#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