• No results found

The preceding study of the effeot of the boundary conditions on solutions of the wave equation, clearly leaves sone questions unanswered. Necessary and sufficient ocnditionst for the solutions of the differential system

to

be bounded, have yet to be

obtain®!. It has been shown, however, that, if time derivatives appear

in

the boundary conditions, problems will arise which give bounded solutions of the differential system, but which caisiot be solved by any

discretisation method, of the type which uses the space derivative

replacement discussed in this chapter. This is in sharp contrast to the position when time derivatives do not appear in the boundary conditions | then the class of boundary conditions, for which the differential

equation has bounded solutions, coincides with the class of boundary conditions, for which the difference-differential aysten is stable.

5

In thia thesis, ini tial*boundary value proulems, and nuaerical

approximations

to them, have been discussed. It has oeen shown that the at&bili-y of a lif erence approximation to such a problem is affected as much by the boundary conditions, as by the aethod of

approximation to the differential operator which is used* I or the heat equation, and for a class of wave operators, the instability observed in the difference approximations, was shown to be oaused by the problems* being improperly posed, in the sense defined in ohapter 0. As far as these problems show, we can say that, if a difference approximation is

unconditionally stable for one properly posed problem, then it is

unconditionally stable for all ooundary conditions for which the proolea is properly posed*

That

this conclusion is not invariably oorreot, was shown by the example of the wav© equation with boundary conditions

involving tim (i*e* tangential) derivatives - there exist properly posed problems for which the difference approximations are unconditionally

unstable*

It has also been noted that the instability

observed, is an asymptotic instability, l,e* as

t <+ «•*

i or this reason, such instability becomes more important in the iterative solutions of equations arising from numerical approximations to, aay, Laplaee*s

equation. It was demonstrated, in chapter}, that, for a certain class of boundary conditions, all the oouaaonly used iterative methods, for the numerical solution of baplaoe*s equation, break down*

A fruitful area for further research, would be the extension of the results of this thesis, to equations with variable

coefficients* and also to non-root angular regions. The former extension is essentially similar In fora to the case of constant coefficients, but* for non-re otangular regions* nothing has yet oeen attempted*

'4-1

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See#

Indust# Appl. Math. J2, 919-935 (1963).

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19] Copaon, #T., and ?• Keast : On a boundary value problem for the equation of heat. J# Inst# Math# Applio# 2, 358*383 (1986)#

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New York (1959)*

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North Holland Publishing Co#* msterdsn* (1964)#

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16>168 (1958)*

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J*

2, 110-114 (1966)*

[26] Eeast, P*, and A*R* Mitchell x J inite difference solution of the third boundary problem in elliptic and parabolic equations* Nub*

i'ath* (1967)> (to appear)*

[27] Kreiss, H*0* x Uber die losung dee luuchyproblems fur lineare

partieile differentialglaichungen nit hilfe von differensengXelchungea* Acta lath. 1Q1* 179-199 (1959).

[28] Kreiss, H*0* : uber nat risen die beschranke halbgruppen ersengen* Math. Soand* J, 71-80 (1959)»

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166*

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[52] Xrelss, H.C. : On difference approximations of the dissipative type for hyperbolic differential equations* Coosa* Pure Appl* Math* 17,

333-353 (1964).

[33) Lax, P*D., and R.D. Riohtmeyer : Survey of the stability of linear f-inite difference equations* Com* Pure Jppl* Math* 2» 267-293 (1956)*

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[35] Lowan, AJI* t The operator approach to problems of stability and convergence* Fcrlpta Mathematic a, New York (1957)«

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The numerical solution of parabolic and elliptic equations* J* oc* Indust* Appl. Math* J, 28-2^1 (1955).

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