Volume 14, 2010
15
ON THE ITERATIVE SOLUTION OF A SYSTEM OF DISCRETE
TIMOSHENKO EQUATIONS
Peradze J. and Tsiklauri Z.
I. Javakhishvili Tbilisi State University, 2, University St., Tbilisi 0186,
Georgia
Georgian Technical University, 77, M. Kostava St., Tbilisi 0175, Georgia
j peradze@yahoo.com; zviad tsiklauri@yahoo.com
Abstract. The variational and difference methods are used respectively for
spatial and time variables to solve a nonlinear integro-differential Timoshenko
dynamic beam equation. The resulting algebraic system of cubic equations is
solved by the iterative method. The iteration process error is estimated.
Key words and phrases: Timoshenko beam equation, Galerkin method, CrankNicolson difference scheme, Jacobi iteration method, Cardano formula, error
estimate
AMS subject classification 2000: 65M60, 65M06, 65Q10, 65M15
1. Statement of the problem
Let us consider the equation
1 (
utt (x, t)+uxxxx (x, t)− huxxtt (x, t)−
λ+
2L
∫
L
)
u2x (x, t) dx
0
0 < x < L,
uxx (x, t) = 0, (1)
0 < t ≤ T,
with the initial boundary condition
u(x, 0) = u0 (x), ut (x, 0) = u1 (x),
u(0, t) = u(L, t) = 0, uxx (0, t) = uxx (L, t) = 0,
0 ≤ x ≤ L, 0 ≤ t ≤ T,
(2)
where h > 0, λ > 0, L, T and u0 (t), u1 (t) are the given constants and functions,
u(x, t) is the function we want to define.
The equation (1), which describes the oscillation of a beam by the Timoshenko theory, is considered in [1], [3] and [7]. In the present paper we introduce
an approximate algorithm for the problem (1),(2) and study the accuracy of
its iteration part. Note that the problem of construction of numerical methods through estimation of algorithm errors was investigated in [5] for other
nonlinear Timoshenko beam equations.
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Bulletin of TICMI
2. Algorithm
a. Galerkin method
An approximate solution will be sought for in the form of a finite sum
un (x, t) =
n
∑
uni (t) sin
i=1
iπ
x,
L
(3)
where, according to the Galerkin method, the coefficients uni (t) are a solution
of the system of ordinary differential equations
(
( )2 )
iπ
1+h
u′′ni (t)
L
[ ( )
]( )
)
n (
2
2
iπ
1 ∑ jπ
iπ
2
+ λ+
+
unj (t)
uni (t) = 0, i = 1, 2, . . . , n, (4)
L
4 j=1 L
L
with the initial condition
u′ni (0) = a1i ,
uni (0) = a0i ,
where
api
2
=
L
∫
L
up (x) sin
0
iπ
x dx,
L
i = 1, 2, . . . , n,
i = 1, 2, . . . , n,
(5)
p = 0, 1.
b. Difference scheme
Let us introduce the functions
yni (t) = u′ni (t),
iπ
uni (t),
L
i = 1, 2, . . . , n,
zni (t) =
and rewrite the system (4),(5) in the new notation as
[
]
(
( )2 )
( )2
n
∑
iπ
iπ
1
iπ
′
2
(t) + λ +
1+h
yni
+
(t)
zni (t) = 0,
znj
L
L
4 j=1
L
′
zni
(t) =
(6)
(7)
iπ
yni (t),
L
yni (0) = a1i ,
zni (0) =
iπ 0
a,
L i
i = 1, 2, . . . , n.
(8)
The problem (7),(8) will be solved by the difference method. On a time
T
interval [0, T ] we introduce a net with step τ = M
and nodes tm = mτ ,
m = 0, 1, . . . , M .
Volume 14, 2010
17
At the m-th layer, i.e., for t = tm , the approximate values yni (t) and zni (t)
m
m
are denoted by yni
and zni
.
We make use of the Crank-Nicolson scheme
(
[
]
( )2 ) m
( )2
n
m−1 2
m 2
m−1
) + (znj
)
iπ
yni − yni
iπ
1 ∑ (znj
1+h
+ λ+
+
L
τ
L
4 j=1
2
m−1
m
iπ zni
+ zni
= 0,
L
2
m−1
m
iπ yni
+ yni
=
, m = 1, 2, . . . , M,
L
2
i = 1, 2, . . . , n,
×
m−1
m
zni
− zni
τ
(9)
with the condition
0
yni
= a1i ,
0
zni
=
iπ 0
a,
L i
i = 1, 2, . . . , n.
(10)
c. Iteration method
We will solve the system (9),(10) layer-by-layer.
Assuming that the solution has been obtained on the (m − 1)th layer, to
find it on the mth layer we will apply the Jacobi iteration method [4]. For the
sake of simplicity we will neglect the error of the final iteration approximation
on the (m − 1)th layer. This means that for fixed m the counting will be
carried out by the formulas
(
( )2 ) m
m−1
yni,k+1 − yni
iπ
1+h
L
τ
[
( )2
m−1
m
2
iπ
1 (zni,k+1 ) + (zni )2
+ λ+
+
(11.1)
L
4 ]
2
n
m−1 2
m−1
m
m
)2 + (znj
) iπ zni,k+1
+ zni
1 ∑ (znj,k
+
= 0,
4 j=1
2
L
2
j̸=i
m−1
m−1
m
m
zni,k+1
− zni
+ yni
iπ yni,k+1
=
,
τ
L
2
m = 1, 2, . . . , M, k = 0, 1, . . . , i = 1, 2, . . . , n,
(11.2)
m−1
m−1
are the known values, i = 1, 2, . . . , n, and
and zni
where yni
0
= a1i ,
yni
0
=
zni
iπ 0
a,
L i
i = 1, 2, . . . , n.
m−1
m−1
m
m
and zni,k+1
, zni
After expressing yni,k+1
in (11.2) through yni
m
− zni
L zni,k+1
+2
iπ
τ
m−1
m
yni,k+1
=
m−1
−yni
,
(12)
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Bulletin of TICMI
and substituting (12) into (11.1), we come to the expression
(
) m
[
( )2
m−1
m−1 2
m
− zni
)2 + (zni
)
2 L
iπ zni,k+1
iπ
1 (zni,k+1
+h
+ λ+
+
τ iπ
L
τ
L
4
2
]
n
m−1 2
m−1
m
m
)2 + (znj
) iπ zni,k+1
+ zni
1 ∑ (znj,k
+
4 j=1
2
L
2
j̸=i
(
2
−
τ
(
1+h
iπ
L
)2 )
m−1
yni
= 0.
(13)
Hence it follows that for each k the iteration process means the realization
of only one formula (13). After obtaining the final iteration approximation
m
m
, we substitute this value into (12) to find an approximation for yni
,
zni,k+1
i = 1, 2, . . . , n.
By the expression (13) we conclude that we have to solve a cubic equation
m
with respect to zni,k+1
at the (k + 1)th iteration step for each i. This equation
is written in the form
m
m
m
(zni,k+1
)3 + ai (zni,k+1
)2 + bi zni,k+1
+ ci = 0,
where
and
32
di = 2
τ
m−1
m−1 2
ai = (zni
, bi = di + (z)ni
) + ei ,
m−1 2
m−1
ci = −di + (zni ) + ei zni − τ di
(
(
h+
L
iπ
)2 )
(
(
, ei = 8 λ +
iπ
L
)2 )
+
iπ
L
m−1
yni
,
n
∑
(
(14)
(15)
)
m−1 2
m
(znj,k
)2 + (znj
) .
j=1
j̸=i
(16)
Let us apply Cardano’s formula [2] to the equation (14) and the relations
(15),(16). Recall that the a priori real root of the equation
w3 + aw2 + bw + c = 0
is equal to
a
w=− +
3
where
r=−
2
∑
p=1
[
s
(−1)p +
2
a2
+ b,
3
s=
(
2
s
r
+
4
27
) 12 ] 13
,
2a3 ab
−
+ c.
27
3
Thus we get
m−1
∑
zni
=−
+
(−1)p+1 σi,p ,
3
p=1
2
m
zni,k+1
k = 0, 1, . . . ,
i = 1, 2, . . . , n,
(17)
Volume 14, 2010
where
[
σi,p
and
si
= (−1)p +
2
(
19
s2i
r3
+ i
4
27
) 12 ] 31
m−1 2
ri = di + 23 (z(ni
) + ei ,
)
m−1 2
2 m−1
si = 3 zni
−2di + 10
(zni
) + ei − τ d i
9
(18)
iπ
L
m−1
yni
.
(19)
The considered algorithm of solution of the problem (1), (2) should be
m
and
understood as the counting carried out by the formula (17). Having zni,k
taking (6) and (3) into consideration, we construct the approximated value of
the function u(x, t) for t = tm as the sum
n
∑
iπ
L m
zni,k sin x.
iπ
L
i=1
(20)
3. Iteration method error
Under the iteration method error ∆um
n,k we will understand the difference between (20) and the sum
n
∑
L m
iπ
zni sin x,
iπ
L
i=1
which would give an approximate value of the function u(x, t) for t = tm if the
difference scheme (9), (10) were solved exactly. Thus
∆um
n,k (x)
n
∑
L m
iπ
m
=
(zni,k − zni
) sin x.
iπ
L
i=1
(21)
Our aim consists in estimating the error ∆um
n,k (x).
Let us represent the system (17) as
m
m
m
m
zni,k+1
= φi (zn1,k
, zn2,k
, . . . , znn,k
)
and consider the Jacobi matrix
J=
(
∂φi
m
∂znj,k
(22)
)n
.
(23)
i,j=1
By virtue of (17)–(19) and (22) the diagonal terms of the matrix J are
equal to zero, while for the nondiagonal terms we have
)( 2
[
(
) 1]
2
m ∑
3 −2
znj,k
1
s
r
∂φi
1
i
i
m−1
m−1
+(−1)p si zni
+ ri2
+
. (24)
=−
2zni
m
2
∂znj,k
9 p=1 σi,p
3
4 27
20
Bulletin of TICMI
By (18) and (19)
σi,1 σi,2
(
ri
= ,
3
3
σi,2
−
3
σi,1
= si ,
s2i
r3
+ i
4
27
) 21
3
3
σi,1
+ σi,2
.
=
2
(25)
From (24) and (25) follows
)−1
ri
∂φi
4 m m−1 ( 2
2
−
z
σ
=
−
z
+
σ
i,1
i,2
m
∂znj,k
9 nj,k ni
3
(
)−1
ri2
2 m
4
4
+ znj,k si σi,1 +
, i ̸= j.
+ σi,2
3
9
(26)
2p
2p
Now we apply the first relation of (25) to the estimate σi,1
+ σi,2
≥ 2(σi,1 σi,2 )p ,
p = 1, 2. We get
( r )p
i
2p
2p
σi,1
+ σi,2
≥2
.
3
Using this inequality, (19) and (16), from (26) we obtain
)
∂φ ( 4
2
i m−1
m
|z
| + 2 |si | |znj,k
|
m ≤
∂znj,k 3ri ni
ri


(
( )2 )−1
3
∑
1
L
1
(p)
m−1
m
|,
(27)
≤  τ2 h +
|zni
|+2 2
|s | |znj,k
24
iπ
ri p=1 i
where
(1)
|si |
=
2
3
[
64
τ2
(
h+
( L )2 )
iπ
]
m−1 2
m−1
+ 49 (zni
) |zni
|,
[ (
]
( iπ )2 ) 2 m−1 2 ∑n ( m 2
) m−1
(2)
m−1 2
2
|si | = 3 8 λ + L
+ 3 (zni ) + j=1 (znj,k ) + (znj )
|zni |,
j̸=i
(
( L )2 ) iπ m−1
(3)
1
|si |
= 32 τ h + iπ
|yni |.
L
ξ
1
We again apply (19) and (16) and also the inequalities (a+ξ)
2 ≤ 4a for ξ > 0,
a > 0 and 4ab ≤ (a + b)2 for arbitrary a and b. The results is as follows
(1)
|si |
ri2
≤
1
24
|si |
≤
ri2
1
192
τ
2
(2)
(3)
|si |
ri2
≤
τ
32
τ
2
(
h+
( L )2 )−1
(
h+
(
λ+
iπ
( L )2 )−1
iπ
( iπ )2 )−1
L
iπ
L
m−1
|zni
|,
m−1
|zni
|,
m−1
|yni
|.
Volume 14, 2010
21
Using these relations in (27) we come to a conclusion that
(
( )2 )−1
∂φ ( 13
L
i m−1
τ2 h +
|zni
|
m ≤
∂znj,k 96
iπ
(
( )2 )−1
)
1
iπ
iπ m−1
m
+ τ λ+
|y
| |znj,k
|.
16
L
L ni
We will need a vector norm equal to ∥v∥1 =
norm for the matrices ∥U ∥1 = max
n
∑
1≤j≤m i=1
n
∑
(28)
|vi | and the corresponding
i=1
|uij |, where v = (vi )ni=1 , U = (uij )ni,j=1 .
By (23) and (27) we get
∥J1 ∥ ≤
(
(
( )2 )−1
n
)( 13 ∑
L
m−1
m
|
max |znj,k
|zni
τ2
|
h+
1≤j≤n
96
iπ
i=1
(
( )2 )−1
)
n
iπ
iπ m−1
1 ∑
λ+
|y
| .
+ τ
16 i=1
L
L ni
(29)
By the principle of compressed mapping let us assume that the condition
m
∥J∥1 ≤ q is fulfilled for 0 < q < 1 and (zni,k
)ni=1 , k = 0, 1, . . ., belonging to the
domain
n
{
∑
n
n
m
(vi )i=1 ∈ R :
|vi − zni,0
|≤
i=1
n
}
1 ∑ m
m
|zni,1 − zni,0
| .
1 − q i=1
As seen from (28), for this it suffices that the restriction
]
1
1 [
2
2
τ≤
−β + (β + 4αγ)
2α
(30)
be fulfilled for the step τ . Here
α=
n
∑
i=1
n
1
6 ∑
1
iπ m−1
m−1
|zni |, β =
|y
|,
iπ 2
L 2
13 i=1 λ + ( L ) L ni
h + ( iπ )
) ]−1
n (
96 [ ∑
1
m
m
m
.
γ=
q
|z − zni,0 |
|zni,0 | +
13
1 − q ni,1
i=1
If this restriction is fulfilled, then the system (9),(10) has a unique solution
m
m
m
m
, i = 1, 2, . . . , n,
, the iteration process (17) converges, lim zni,k
= zni
, zni
yni
k→∞
and the convergence rate is defined by the inequality
n
∑
i=1
m
|zni,k
−
m
zni
|
n
qk ∑ m
m
≤
|z − zni,0
|.
1 − q i=1 ni,1
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Bulletin of TICMI
Using this relation in (21), we come to a conclusion that if the condition
(30) is fulfilled, then for the L2 (0, L)-norm of the iteration method error we
have the estimate
( )1−p √
n
dp
L qk ∑ m
L
m
m
|z − zni,0
|,
≤
p ∆un,k (x) 2
dx
π
2 1 − q i=1 ni,1
L (0,L)
p = 0, 1,
m = 1, 2, . . . , M,
k = 1, 2, . . . .
The question of accuracy of the Galerkin method for the problem (1),(2) is
studied in [6].
References
[1] E. Henriques de Brito, A nonlinear hyperbolic equation, Internat. J. Math. Math.
Sci., 3 (1980), no. 3, 505–520.
[2] N. Jacobson, Basic algebra. I. Second edition. W. H. Freeman and Company, New
York (1985).
[3] G. P. Menzala and E. Zuazua, Timoshenko’s beam equation as a limit of a nonlinear
one-dimensional von Karman system, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), no.
4, 855–875.
[4] J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in
several variables, Reprint of the 1970 original. Classics in Applied Mathematics, 30. Society
for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
[5] J. Peradze, The existence of a solution and a numerical method for the Timoshenko
nonlinear wave system, M2AN Math. Model. Numer. Anal. 38 (2004), no. 1, 1–26.
[6] J. Peradze, On the accuracy of the Galerkin method for one nonlinear beam equation,
Math. Meth. Appl. Sci., 8 pp., 2010 (submitted).
[7] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation,
Math. Methods Appl. Sci. 19 (1996), no. 11, 897–907.
Received 1, 06, 2010; revised 13, 10, 2010; accepted 02.12.2010
Bibliography
[1] E. Henriques de Brito, A nonlinear hyperbolic equation, Internat. J. Math. Math. Sci.,
3 (1980), no. 3, 505–520.
[2] N. Jacobson, Basic algebra. I. Second edition. W. H. Freeman and Company, New York
(1985).
[3] G. P. Menzala and E. Zuazua, Timoshenko’s beam equation as a limit of a nonlinear
one-dimensional von Karman system, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000),
no. 4, 855–875.
[4] J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Reprint of the 1970 original. Classics in Applied Mathematics, 30. Society
for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
[5] J. Peradze, The existence of a solution and a numerical method for the Timoshenko
nonlinear wave system, M2AN Math. Model. Numer. Anal. 38 (2004), no. 1, 1–26.
[6] J. Peradze, On the accuracy of the Galerkin method for one nonlinear beam equation,
Math. Meth. Appl. Sci., 8 pp., 2010 (submitted).
[7] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation,
Math. Methods Appl. Sci. 19 (1996), no. 11, 897–907.
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