this problem in terms of the Jacobian elliptic function has been given [4] as u(x) =
2
γ
sinh
−1
u′ (0) 2
1 sc γ x|1 − u′ (0)2 4
,
(2)
√
where 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
γ 2
√
1−m
= sc(γ |m),
(3)
where the Jacobian elliptic function sc(γ |m) is defined by sc(γ |m) =
γ =
0
∗
φ
1
1 − m sin2 θ
sin φ , cos φ
where φ, γ are related by the integral
dθ .
Corresponding author. Tel.: +98 4515514702; fax: +98 4515514701. E-mail addresses: [email protected] (M. Zarebnia), [email protected] (M. Sajjadian).
0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.11.071
(4)
M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
219
From (2), it was noticed that a pole of u(t ) occurs at a pole of sc γ x|1 − 41 u′ (0)2 . It was also noticed that the pole occurs at x≈
1
2γ
16
ln
.
1−m
It also has an equivalent definition given in terms of a lattice. Troesch’s problem has been solved by another method. Xinlong Fen et al. [4] have introduced an efficient algorithm for solving this problem; their method is based on the modified homotopy perturbation technique. They have compared this method with the variational iteration method and the Adomian decomposition method. The variational iteration method is applied for solving Troesch’s problem by Chang [5]. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. In another article, Chang [6] has used simple shooting method. The paper is organized into five sections. Section 2 outlines some of the main properties of sinc functions and sinc method that are necessary for the formulation of the discrete Troesch’s problem. In Section 3, we illustrate how the sinc–Galerkin method may be used to replace Eq. (1) by an explicit system of nonlinear algebraic equations. In Section 4, the convergence analysis of the method has been discussed. It is shown that the sinc procedure converges to the solution at an exponential rate. Finally, we report our numerical results and demonstrate the efficiency and accuracy of the proposed numerical scheme by considering some numerical examples in Section 5. 2. Survey of some properties of the sinc function In this section, we will review sinc function properties, sinc quadrature rule, and the sinc method. These are discussed thoroughly in [7,8]. For solving Troesch’s equation (1), these properties will be used extensively in Section 3. The sinc function. The sinc function is defined on the whole real line, −∞ < x < ∞, by
sinc(x) =
sin(π x)
1,
πx
,
x ̸= 0;
(5)
x = 0.
For any h > 0, the translated sinc functions with evenly spaced nodes are given as S (j, h)(z ) = sinc
z − jh h
,
j = 0, ±1, ±2, . . . .
(6)
The sinc functions are cardinal for the interpolating points zk = kh in the sense that (0)
S (j, h)(kh) = δjk =
1, 0,
k = j; k ̸= j.
(7)
If f is defined on the real line, then for h > 0 the series C (f , h)(z ) =
∞
f (jh)sinc
z − jh
j=−∞
h
,
(8)
is called the Whittaker cardinal expansion of f whenever this series converges. They are based on the infinite strip Ds in the complex plane
Ds = w = u + iv : |v| < d ≤
π 2
.
(9)
To construct approximation on the interval (0, 1), we consider the conformal map
φ(z ) = ln
z 1−z
.
(10)
The map φ carries the eye-shaped region
z π DE = z = x + iy : arg .
This is shown in Fig. 1.
(11)
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
Fig. 1. The domains DE and DS .
For the sinc method, the basis functions on the interval (0, 1) for z ∈ DE are derived from the composite translated sinc functions, Sj (z ) = S (j, h) ◦ φ(z ) = sinc
φ(z ) − jh
h
.
(12)
The function z = φ −1 (w) =
ew
(13)
1 + ew
is an inverse mapping of w = φ(z ). We define the range of φ −1 on the real line as
Γ = {ψ(u) = φ −1 (u) ∈ DE : −∞ < u < ∞}. (14) ∞ The sinc grid points zk ∈ (0, 1) in DE will be denoted by xk because they are real. For the evenly spaced nodes {kh}k=−∞ on the real line, the image which corresponds to these nodes is denoted by xk = φ −1 (kh) =
ekh 1 + ekh
,
k = ±1, ±2, . . . .
(15)
Sinc interpolation and quadrature rules. For further explanation of the procedure, the important class of functions is denoted by B(DE ). The properties of functions in B(DE ) and detailed discussions are given in [7,8]. We recall the following definitions and theorems for our purpose. Definition 1. Let B(DE ) denote the family of functions F which are analytic in DE and satisfy
ψ(u+L)
|F (z )dz | → 0,
u → ±∞,
where L = {iv : |v| < d ≤ π2 }, and on the boundary of DE (denoted by ∂ DE ) satisfy
N (F ) ≡
∂ DE
|F (z )dz | < ∞.
Theorem 1. If φ ′ ∈ B(DE ), then for all x ∈ Γ
∞ N (F φ ′ ) F (xk )S (k, h)oφ(x) ≤ F (x) − 2π d sinh(π d/h) k=−∞ ≤
2N (F φ ′ ) −π d/h e . πd
Moreover, if |F (x)| ≤ C1 e−α|φ(x)| , x ∈ Γ , for some positive constants C1 and α , and also h = F (x) =
N
1
F (xk )S (k, h)oφ(x) + O exp −(π dα N ) 2
.
√ π d/α N, then (16)
k=−N
In this paper, we assume that the solution to Troesch’s problem satisfies the hypotheses of Theorem 1, which mandates that this solution vanishes at the endpoints of the interval Γ = [a, b]. Nonhomogeneous boundary conditions require a
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221
slight modification of the techniques of this paper. For example, Troesch’s problem with boundary conditions u( a) = ua ,
u(b) = ub
where at least one of ua or ub is non-zero, we could proceed by using the space Mα (D) [7, p. 180] or substituting ua + ρ(x)ub
u(x) = y(x) +
1 + ρ(x)
ρ(x) = eφ(x) ,
,
in Troesch’s problem (1) and thus derive a new boundary value problem with homogeneous boundary conditions. Theorem 2. Let F ∈ B(DE ) and φ be a conformal map with constants α and C2 so that
F (z ) −α|φ(z )| , φ ′ ( z ) ≤ C2 e
z ∈ Γ,
then the sinc trapezoidal quadrature rule is
Γ
F (z )dz = h
N F (zj ) + O(exp(−α Nh)) + O(exp(−π d/h)). ′ (z ) φ j j=−N
(17)
Hence, by selecting
h=
πd αN
12 ,
(18) 1
the exponential order of the sinc trapezoidal quadrature rule in Eq. (17) is O(exp(−(π dα N ) 2 )). By applying Theorem 2, we conclude the following corollary that is a special case of (17). Corollary. Let F ∈ B(DE ), and let h be selected by Eq. (18), then
Γ
F (zk ) 1 F (z )S (k, h) ◦ φ(z )dz = h ′ + O exp −(π dα N ) 2 . φ (z )
(19)
k
Theorem 3. Let φ be a conformal one-to-one map of the simply connected domain DE onto Ds . Then
δjk(0) = [S (j, h)oφ(x)]|x=xk = δjk(1) = h
d dφ
1, 0,
[S (j, h)oφ(x)]|x=xk
k = j; k ̸= j,
(20)
0, = (−1)(k−j) , (k − j)
k = j; k ̸= j,
(21)
and
δjk(2) = h2
d2 dφ 2
[S (j, h)oφ(x)]|x=xk =
−π 2 ,
3 (k−j) −2(−1)
(k − j)2
k = j; (22)
,
k ̸= j.
3. The sinc–Galerkin method Consider Troesch’s problem that is a nonlinear boundary value problem as: u′′ (x) − γ sinh(γ u(x)) = 0,
u(0) = 0,
u(1) = 1.
(23)
Let u(x) be the solution of boundary value problem (23) and u(x) ∈ B(DE ). We approximate the solution of Eq. (23) by the sinc–Galerkin method. For our purpose, first we convert the nonhomogeneous boundary conditions (23) to homogeneous conditions by considering the transformation y(x) = u(x) − x. This change of variable yields the following boundary value problem y′′ (x) − γ sinh(γ (y(x) + x)) = 0,
(24)
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
with boundary conditions y(0) = 0,
y(1) = 0.
(25)
In this case, consider sinc approximation by the formula N
y(x) ≈ yN (x) =
wj S (j, h)oφ(x).
(26)
j=−N
Our purpose is applying the Galerkin method based on sinc functions. Therefore, consider inner product for arbitrary functions f and g in the following form
⟨f , g ⟩ =
Γ
w(x)f (x)g (x)dx,
(27)
where
w(x) =
1 1
[φ ′ (x)] 2
.
Considering Eqs. (24)–(27), we have the sinc–Galerkin method as follows
⟨y′′ (x), S (k, h)oφ(x)⟩− < γ sinh(γ (y(x) + x)), S (k, h)oφ(x) >= 0.
(28)
Multiplying both sides of Eq. (28) in h and considering Eqs. (27) and (28), we obtain:
h Γ
S (k, h)oφ(x) ′′ yN (x)dx − hγ 1 [φ ′ (x)] 2
S (k, h)oφ(x)
sinh(γ (yN (x) + x))dx = 0.
1
[φ ′ (x)] 2
Γ
(29)
We apply part by part integration for the first term of Eq. (29) and then we get
h Γ
S (k, h)oφ(x) ′′ yN dx = h 1 [φ ′ (x)] 2
S (k, h)oφ(x)
Γ
[φ ′ (x)] 2
′′ yN (x)dx.
1
(30)
By using Eqs. (19) and (26), we can write
S (k, h)oφ(x)
Γ
[φ ′ (x)] 2
h
′′
1
yN (x)dx = h
2
N
S (k, h)oφ(xj )
′′
1
[φ ′ (xj )] 2
j=−N
.
1
wj .
φ ′ (xj )
(31)
Having used Eqs. (20)–(22), we have
[S (j, h)oφ(x)]|x=xk = φ ′
d dφ
′ = φ [S (j, h)oφ(x)] dφ
[S (j, h)oφ(x)] |x=xk ′′
[S (j, h)oφ(x)]|x=xk = φ ′ (xk )h−1 δjk(1) ,
′
d
(32)
= φ ′′ (xk )h−1 δjk(1) + [φ ′ (xk )]2 h−2 δjk(2) .
(33)
x =x k
From the above relations and Eq. (31), we obtain h Γ
(2) S (k, h)oφ(x) ′′ 1 yN (x)dx = δk,j [φ ′ (zj )] 2 + hδk(1,j) 1 ′ [φ (x)] 2 j=−N N
+
(0) h2 k,j
δ
−1
[φ ′ (zj ) 2 ]′′ φ ′ (zj )
′
φ ′′ (zj )
+ 2 [φ (zj )] ′
3
[φ ′ (zj )] 2
−1 2
wj .
(34)
Now, consider the second term of Eq. (29). By applying Eq. (19), we can obtain hγ
S (k, h)oφ(x)
Γ
[φ ′ (x)]
1 2
sinh γ (yN (x) + x)dx = γ h2
= γh
sinh γ (yN (xk ) + xk )
×
1
k) [φ ′ (xk )] N sinh(γ j=−N wj S (j, h)oφ(xk ) + xk ) 2 1 2
3
[φ ′ (xk )] 2
φ ′ (x
.
(35)
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By using the properties of the sinc function and applying Eq. (20), we rewrite Eq. (35) as hγ
S (k, h)oφ(x)
1
[φ ′ (x)] 2
Γ
sinh γ (yN (x) + x) = γ h2
sinh γ (wk + xk ) 3
[φ ′ (xk )] 2
.
(36)
Substituting (34) and (36) in (29), we get the Galerkin results as follows: N
1 δk(2,j) [φ ′ (zj )] 2
+
(1) hδk,j
sinh γ (wk + xk ) 3
[φ ′ (xk )] 2
3
= 0,
+ 2([φ (zj )] ′
[φ ′ (zj )] 2
j=−N
− γ h2
φ ′′ (zj )
−1 ′ 2
)
+
(0) h2 k,j
δ
−1
[φ ′ (zj ) 2 ]′′ φ ′ (zj )
wj
k = −N , −N + 1, . . . , N .
(37) (2)
Eq. (37) is a nonlinear system with (2N + 1) equations and (2N + 1) unknowns wj , j = −N , . . . , N. We set I (2) = [δkj ], (2)
where δkj denotes the (k, j)th element of the matrix I (2) . Also, we define the diagonal matrix D(g ) as follows: g (x−N ) 0
0 0
.. .
0 0
··· ···
g (x−N +1 )
.. .
D(g ) =
··· ··· .. .
0
0 0
0 0
.. .
.. . , 0
g (xN −1 ) 0
g (xN )
(2)
where I and D(g ) are square matrices of order (2N + 1) × (2N + 1). In order to determine (2N + 1) unknowns wj , j = −N , . . . , N in nonlinear system of Eq. (37), we rewrite this system in matrix form as: AW − γ h2 D
1
F (W ) = 0,
3
[φ ′ ] 2
(38)
where
A=
I
(2)
+h D
1
2
′′
1
×D
√ ′ φ
3
φ′ 2
φ
1
′2
,
(39)
F (W ) = sinh(γ (w−N + x−N )), sinh(γ (w−N +1 + x−N +1 )), . . . , sinh(γ (wN + xN )) W = w−N , w−N +1 , . . . , wN
T
T
,
.
Solving the nonlinear system (38) by Newton’s method we obtain an approximate solution yj , j = −N , . . . , N, then we can obtain an approximation to the solution of Troesch’s problem (1): uN (x) =
N
wj S (j, h)oφ(x) + x.
(40)
j=−N
4. Convergence analysis In this section, the convergence of the sinc–Galerkin method for Troesch’s problem (1) is discussed. For this purpose, we apply the sinc–Galerkin method for the linear boundary value problem
(Ly)(x) = y′′ (x) + µy′ (x) + ν y(x) = σ (x), y(a) = 0, y(b) = 0.
a < x < b, (41)
Consider the approximate solution (26). As may be expected in view of Section 3, by using the inner product (27), we can obtain a similar result. We select yN (x) as in (26) and then we evaluate the integrals
⟨LyN , S (k, h)oφ⟩ = 0,
k = −N , . . . , N .
After integrating by parts to eliminate derivatives in y, and dropping the intermediate boundary terms, which can be shown to vanish under assumptions which we shall state below. Then the discrete sinc–Galerkin system for the determination of the unknown coefficients {wj }Nj=−N is given by
I (2) D(φ ′ w) + hI (1) D
φ ′′ w φ′
+ 2w′ − µw + +h2 D
w ′′ − (µw)′ + νw φ′
where 1 is a (2N + 1)-vector each of whose components are 1 and w(x) = (φ ′ (x))
W = h2 D
−1 2
.
σw 1, φ
(42)
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
The error in approximating the exact solution of boundary value problem (41) by (26) is given in the following theorem [9]. Theorem 4. Assume that the coefficients µ, ν and σ are analytic in the region DE and that problem (41) has a unique solution y(x) which is analytic in DE . Assume also that
′′ ′ y (z ) y (z ) ′ dz < ∞, ′ dz < ∞, ∂ D φ (z ) ∂ D φ (z ) ν(z )y(z ) σ (z ) φ ′ (z ) dz < ∞, ′ dz < ∞, ∂D ∂ D φ (z )
and for positive α ,
|y(x)| ≤ K exp(−α|φ(x)|),
x ∈ Γ.
1 2
If h = (π d/α N ) and the coefficients {wj }Nj=−N in (26) are determined from (42) with w(x) = 1
3
|y(x) − yN (x)| ≤ C3 N 2 e−(2π dαN ) 2 ,
1 1
(φ ′ (x)) 2
, then
x ∈ Γ.
Now, we discuss convergence of the sinc–Galerkin method for Troesch’s equation and obtain a bound on the error y(x) − yN (x) in the maximum norm, where y(x) is the exact solution and yN (x) is the approximate solution (26). Lemma 1. Let y be the exact solution of the nonlinear equation Ly − γ G(y) = 0, where (Ly)(x) = y′′ (x) is a second-order linear operator and G(y) = sinh(γ y(x) + x) is a nonlinear operator. Let y ∈
−1
B(DE ), G′ (y) = ∂∂Gy , and G′′ (y) = ∂∂ yG2 be well defined and bounded on the ball B(y0 , r ). Also, let L − γ G′ (y) 2
−1 L − γ G (y ) γ G(y0 ) − Ly0 be bounded on B(y0 , r ), and −1 ∥γ G(y0 ) − Ly0 ∥∞ ≤ H0 , ∥ L − γ G′ (y) ∥∞ ≤ H1 ,
′
and
0
∥γ G′′ (y)∥∞ ≤ H2 ,
y ∈ B(y0 , r ),
y ∈ B(y0 , r ).
(43)
If h˜ = H12 H2 H0 < 2,
(44)
and r > H0 H1
2k −1 ∞ h˜ k=0
2
,
then the sequence
yn+1 = yn + L − γ G′ (yn )
−1 γ G(yn ) − Lyn
(45)
is well defined. Also yn+1 ∈ B(DE ) for every positive integer n and the sequence yn converges to y∗ . Furthermore, n
∥yn − y∗ ∥∞ ≤ H1 H0
(h˜ /2)2 −1 . n 1 − (h˜ /2)2
(46)
Proof. By applying Kantorovich’s Theorem [10], we can conclude the existence of the sequence {yn }n≥0 and the bound (46). √
From Lemma 1, we can show that the sinc–Galerkin method converges at the rate of O(e−k
N
), k > 0.
Theorem 5. Let us consider all assumptions in Lemma 1 and let, the discrete equivalent of γ G′ (y) and γ G′′ (y) be well defined and bounded on the ball B(y0 , r ), and also the discrete equivalent of (L − γ G′ (y))−1 be bounded on B(y0 , r ). Let the sequence νNn be the discrete equivalent of (45). Then, (a) {νNn }n≥0 converges to νN∗ and νNn − νN∗ has a bound as defined in (46).
M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
225
Table 1 Numerical solutions of Troesch’s problem for the case γ = 0.5. x
Exact [11]
SGM
HPM [11]
LM [12]
PM [12]
SM [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.095176902 0.190633869 0.286653403 0.383522929 0.481537385 0.581001975 0.682235133 0.785571787 0.891366988
0.095944347 0.192128740 0.288794409 0.386184841 0.484547165 0.584133254 0.685201142 0.788016528 0.892854218
0.095948026 0.192135797 0.288804238 0.386196642 0.4845599 0.584145785 0.685212297 0.788025104 0.892859085
0.0959444 0.1921288 0.2887944 0.3861849 0.4845472 0.5841333 0.6852012 0.7880166 0.8928542
0.095995 0.192229 0.288944 0.386380 0.484782 0.584395 0.685468 0.788254 0.893011
0.095944 0.192128 0.288793 0.386380 0.484546 0.584132 0.685200 0.788015 0.892853
(b) There exists a constant C3 independent of N such that
3
1
sup |νN∗ − y∗ | ≤ C4 N 2 exp −(π dα N ) 2
x∈Γ
.
(47)
Proof. (a) Let {νNn }n≥0 be the discrete sequence by the sinc–Galerkin method that defined by the discrete equivalent of (45). Similarly, by using Lemma 1, the sequence {νNn }n≥0 exists and converges to νN∗ and moreover we have n
∥ν − νN ∥∞ n N
∗
(h˜ /2)2 −1 , ≤ H1 H0 n 1 − (h˜ /2)2
(48)
where H0 , H1 and h˜ are defined in (43) and (44). (b) Let the sequence yn defined by (45). By using Lemma 1, we know that the sequence yn exists and converges to y∗ and also n
(h˜ /2)2 −1 . n 1 − (h˜ /2)2 −1 By considering bounded inverse of L − γ G′ (y) on the ball B(y0 , r ) and Theorem 4, we have 3 3 sup |νNn − yn | ≤ C5 N 2 exp −(π dα N ) 2 . ∥yn − y∗ ∥∞ ≤ H1 H0
x∈Γ
(49)
(50)
Now, we consider the following inequality
|νn∗ − y∗ | ≤ |νN∗ − νNn | + |νNn − yn | + |yn − y∗ |.
(51)
By considering assumptions in Lemma 1 and h˜ < 2, the following inequality can be made for n large enough, n
H1 H0
(h˜ /2)2 −1 3 1 ≤ C6 N 2 exp{−(π dα N ) 2 }. n 1 − (h˜ /2)2
(52)
Having applied the relations (48)–(50) and (52), and also by having considered the maximum norm bounds for νN∗ −νNn , νNn − yn and yn − y∗ , we obtain 3
1
sup |νN∗ − y∗ | ≤ C4 N 2 exp{−(π dα N ) 2 }.
(53)
x∈Γ
5. Numerical examples In order to show numerical results of solving Troesch’s equation based on the sinc–Galerkin method and discuss the accuracy of the method, we consider the absolute errors of this method and other methods that are tabulated and compared in tables. In these examples, we suppose d = π2 and α = Error = |u(xj ) − uN (xj )|,
1 2
1
which yields h = π ( N1 ) 2 . The error function is given by
j = −N , . . . , N ,
where u and uN represent the exact and approximate solutions, respectively. These examples are solved for different values of γ . We solved Troesch’s problem (1) by using the sinc–Galerkin method(SGM). Solving the nonlinear system (38), we obtain the approximate solution uN (x). We calculated the approximate solution at the uniform grid points xi = ihu , i = 1, . . . , 9, , hu = 0.1. The results are tabulated in Table 1. Also, Table 1, shows the numerical solution of Troesch’s equation using Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods for γ = 0.5.
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M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228 Table 2 Errors of Troesch’s problem for the case γ = 0.5. x
SGM.E
HPM.E [11]
LM.E [12]
PM.E [12]
SM.E [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.000767445 0.001494871 0.002141006 0.002661912 0.003009780 0.003131279 0.002966009 0.002444741 0.001487230
0.000771124 0.001501928 0.002150835 0.002673713 0.003022515 0.00314381 0.002977164 0.002453317 0.001492098
7.7 × 10−4 1.5 × 10−3 2.1 × 10−3 2.7 × 10−3 3.0 × 10−3 3.1 × 10−3 3.0 × 10−3 2.4 × 10−3 1.5 × 10−3
8.2 × 10−4 1.6 × 10−3 2.3 × 10−3 2.9 × 10−3 3.2 × 10−3 3.4 × 10−3 3.2 × 10−3 2.7 × 10−3 1.6 × 10−3
7.7 × 10−4 1.5 × 10−3 2.1 × 10−3 2.7 × 10−3 3.0 × 10−3 3.1 × 10−3 3.0 × 10−3 2.4 × 10−3 1.5 × 10−3
Table 3 Numerical solutions of Troesch’s problem for the case γ = 1. x
Exact [11]
SGM
HPM [11]
LM [12]
PM [12]
SM [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.081796997 0.164530871 0.249167361 0.336732209 0.428347161 0.52527403 0.628971143 0.741168378 0.863970021
0.084661250 0.170171338 0.257393933 0.347222839 0.440599836 0.538534416 0.642128589 0.752608114 0.871362527
0.084934415 0.170697546 0.258133224 0.348116627 0.44157274 0.539498234 0.642987984 0.753267551 0.871733059
0.0846631 0.1701750 0.2573995 0.3472304 0.4406094 0.5385460 0.6421421 0.7526227 0.8713749
0.085417 0.171669 0.259603 0.350053 0.444010 0.542315 0.645984 0.756064 0.873671
0.084655 0.170160 0.257377 0.347201 0.440575 0.538508 0.642103 0.752586 0.871349
Table 4 Errors of Troesch’s problem for the case γ = 1. x
SGM.E
HPM.E [11]
LM.E [12]
PM.E [12]
SM.E [12]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.002864253 0.005640467 0.008226572 0.010490630 0.012252675 0.013260386 0.013157446 0.011439736 0.007392506
0.003137419 0.006166675 0.008965863 0.011384418 0.013225579 0.014224205 0.01401684 0.012099173 0.007763039
2.9 × 10−3 5.9 × 10−3 8.2 × 10−3 1.0 × 10−2 1.2 × 10−2 1.3 × 10−2 1.3 × 10−2 1.1 × 10−2 7.4 × 10−3
3.6 × 10−3 7.1 × 10−2 1.0 × 10−2 1.3 × 10−2 1.6 × 10−2 1.7 × 10−2 1.7 × 10−2 1.5 × 10−2 9.7 × 10−3
2.8 × 10−3 5.6 × 10−3 8.2 × 10−3 1.0 × 10−2 1.2 × 10−2 1.3 × 10−2 1.3 × 10−2 1.1 × 10−2 7.4 × 10−3
Table 5 Numerical solution of Troesch’s problem for the case γ = 0.5 and different values of N. x
Exact [12]
SGM N =5
SGM N = 10
SGM N = 20
BSM N =5
BSM N = 10
BSM N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.00000000 0.19063387 0.38352292 0.58100197 0.78557179 1.00000000
0.00000000 0.19211095 0.38602762 0.58425183 0.78806744 1.00000000
0.00000000 0.19214588 0.38616848 0.58414841 0.78801133 1.00000000
0.00000000 0.19212882 0.38618437 0.58413371 0.78801652 1.00000000
0.000000 0.192116 0.386162 0.584104 0.787996 1.000000
0.000000 0.192125 0.386179 0.584126 0.788011 1.000000
0.000000 0.192128 0.386183 0.584132 0.788015 1.000000
In Table 2, we compare the absolute errors of the sinc–Galerkin(SGM) with Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods for γ = 0.5. In Table 3, the sinc–Galerkin solutions are calculated for grid points xi = ihu , i = 1, . . . , 9, hu = 0.1 and γ = 1. The resulting solution is also contrasted with the other numerical techniques, namely with Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods. In Table 4, the absolute errors obtained by the sinc–Galerkin method at the mesh points x = 0, 0.1, 0.2, . . . , 0.9, 1.0 for the case γ = 1 are compared with the absolute errors of the Laplace(LM), HPM, Perturbation(PM) and Spline(SM) methods. We solved Troesch’s problem for different values N = 5, 10, 20 by presented method. Solving the nonlinear system (38), we obtain the approximate solution uN (x), the numerical results on the mesh points x = 0, 0.2, . . . , 0.8, 1 for the case γ = 0.5 are tabulated in Table 5. Also the results with the B-Spline Method (BSM) [12] have been shown for the same equation and parameters as in Table 5. Table 6, exhibits the absolute errors of the B-Spline(BSM) [12] and the sinc–Galerkin method for different values of N and γ = 0.5.
M. Zarebnia, M. Sajjadian / Mathematical and Computer Modelling 56 (2012) 218–228
227
Table 6 Errors of Troesch’s problem for the case γ = 0.5 and different values of N. x
SGM.E N =5
SGM.E N = 10
SGM.E N = 20
BSM.E N =5
BSM.E N = 10
BSM.E N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.00000000 0.00147708 0.00250470 0.00324986 0.00249565 0.00000000
0.00000000 0.00151201 0.00264556 0.00314644 0.00243954 0.00000000
0.00000000 0.00149495 0.00266145 0.00313174 0.00244473 0.00000000
0.000000 0.001482 0.002639 0.003102 0.002424 0.000000
0.000000 0.001491 0.002656 0.003124 0.002439 0.000000
0.000000 0.001494 0.002660 0.003130 0.002443 0.000000
Table 7 Numerical solution of Troesch’s problem for the case γ = 1 and different values of N. x
Exact [12]
SGM N =5
SGM N = 10
SGM N = 20
BSM N =5
BSM N = 10
BSM N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.0000000000 0.1645308709 0.3367322092 0.5252740296 0.7411683782 1.0000000000
0.0000000000 0.1701116707 0.3466496482 0.5390616505 0.7527987047 1.0000000000
0.0000000000 0.1702320815 0.3471631282 0.5385947256 0.7525814007 1.0000000000
0.0000000000 0.1701715950 0.3472212611 0.5385360476 0.7526080999 1.0000000000
0.000000 0.169980 0.346874 0.538106 0.752251 1.000000
0.000000 0.170124 0.347137 0.538429 0.752520 1.000000
0.000000 0.170160 0.347201 0.538508 0.752586 1.000000
Table 8 Errors of Troesch’s problem for the case γ = 1 and different values of N. x x
SGM.E N =5
SGM.E N = 10
SGM.E N = 20
BSM.E N =5
BSM.E N = 10
BSM.E N = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.0000000000 0.0055807998 0.0099174390 0.0137876209 0.0116303265 0.0000000000
0.0000000000 0.0057012106 0.0104309190 0.0133206960 0.0114130225 0.0000000000
0.0000000000 0.0056407241 0.0104890519 0.0132620180 0.0114397217 0.0000000000
0.000000 0.005449 0.010142 0.012832 0.011083 0.000000
0.000000 0.005593 0.010405 0.013155 0.011352 0.000000
0.000000 0.005629 0.010469 0.013234 0.011418 0.000000
Fig. 2. Sinc–Galerkin solutions of Troesch’s problem for γ ≤ 5.
Table 7 shows comparison of the exact solutions, numerical solutions of the sinc–Galerkin method and the solutions of the B-spline method for different values of N in mesh points x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 for γ = 1. Table 8, indicates the absolute errors of Troesch’s problem obtained by the B-spline and the sinc–Galerkin method for N = 5, 10, 20, γ = 1 and x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. Fig. 2 displays solutions of the sinc–Galerkin method for γ = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5. 6. Conclusions The sinc–Galerkin method is used to solve the nonlinear two point boundary value problem with application to Troesch’s equation. The results obtained here were compared with the exact solution, HPM, spline, perturbation, B-spline, Laplace methods. The method is computationally attractive and applications are demonstrated through illustrative examples.
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Acknowledgments We thank the referees for their valuable comments that helped us in revising this paper. References [1] E.S. Weibel, On the confinement of a plasma by magnetostatic fields, Physics of Fluids 2 (1) (1959) 52–56. [2] V.S. Markin, A.A. Chernenko, Y.A. Chizmadehev, Y.G. Chirkov, Aspects of the theory of gas porous electrodes, in: V.S. Bagotskii, Y.B. Vasilev (Eds.), Fuel Cells: Their Electrochemical Kinetics, Consultants Bureau, New York, 1966, pp. 21–33. [3] D. Gidaspow, B.S. Baker, A model for discharge of storage batteries, Journal of The Electrochemical Society 120 (1973) 1005–1010. [4] Xinlong Feng, Liquan Mei, Guoliang He, An efficient algorithm for solving Troesch’s problem, Applied Mathematics and Computation 189 (2007) 500–507. [5] Shih-Hsiang Chang, A variational iteration method for solving Troesch’s problem, Journal of Computational and Applied Mathematics 234 (2010) 3043–3047. [6] Shih-Hsiang Chang, Numerical solution of Troesch’s problem by simple shooting method, Applied Mathematics and Computation 216 (2010) 3303–3306. [7] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York, 1993. [8] J. Lund, K. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, PA, 1992. [9] J. Lund, Symmetrization of the sinc-Galerkin method for boundary value problems, Mathematics of Computation 47 (1986) 571–588. [10] L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces, The Macmillan Company, New York, 1964. [11] S.H. Mirmoradia, I. Hosseinpoura, S. Ghanbarpourb, A. Bararia, Application of an approximate analytical method to nonlinear Troesch’s problem, Applied Mathematical Sciences 3 (32) (2009) 1579–1585. [12] S.A. Khuri, A. Sayfy, Troesch’s problem: A B-spline collocation approach, Mathematical and Computer Modelling 54 (9–10) (2011) 1907–1918.