1) a) Given the PDE where and , and are constant coefficients, determine the exact solution if we assume that the solution takes the form

(b) Discuss the effect of the coefficients of each term on the solution.

(c) Define modified wavenumbers for numerical approximations of the various derivatives appearing in the above equation. (Note that you should keep this general; you do not need to assume a specific approximation.)

(d) Using the modified wavenumbers for each term to take into account the effect of numerical errors, determine the solution using exact integration in time. Split each modified wavenumber into real and imaginary parts and clearly explain their effects on the solution in terms of amplitude and phase errors.