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A cardinal function method of solution of the equation $ \Delta u=u-u\sp{3}$


Author: L. R. Lundin
Journal: Math. Comp. 35 (1980), 747-756
MSC: Primary 65P05
MathSciNet review: 572852
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Abstract: The steady-state form of the Klein-Gordon equation is given by $ (^\ast)$

$\displaystyle \Delta u = u - {u^3},\quad u = u(X),\quad X \in {R^3}.$

For solutions which are spherically symmetric, $ (^\ast)$ takes the form $ \ddot u + 2\dot u/r = u - {u^3}$, $ u = u(r)$, where r is the distance from the origin in $ {R^3}$. The function $ y = ru$ satisfies $ {(^\ast}^\ast)$

$\displaystyle \ddot y = y - {y^3}/{r^2}.$

It is known that $ {(^\ast}^\ast)$ has solutions $ \{ {y_n}\} _{n = 0}^\infty $, where $ {y_n}$ has exactly n zeros in $ (0,\infty )$, and where $ y(0) = y(\infty ) = 0$.

In this paper, an approximation is obtained for the solution $ {y_0}$ by minimizing a certain functional over a class of functions of the form

$\displaystyle \sum\limits_{k = - m}^m {{a_k}\;} {\operatorname{sinc}}\left[ {\frac{{r - k{h_m}}}{{{h_m}}}} \right].$

It is shown that the norm of the error is $ O({m^{3/8}}\exp ( - \alpha m^{1/2}))$ as $ m \to \infty $, where $ \alpha $ is positive.

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DOI: http://dx.doi.org/10.1090/S0025-5718-1980-0572852-1
Article copyright: © Copyright 1980 American Mathematical Society