Abstract
AbstractFirst, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form $$(Q(\partial ')-a^2\partial _n^2)(Q(\partial ')-b^2\partial _n^2)$$
(
Q
(
∂
′
)
-
a
2
∂
n
2
)
(
Q
(
∂
′
)
-
b
2
∂
n
2
)
(where $$\partial '=(\partial _1,\dots ,\partial _{n-1})$$
∂
′
=
(
∂
1
,
⋯
,
∂
n
-
1
)
and $$a>0,b>0,a\ne b)$$
a
>
0
,
b
>
0
,
a
≠
b
)
with respect to Dirichlet boundary conditions at $$x_n=0.$$
x
n
=
0
.
The Green function $$G_\xi $$
G
ξ
is represented by a linear combination of fundamental solutions $$E^c$$
E
c
of $$Q(\partial ')(Q(\partial ')-c^2\partial _n^2),$$
Q
(
∂
′
)
(
Q
(
∂
′
)
-
c
2
∂
n
2
)
,
$$c\in \{a,b\},$$
c
∈
{
a
,
b
}
,
that are shifted to the source point $$\xi ,$$
ξ
,
to the mirror point $$-\xi ,$$
-
ξ
,
and to the two additional points $$-\frac{a}{b}\xi $$
-
a
b
ξ
and $$-\frac{b}{a}\xi ,$$
-
b
a
ξ
,
respectively.
Funder
University of Innsbruck and Medical University of Innsbruck
Publisher
Springer Science and Business Media LLC
Reference17 articles.
1. Bernstein, I.N.: Modules over a ring of differential operators. Study of fundamental solutions of equations with constant coefficients. Funct. Anal. Appl. 5, 89–101 (1971)
2. More special functions;YA Brychkov,1990
3. Cheng, S., He, F.B.: Theory of orthotropic and composite cylindrical shells, accurate and simple fourth-order governing equations. ASME J. Appl. Mech. 51, 736–744 (1984)
4. Delache, S., Leray, J.: Calcul de la solution élémentaire de l’opérateur d’Euler-Poisson-Darboux et de l’opérateur de Tricomi-Clairaut, hyperbolique, d’ordre 2. Bull. Soc. Math. France 99, 313–336 (1971)
5. Friedlander, G., Joshi, M.: Introduction to the theory of distributions, 2nd edn. Cambridge University Press, Cambridge (1998)