5.
We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives of
arbitrary order, posed on the domain {
t > 0, 0 <
x <
L}. We show that the solution can be expressed as an integralin the complex
k-plane. This integral is defined in terms ofan
x-transform of the initial condition and a
t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the
global relation, which is an algebraicrelation defined in the complex
k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for
even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which
cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite interval
cannot be expressed in terms of an infinite series.
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