Volume 1, 1997 18

[4] T.S. Vashakmadze, A Generalized Finite-Difference Method, Differential Equation, vol. II, N 5, 1966, 614-618 pp.

ON A STABILITY OF MULTI-STEP METHOD FOR AN

ABSTRACT PARABOLIC EQUATION

J. Rogava

Georgian Technical University

In the Hilbert space H for the Cauchy problem

(1)   u'(t)+Au(t)=f(t), t>0, u(0)=j ,

where A is self-adjoint, positive definite operator, the multi-step method defined by means of the following scheme is considered:

 (2)

where q ³ 1, >0; uk is the approximate solution of the problem (1) in t=tk=k; u0, u1,..., uq-1 are given initial vectors from H; a i and b i (i=0,1,...,q) are real numbers, and a q >0, b q >0.

In order to study the stability problem for the scheme (2) we introduce a special class at polinomials, which is defined by the following recurent dependence:

Uk=x1Uk-1+x2Uk-2+...+xqUk-q,

U0=1, U-1=...=U1-q=0.

On the basis of the properties of these polinomials the following theorems are proved.

Theorem 1. If the multi-step method (2) is stable, then all the roots of the characteristic equation

(3  ) l 2-x1(s) l q-1-...-xq-1(s) l -xq(s)=0, sÎ [0, +¥ )

where xi(s)=( a q-1+b q-1 s)( a q +b q s)-1, i=1 ...,q, belong to the unit circle.

Theorem 2. Let for arbitrary sÎ [0, +¥ ), all the roots of the equation (3) but probably one, belong to the same circle which is inside the unit circle. Then the multi-step method (2) is stable.

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