Mon Feb 16 10:47:53 MST 1998 aquarius% mupad *----* MuPAD 1.4.0 -- Multi Processing Algebra Data Tool /| /| *----* | Copyright (c) 1997 - 98 by SciFace Software GmbH | *--|-* All rights reserved. |/ |/ *----* Licensed to: Michael Wester >> # ----------[ M u P A D ]---------- # >> # ---------- Initialization ---------- # >> TEXTWIDTH:= 80: >> read("../../Time.mupad"): >> # ---------- Partial Differential Equations ---------- # >> # A very simple PDE => g(x) + h(y) for arbitrary functions g and h # >> diff(f(x, y), x, y) = 0; diff(f(x, y), x, y) = 0 Time: 90 msec Type: "_equal" >> # Heat equation: the fundamental solution is 1/sqrt(4 pi t) exp(-x^2/[4 t]). &> If f(x, t) and a(x, t) are solutions, the most general solution obtainable &> from f(x, t) by group transformations is of the form u(x, t) = a(x, t) &> + 1/sqrt(1 + 4 e6 t) exp(e3 - [e5 x + e6 x^2 - e5^2 t]/[1 + 4 e6 t]) &> f([e^(-e4) (x - 2 e5 t)]/[1 + 4 e6 t] - e1, [e^(-2 e4) t]/[1 + 4 e6 t] - e2) &> See Peter J. Olver, _Applications of Lie Groups to Differential Equations_, &> Second Edition, Springer Verlag, 1993, p. 120 (an excellent book). See also &> Heat.mupad # >> diff(u(x, t), t) = diff(u(x, t), x$2); diff(u(x, t), t) = diff(u(x, t), x, x) Time: 80 msec Type: "_equal" >> # Potential equation on a circular disk---a separable PDE &> => v(r, theta) = a[0] + sum(a[n] r^n cos(n theta), n = 1..infinity) &> + sum(b[n] r^n sin(n theta), n = 1..infinity) # >> 1/r * diff(r * diff(v(r, theta), r), r) &> + 1/r^2 * diff(v(r, theta), theta$2) = 0; diff(v(r, theta), theta, theta) ------------------------------- + 2 r diff(v(r, theta), r) + r diff(v(r, theta), r, r) ------------------------------------------------ = 0 r Time: 70 msec Type: "_equal" >> # ---------- Quit ---------- # >> quit real 2.64 user 2.10 sys 0.45