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Mô tả chi tiết
FINITE-ELEMENT APPROXIMATION OF A BOUNDARY VALUE PROBLEM
FOR FOURTH ODER DIFFERENTIAL EQUATION
Nguyen Thanh Huong1
and Vu Vinh Quang2
1
Thainguyen University, College of Sciences
2
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 2.1 and correct a mistake in theorem
2.2. 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) − εy00 −
2
π
Z π
0
y
0
2
dx
y
00 = p(x), 0 ≤ x ≤ π,
y(0) = y(π) = y
00(0) = y
00(π) = 0,
(1.1)
where > 0 is a constant, p(x) is a continuous function and nonpositive on [0, π].
Letting φ = −y
00, problem (1.1) is reduced to the system of two second order equations:
−φ
00 + εφ +
2
π
Z π
0
y
0
2
dx
φ = p(x), 0 ≤ x ≤ π,
−y
00 − φ = 0,
y(0) = y(π) = φ(0) = φ(π) = 0.
(1.2)
Letting H1
0 denote the Sobolev space of L
2
(0, π) functions with first derivatives
in L
2
(0, π) and vanishing at 0, π, the variational formulation of (1.2) can be given as
follows. Find (φ, y) ∈ H1
0 × H1
0
such that
φ
0
, ϕ0
+ ε(φ, ϕ) + 2
π
Z π
0
y
0
2
dx
(φ, ϕ) = (p, ϕ), ϕ ∈ H
1
0
,
y
0
, η0
− (φ, η) = 0, η ∈ H
1
0
,
(1.3)
where (f, g) = R π
0
f(x)g(x)dx.
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