THERE IS AN ERROR IN THIS ASSIGNMENT.  IN PROBLEM 1, YOU SHOULD SHOW THAT f(x+ct) + g(x-ct)  [NOT f(t+cx)+g(t-cx))] IS A SOLUTION.    OR, IF YOU PREFER, THAT f(t + x/c) + g(t-x/c) IS A SOLUTION.

LIKEWISE, YOU MAY WISH TO MODIFY PROBLEM 3, CHANGING THE EQUATION TO   4 u_t (t,x)  = u _xx (t,x).  (THIS ISN'T NECESSARY, BUT MAKES THE NUMBERS COME OUT SIMPLER).

The main goal of the assignment is to make you familiar with some of the more usual solutions to the advection and wave equations.  Note that partial derivatives in this assignment are written with subscripts.
 

This means that  the symbol u_t (t,x)  stands for the partial derivative of the function  u(t,x) in the direction of t, which you compute by fixing x and taking the derivative of u in the direction t.    So, in fact,   u_t(t,x) =  d/dt u(t.x).     u_x(t,x) = d/dx u(t,x)  etc.
 

1.   Chapter  13 introduced the wave equation which travels to the left, which was   u_t(t,x) + c  u _x(t,x)  = 0.     Suppose  u is a solution of this equation.   Show that there is a  postive constant   k   such that  u(t,x) also satisfies

                          u_{tt (}t,x)  - k u_xx(t,x) =   0.

What is the relation of k to c?        Show that     f( t  + cx)  + g(t - cx)    solves this equation of second order  (this is the wave equation which allows waves to travel both to the right and to the left).
 

2.    Find   functions f(t)  and g(t)  so that   u(t,x) =   f(t)sin(x) + g(t) cos x solves  the equation    u _t(t,x) - 3 u_x (t,x) = 0.  Hint:  Plug in and find a relationship between the derivatives of  f and g and the function f and g.
 

3.    Find a function  h(t)   such that   the function u(t,x) = h(t) sin(2x)  solves the  diffusion equation
 

                                  u_t (t,x)  = 4 u _xx (t,x).
 
 

4.    A very important solution of the   diffuction equation is a Gaussian function   u(t,x) =  t ^-1/2 exp  (- a x ² / t).     If the diffusion equation is  the one  in problem 3 (with the diffusion constant 4), find the constant a.    This is a tricky calculation.  You might find it similiar to the problems  following chapter 15 on page 245.
 
 

5.      Suppose that you have an output which you guess to be   near  (maybe by best curve-fitting)    the function   e ^-3t  1/ ( 1 + (2x - t))². Find an advection equation which this function actually solves.   the equation should be one of the type described in chapter 13

                                         u_t (t,x) =  c u _x (t,x)  - ru(t,x).

6*(extra credit)  On page 349 of Taubes  Number 1.    Indicate whether modeling with an advection equation or a diffusion equation
is more appropriate in each of the following cases.

        a)   The concentration of venom in the blood after a snake bite.

         b)  The spread of geneticaly engineered mosquitos released from a research station.

          c)  The spread of an oil slick in the Gulf Stream

           d)   The spread of an oil slick in Lake Michigan.

7*(extra credit)   For each of the the two basic examples of a diffusion and an advection equation, give an example of a situation which we  have not yet discussed in which it might be appropriate to use  the equation  in the model.