A transform method for solving initial-boundary-value problems (IBVPs) for linear and integrable nonlinear partial differential equations was recently developed by A. S. Fokas. Here we demonstrate that Fokas' method can be extended to solve IBVPs for differential-di?erence (semi-discrete) and difference-difference (fully discrete) evolution equations (DDEEs). We show how the method can be used to solve the IBVP for DDEEs. As in the continuum case, the method for the solution of the IBVP employs the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the symmetries of the equation as well as by the global algebraic relation, which couples all known and unknown boundary values.
Selected References:
1. A. S. Fokas, A new transform method for evolution partial
differential equations, IMA J. Appl. Math. 67, 559-590 (2002).
2. A. S. Fokas, On the integrability of certain linear and nonlinear
partial differential equations, J. Math. Phys. 41, 4188-4237 (2000).
3. G. Biondini and G. Hwang, Initial-boundary value problems for
diferential-difference evolution equations: discrete linear and integrable
nonlinear Schrodinger equations, to appear in Inv. Probl.
4. G. Biondini and D. Wang, Initial-boundary-value problems for
linear discrete evolution equations, in preparation.