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We describe a new form of diagonalization for linear two point constantcoefficient differential operators with arbitrary linear boundary conditions.Although the diagonalization is in a weaker sense than that usually employed tosolve initial boundary value problems (IBVP), we show that it is sufficient tosolve IBVP whose spatial parts are described by such operators. We argue thatthe method described may be viewed as a reimplementation of the Fokas transformmethod for linear evolution equations on the finite interval. The results areextended to multipoint and interface operators, including operators defined onnetworks of finite intervals, in which the coefficients of the differentialoperator may vary between subintervals, and arbitrary interface and boundaryconditions may be imposed; differential operators with piecewise constantcoefficients are thus included. Both homogeneous and inhomogeneous problems aresolved.

Fokas diagonalization of piecewise constant coefficient linear differential operators on finite intervals and networks