#1:  " ----------[ D e r i v e ]---------- "                User


        #2:  " ---------- Initialization ---------- "               User


        #3:  " ---------- Partial Differential Equations ---------- "User


        User

        #4:  " A very simple PDE => g(x) + h(y) for arbitrary functions g and

               h "


        #5:  F(x, y) :=                                             User


             d  d             
        #6:  -- -- F(x, y) = 0                                      User
             dy dx            


             d  d             
        #7:  -- -- F(x, y) = 0                                  Simp(#6)
             dy dx            


        #8:  f :=                                                   User


        User

        #9:  " Heat equation: the fundamental solution is 1/sqrt(4 pi t)

               exp(-x^2/[4 t]). "


        User

        #10: " If f(x, t) and a(x, t) are solutions, the most general

               solution obtainable "


        User

        #11: " from f(x, t) by group transformations is of the form u(x, t) =

               a(x, t) "


        User

        #12: " + 1/sqrt(1 + 4 e6 t) exp(e3 - [e5 x + e6 x^2 -  e5^2 t]/[1 + 4

               e6 t]) "


        User

        #13: " f([e^(-e4) (x - 2 e5 t)]/[1 + 4 e6 t] - e1, [e^(-2 e4) t]/[1 +

               4 e6 t] - e2) "

        User

        #14: " See Peter J. Olver, _Applications of Lie Groups to

               Differential Equations_, "


        User

        #15: " Second Edition, Springer Verlag, 1993, p. 120 (an excellent

               book).  See also "


        #16: " Heat.mth "                                           User


        #17: U(x, t) :=                                             User


             d            /d \2        
        #18: -- U(x, t) = |--|  U(x, t)                             User
             dt           \dx/         


             d            /d \2        
        #19: -- U(x, t) = |--|  U(x, t)                        Simp(#18)
             dt           \dx/         


        #20: u :=                                                   User


        User

        #21: " Potential equation on a circular disk---a separable PDE "


        User

        #22: " => v(r, theta) = a[0] + sum(a[n] r^n cos(n theta), n =

               1..infinity) "


        User

        #23: "                       + sum(b[n] r^n sin(n theta), n =

               1..infinity) "


        #24: V(r, theta) :=                                         User


        User

              1  d  /  d             \     1  /   d   \2                
             ---*-- |r*-- V(r, theta)| + ----*|-------|  V(r, theta) = 0
        #25:  r  dr \  dr            /     2  \d theta/                 
                                          r                             


        Simp(#25)


               2 /d \2                 d                /   d   \2          ~
              r *|--|  V(r, theta) + r*-- V(r, theta) + |-------|  V(r, thet~
                 \dr/                  dr               \d theta/           ~
        #26: ---------------------------------------------------------------~
                                              2                             ~
                                             r                              ~


                      
               a)     
                      
               --- = 0
                      
                      


        #27: v :=                                                   User


        #28: " ---------- Quit ---------- "                         User