tailieunhanh - Finite element approximation of a boundary value problem for fourth oder differential equation
A finite-element approximation of a boundary value problem for fourth oder differential equation is given in work of J. Y. Shin [1]. In this paper, we introduce some results in [1] with a different way in the proof of lemma and correct a mistake in theorem . After that, we provide a remark of choosing an appropriate parameter ω which will guarantee the convergence of the iteration. | FINITE-ELEMENT APPROXIMATION OF A BOUNDARY VALUE PROBLEM FOR FOURTH ODER DIFFERENTIAL EQUATION Nguyen Thanh Huong 1 2 1 and Vu Vinh Quang 2 Thainguyen University, College of Sciences Thainguyen University, Information and Communication Technology ABSTRACT A finite-element approximation of a boundary value problem for fourth oder differential equation is given in work of J. Y. Shin [1]. In this paper, we introduce some results in [1] with a different way in the proof of lemma and correct a mistake in theorem . After that, we provide a remark of choosing an appropriate parameter ω which will guarantee the convergence of the iteration. Key words: Fourth oder differential equation, Finite-element approximation. 1. Introduction In the study of transverse vibrations of a hinged beam there arises the following boundary value problem fourth order differential eqution: y (4) 2 − εy − π 00 Z π 2 y 0 dx y 00 = p(x), 0 ≤ x ≤ π, () 0 00 00 y(0) = y(π) = y (0) = y (π) = 0, where > 0 is a constant, p(x) is a continuous function and nonpositive on [0, π]. Letting φ = −y 00 , problem () is reduced to the system of two second order equations: Z 2 π 0 2 00 −φ + εφ + y dx φ = p(x), 0 ≤ x ≤ π, π 0 () −y 00 − φ = 0, y(0) = y(π) = φ(0) = φ(π) = 0. Letting H01 denote the Sobolev space of L2 (0, π) functions in L2 (0, π) and vanishing at 0, π, the variational formulation of follows. Find (φ, y) ∈ H01 × H01 such that Z 2 π 0 2 0 0 y dx (φ, ϕ) = (p, ϕ), φ , ϕ + ε(φ, ϕ) + π 0 y 0 , η 0 − (φ, η) = 0, Rπ where (f, g) = 0 f (x)g(x)dx. Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên with first derivatives () can be given as ϕ ∈ H01 , η∈ H01 , () Given a uniform partition of [0, π], let S h be the space of piecewise linear polynomials on the grid such that S h ⊂ H01 . Then the standard finite-element approximation to () can be given as follows. Find (φh , yh ) ∈ S h × S h such that Z 2 π
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