3. (a) Given the PDE nload A Alte Ot +a =a- +B. 2 846 +7 3 4 (3) where a, a, , and y are constant coefficients, determine the exact solution if we assume that the solution takes the form (x,t) = (t) eik. (b) Discuss the effect of the coefficients of each term on the solution. (c) Based on what we covered in class, define modified wavenumbers for numerical approx- imations 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 numer- ical 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 solu- tion in terms of amplitude and phase errors.