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It is well-known that the basic difficulty in studying the initial boundary value problems for linear and nonlinear PDEs is the presence, in any method of solution, of unknown boundary values. In the first part of this paper we review two spectral methods in which the above difficulty is faced in different ways. In the first method one uses the analyticity properties of the x-scattering matrix S(k, t) to replace the unknown boundary values by elements of the scattering matrix itself, thusdoi:10.2991/jnmp.2005.12.s1.19 fatcat:rairzwly2vfz7e5l7w4fk3t3li