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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 [email protected]; zviad [email protected] 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. 16 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) 18 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 22 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. 23