Phương pháp lặp giải một bài toán biến đổi với phương trình kiểu song điều hòa.

Phương pháp lặp giải một bài toán biến đổi với phương trình kiểu song điều hòa. Ý tưởng cốt lõi của điều khiển học là hình thành lĩnh vực tập trung về tính mục đích: sự định hướng mục đích là do các vòng phản hồi âm giảm bớt sự chênh lệch giữa mục đích - tình trạng mong muốn với trạng thái đã đạt được.
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Phương pháp lặp giải một bài toán biến đổi với phương trình kiểu song điều hòa. Journal of Computer Science and Cybernetics, Vol.22, No.3 (2006), 229—234 ITERATIVE METHOD FOR SOLVING A BOUNDARY VALUE PROBLEM FOR BIHARMONIC TYPE EQUATION* DANG QUANG A1 , LE TUNG SON2 1 Institute of Information Technology 2 Thai Nguyen Pedagogic CollegeAbstract. The solution of boundary value problems (BVP) for fourth order differential equationsby their reduction to BVP for second order equations with the aim to use the achievements for thelatter ones attracts attention from many researchers. In this paper, using the technique developedby ourselves in recent works, we construct iterative method for a BVP for biharmonic type equation.The performed numerical experiments show very fast convergence of the proposed algorithm.T´m t˘t. Viˆc giai c´c b`i to´n biˆn dˆi v´.i phu.o.ng tr` dao h`m riˆng cˆ p bˆn b˘ ng c´ch du.a o ´ a e . ’ a a a ´ e o o ınh . a e a ´ ` ´ o a ach´ng vˆ c´c b`i to´n biˆn dˆi v´.i phu.o.ng tr` cˆ p hai d˜ thu h´t su. quan tˆm cua nhiˆu t´c gia. u ` a a e a ´ e o o ınh a´ a u . a ’ ` a e ’Trong b`i b´o n`y, su . a a a ’. dung k˜ thuˆt do ch´ng tˆi ph´t triˆ n trong nhi`u cˆng tr` m´.i dˆy, mˆt y a . u o a e’ e o ınh o a o .phu.o.ng ph´p l˘p giai mˆt b`i to´n biˆn cho phu.o.ng tr` song kiˆ u diˆu h`a d˜ du.o.c dˆ xuˆ t. Su. a a . ’ o a . a e ınh e’ ` e o a . ` e a ´ .hˆi tu nhanh cua phu.o.ng ph´p d˜ du.o.c ch´.ng to trˆn nhiˆu thu.c nghiˆm t´ to´n. o . . ’ a a . u ’ e `e . e . ınh a 1. INTRODUCTION The solution of fourth order differential equations by their reduction to boundary valueproblems (BVP) for the second order equations, with the aim of using efficient algorithms forthese, attracts attention from many researchers. Namely, for the biharmonic equation with theDirichlet boundary condition, there is intensively developed the iterative method, which leadsthe problem to two problems for the Poisson equation at each iteration (see e.g. [4, 9, 11, 12]).Recently, Abramov and Ulijanova [1] proposed an iterative method for the Dirichlet problemfor the biharmonic type equation, but the convergence of the method is not proved. Inour previous works [6, 7] with the help of boundary or mixed boundary-domain operatorsappropriately introduced, we constructed iterative methods for biharmonic and biharmonictype equations associated with the Dirichlet boundary condition. Recently, in [8] for solvingthe Neumann BVP we introduced a purely domain operator for studying the convergence ofan iterative method. It is proved that the methods are convergent with the rate of geometricprogression. In this paper we develop our technique in [4] - [8] for the following problem 2 Lu ≡ u − a u + bu = f in , (1) ∂ u u = g0 , = g1 on Γ, (2) ∂νwhere is the Laplace operator, is a bounded domain in Rn (n ≥ 2), Γ is the sufficiently∗ This work is supported in part by the National Basic Research Program in Natural Sciences, Vietnam230 DANG QUANG A, LE TUNG SONsmooth boundary of , ν is the outward normal to Γ and a, b are positive constants. Theequation (1) with other boundary conditions are met, for example, in [2, 3, 8]. An iterativemethod reducing the problem to a sequence of Neumann and Dirichlet problems for secondorder equations will be proposed and investigated by experimental way. 2. CONSTRUCTION OF ITERATIVE METHOD Set u = v and ...