1. Statement of the problem

Transcription

1 Volue 14, ON THE ITERATIVE SOUTION OF A SYSTEM OF DISCRETE TIMOSHENKO EQUATIONS Peradze J. and Tsklaur Z. I. Javakhshvl Tbls State Uversty,, Uversty St., Tbls 0186, Georga Georgan Techcal Uversty, 77, M. Kostava St., Tbls 0175, Georga j peradze@yahoo.co; zvad tsklaur@yahoo.co Abstract. The varatonal and dfference ethods are used respectvely for spatal and te varables to solve a nonlnear ntegro-dfferental Toshenko dynac bea equaton. The resultng algebrac syste of cubc equatons s solved by the teratve ethod. The teraton process error s estated. Key words and phrases: Toshenko bea equaton, Galerkn ethod, Crank- Ncolson dfference schee, Jacob teraton ethod, Cardano forula, error estate AMS subject classfcaton 000: 65M60, 65M06, 65Q10, 65M15 1. Stateent of the proble et us consder the equaton u tt x, t)+u xxxx x, t) hu xxtt x, t) 1 λ + wth the tal boundary condton 0 ) u xx, t) dx u xx x, t)=0, 1) 0 < x <, 0 < t T, ux, 0) = u 0 x), u t x, 0) = u 1 x), u0, t) = u, t) = 0, u xx 0, t) = u xx, t) = 0, 0 x, 0 t T, ) h > 0, λ > 0,, T and u 0 t), u 1 t) are the gven constants and functons, ux, t) s the functon we want to defne. The equaton 1), whch descrbes the oscllaton of a bea by the Toshenko theory, s consdered n 1], ] and 7]. In the present paper we ntroduce an approxate algorth for the proble 1),) and study the accuracy of ts teraton part. Note that the proble of constructon of nuercal ethods through estaton of algorth errors was nvestgated n 5] for other nonlnear Toshenko bea equatons.

2 16 Bulletn of TICMI. Algorth a. Galerkn ethod An approxate soluton wll be sought for n the for of a fte su u n x, t) = u t) sn x, ), accordng to the Galerkn ethod, the coeffcents u t) are a soluton of the syste of ordnary dfferental equatons ) 1+h u t) + λ ) ] jπ u njt) u t)=0, = 1,,..., n, 4) wth the tal condton u 0) = a 0, u 0) = a 1, = 1,,..., n, 5) a p = 0 u p x) sn x dx, = 1,,..., n, p = 0, 1. b. Dfference schee et us ntroduce the functons y t) = u t), z t) = u t), 6) = 1,,..., n, and rewrte the syste 4),5) n the new notaton as ) ) ) ] 1 + h y t) + λ z 4 njt) z t) = 0, 7) z t) = y t), y 0) = a 1, z 0) = a0, = 1,,..., n. 8) The proble 7),8) wll be solved by the dfference ethod. On a te nterval 0, T ] we ntroduce a net wth step τ = T M and nodes t = τ, = 0, 1,..., M.

3 Volue 14, At the -th layer,.e., for t = t, the approxate values y t) and z t) are denoted by y and z. We ake use of the Crank-Ncolson schee 1 + h ) wth the condton z z 1 τ y y 1 τ c. Iteraton ethod + λ z nj + z 1 nj z + z 1 = 0, 9) = y + y 1, = 1,,..., M, = 1,,..., n, y 0 = a 1, z 0 = a0, = 1,,..., n. 10) We wll solve the syste 9),10) layer-by-layer. Assung that the soluton has been obtaned on the 1)th layer, to fnd t on the th layer we wll apply the Jacob teraton ethod 4]. For the sake of splcty we wll neglect the error of the fnal teraton approxaton on the 1)th layer. carred out by the forulas Ths eans that for fxed the countng wll be 1 + h + λ znj,k ) + z 1 j ) y,k+1 y 1 τ nj z,k+1 ) + z 1 ] z,k+1 + z 1 = 0, ] 11.1) y 1 z,k+1 z 1 = y,k+1 + y 1, τ = 1,,..., M, k = 0, 1,..., = 1,,..., n, and z 1 are the known values, = 1,,..., n, and y 0 = a 1, z 0 = a0, = 1,,..., n. 11.) After expressng y,k+1 n 11.) through y 1, z 1 and z,k+1 y,k+1 = y 1 + z,k+1 z 1 τ, 1)

4 18 Bulletn of TICMI and substtutng 1) nto 11.1), we coe to the expresson τ + h ) z,k+1 z 1 + λ τ znj,k ) + z 1 j τ 1 + h nj z,k+1 ) + z 1 ] z,k+1 + z 1 ) y 1 = 0. 1) Hence t follows that for each k the teraton process eans the realzaton of only one forula 1). After obtang the fnal teraton approxaton z,k+1, we substtute ths value nto 1) to fnd an approxaton for y, = 1,,..., n. By the expresson 1) we conclude that we have to solve a cubc equaton wth respect to z,k+1 at the k + 1)th teraton step for each. Ths equaton s wrtten n the for and d = τ h + z,k+1) + a z,k+1 + b z,k+1 + c = 0, 14) a = z 1, b = d + z 1 + e, c = ) d + z 1 + e z 1 τd y 1, ) ), e = 8 λ + ) + j 15) z nj,k + z 1 nj ). 16) et us apply Cardano s forula ] to the equaton 14) and the relatons 15),16). Recall that the a pror real root of the equaton w + aw + bw + c = 0 s equal to Thus we get w = a + 1) p s s + p=1 r = a z,k+1 = z r 7 + b, s = a 7 ab + c. + ) 1 ] 1, 1) p+1 σ,p, 17) p=1 k = 0, 1,..., = 1,,..., n,

5 Volue 14, and σ,p = 1) p s + r = d + z 1 + e, s = z 1 d z 1 s 4 + r 7 ) 1 ] 1 ) + e τd y 1. 18) 19) The consdered algorth of soluton of the proble 1), ) should be understood as the countng carred out by the forula 17). Havng z,k and takng 6) and ) nto consderaton, we construct the approxated value of the functon ux, t) for t = t as the su z,k sn x. 0). Iteraton ethod error Under the teraton ethod error u n,k we wll understand the dfference between 0) and the su z sn x, whch would gve an approxate value of the functon ux, t) for t = t f the dfference schee 9), 10) were solved exactly. Thus u n,kx) = z,k z ) sn x. 1) Our a conssts n estatng the error u n,k x). et us represent the syste 17) as z,k+1 = φ z n1,k, z n,k,..., z nn,k) ) and consder the Jacob atrx φ J = z nj,k ) n,. ) By vrtue of 17) 19) and ) the dagonal ters of the atrx J are equal to zero, whle for the nondagonal ters we have φ z nj,k = z nj,k 9 1 σ p=1,p z 1 + 1) p s z r ) ) s 1 ] 4 + r. 4) 7

6 0 Bulletn of TICMI By 18) and 19) σ,1 σ, = r, σ, σ,1 = s, ) 1 s 4 + r σ,1 + σ, =. 5) 7 Fro 4) and 5) follows φ z nj,k = 4 9 z nj,kz 1 σ,1 r + z nj,ks σ 4,1 + r 9 + σ4, ) 1 + σ, ) 1, j. 6) Now we apply the frst relaton of 5) to the estate σ p,1 + σp, σ,1σ, ) p, p = 1,. We get σ p,1 + σp, r ) p. Usng ths nequalty, 19) and 16), fro 6) we obtan s 1) 1 4 τ h + = s ) = 8 λ + φ z nj,k 4 z 1 + r r ) 1 z r 64 τ h + ) + z 1 s ) = 1 τ s p) p=1 ) s znj,k ) ) ] z 1 z 1, + n j ) h + z nj,k, 7) ] z nj,k + z 1 nj ) z 1, y 1. ξ a+ξ 1 4a We agan apply 19) and 16) and also the nequaltes for ξ > 0, a > 0 and 4ab a + b for arbtrary a and b. The results s as follows s 1) r s ) r s ) r h 1 τ + ) ) 1 4 z 1, 1 19 τ τ h + λ + ) 1 z 1, ) 1 y 1.

7 Volue 14, Usng these relatons n 7) we coe to a concluson that φ ) 1 1 znj,k 96 τ h + z ) 1 ) 16 τ λ + y 1 znj,k. 8) We wll need a vector nor equal to v 1 = n v and the correspondng nor for the atrces U 1 = ax u j, v = v ) n, U = u j ) n,. By ) and 7) we get 1 j ) 1 J 1 ax 1 j n z nj,k 96 τ τ λ + h + ) 1 z 1 ) 1 ) y 1. 9) By the prncple of copressed appng let us assue that the condton J 1 q s fulflled for 0 < q < 1 and z,k )n, k = 0, 1,..., belongng to the doan { v ) n R n : v z,0 1 1 q } z,1 z,0. As seen fro 8), for ths t suffces that the restrcton τ 1 α be fulflled for the step τ. Here α = 1 h + z 1, β = 6 ) 1 γ = 96 1 q ] β + β + 4αγ) 1 1 λ + ) y 1, z, q z,1 z,0 ) ] 1. 0) If ths restrcton s fulflled, then the syste 9),10) has a uque soluton y, z, the teraton process 17) converges, l z,k = z k, = 1,,..., n, and the convergence rate s defned by the nequalty z,k z qk 1 q z,1 z,0.

8 Bulletn of TICMI Usng ths relaton n 1), we coe to a concluson that f the condton 0) s fulflled, then for the 0, )-nor of the teraton ethod error we have the estate dp dx p u n,kx) 0,) ) 1 p π q k 1 q z,1 z,0, p = 0, 1, = 1,,..., M, k = 1,,.... The queston of accuracy of the Galerkn ethod for the proble 1),) s studed n 6]. R e f e r e n c e s 1] E. Henrques de Brto, A nonlnear hyperbolc equaton, Internat. J. Math. Math. Sc., 1980), no., ] N. Jacobson, Basc algebra. I. Second edton. W. H. Freean and Copany, New York 1985). ] G. P. Menzala and E. Zuazua, Toshenko s bea equaton as a lt of a nonlnear one-densonal von Karan syste, Proc. Roy. Soc. Ednburgh Sect. A, ), no. 4, ] J. M. Ortega and W. C. Rhenboldt, Iteratve soluton of nonlnear equatons n several varables, Reprnt of the 1970 orgnal. Classcs n Appled Matheatcs, 0. Socety for Industral and Appled Matheatcs SIAM), Phladelpha, PA, ] J. Peradze, The exstence of a soluton and a nuercal ethod for the Toshenko nonlnear wave syste, MAN Math. Model. Nuer. Anal ), no. 1, ] J. Peradze, On the accuracy of the Galerkn ethod for one nonlnear bea equaton, Math. Meth. Appl. Sc., 8 pp., 010 subtted). 7] M. Tucsnak, Se-nternal stablzaton for a non-lnear Bernoull-Euler equaton, Math. Methods Appl. Sc ), no. 11, Receved 1, 06, 010; revsed 1, 10, 010; accepted

9 Bblography 1] E. Henrques de Brto, A nonlnear hyperbolc equaton, Internat. J. Math. Math. Sc., 1980), no., ] N. Jacobson, Basc algebra. I. Second edton. W. H. Freean and Copany, New York 1985). ] G. P. Menzala and E. Zuazua, Toshenko s bea equaton as a lt of a nonlnear one-densonal von Karan syste, Proc. Roy. Soc. Ednburgh Sect. A, ), no. 4, ] J. M. Ortega and W. C. Rhenboldt, Iteratve soluton of nonlnear equatons n several varables, Reprnt of the 1970 orgnal. Classcs n Appled Matheatcs, 0. Socety for Industral and Appled Matheatcs SIAM), Phladelpha, PA, ] J. Peradze, The exstence of a soluton and a nuercal ethod for the Toshenko nonlnear wave syste, MAN Math. Model. Nuer. Anal ), no. 1, ] J. Peradze, On the accuracy of the Galerkn ethod for one nonlnear bea equaton, Math. Meth. Appl. Sc., 8 pp., 010 subtted). 7] M. Tucsnak, Se-nternal stablzaton for a non-lnear Bernoull-Euler equaton, Math. Methods Appl. Sc ), no. 11,