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A cardinal function method of solution of the equation Δ = -
The steady-state form of the Klein-Gordon equation is given by ( ∗ ) (^\ast ) \[ Δ u = u − u 3 , u = u ( X ) , X ∈ R 3 . \Delta u = u - {u^3},\quad u = u(X),\quad X \in {R^3}. \] For solutions which are spherically symmetric, ( ∗ ) (^\ast ) takes the form u ¨ + 2 u ˙ / r = u − u 3 \ddot u + 2\dot u/...
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| Published in: | Mathematics of computation 1980-01, Vol.35 (151), p.747-756 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
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| Summary: | The steady-state form of the Klein-Gordon equation is given by
(
∗
)
(^\ast )
\[
Δ
u
=
u
−
u
3
,
u
=
u
(
X
)
,
X
∈
R
3
.
\Delta u = u - {u^3},\quad u = u(X),\quad X \in {R^3}.
\]
For solutions which are spherically symmetric,
(
∗
)
(^\ast )
takes the form
u
¨
+
2
u
˙
/
r
=
u
−
u
3
\ddot u + 2\dot u/r = u - {u^3}
,
u
=
u
(
r
)
u = u(r)
, where
r
is the distance from the origin in
R
3
{R^3}
. The function
y
=
r
u
y = ru
satisfies
(
∗
∗
)
{(^\ast }^\ast )
\[
y
¨
=
y
−
y
3
/
r
2
.
\ddot y = y - {y^3}/{r^2}.
\]
It is known that
(
∗
∗
)
{(^\ast }^\ast )
has solutions
{
y
n
}
n
=
0
∞
\{ {y_n}\} _{n = 0}^\infty
, where
y
n
{y_n}
has exactly
n
zeros in
(
0
,
∞
)
(0,\infty )
, and where
y
(
0
)
=
y
(
∞
)
=
0
y(0) = y(\infty ) = 0
. In this paper, an approximation is obtained for the solution
y
0
{y_0}
by minimizing a certain functional over a class of functions of the form
\[
∑
k
=
−
m
m
a
k
sinc
[
r
−
k
h
m
h
m
]
.
\sum \limits _{k = - m}^m {{a_k}\;} {\operatorname {sinc}}\left [ {\frac {{r - k{h_m}}}{{{h_m}}} \right ].
\]
It is shown that the norm of the error is
O
(
m
3
/
8
exp
(
−
α
m
1
/
2
)
)
O({m^{3/8}}\exp ( - \alpha m^{1/2}))
as
m
→
∞
m \to \infty
, where
α
\alpha
is positive. |
|---|---|
| ISSN: | 0025-5718 1088-6842 |
| DOI: | 10.1090/S0025-5718-1980-0572852-1 |