A cardinal function method of solution of the equation Δ

The steady-state form of the Klein-Gordon equation is given by (*) Au =u-u3, u== u(X), X E R 3~~~~~~~~ For solutions which are spherically symmetric, (*) takes the form u + 2ut/r = u -u3 u = u(r), where r is the distance from the origin in R3. The function y = ru satisfies (**) y -= y 3Ir2 It is known that (**) has solutions {yn} where has exactly n zeros in (0, 0), and where y(0) = y(oo) = 0. In this paper, an approximation is obtained for the solution y0 by minimizing a certain functional over a class of functions of the form m Fr-khm E ak sinC _. k=-m hm It is shown that the norm of the error is 0(m3/8exp(-om /)) as m oc, where ae is positive.

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