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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | --Copyright The Numerical Algorithms Group Limited 1996.
-- ----------[ A x i o m ]----------
-- ---------- Initialization ----------
)set messages autoload off
)set quit unprotected
-- This is the heat equation
u:= operator('u)
heat:= D(u(x, t), t) - D(u(x, t), x, 2) = 0
-- This is the similarity form of the proposed solution
f:= operator('f)
s:= rule(u(x, t) == f(x/sqrt(t))/sqrt(t))
-- Apply s to the heat equation
s(lhs(heat)) = 0
-- Change to the similarity variable z = x/sqrt(t)
subst(lhs(%), x = z*sqrt(t)) = 0
-- Eliminate the denominator
% * denom(lhs(%))
-- Now, solve the ordinary differential equation
eq:=%
solve(%, f, z=0,[k1,k2])
-- Finally, transform back to the original variables
subst(%, z = x/sqrt(t))/sqrt(t)
-- If we set k2 = 0 and k1 = 1/(2*sqrt(%pi)) in the previous expression,
-- we will obtain the usual fundamental solution of the heat equation
subst(%, [k2 = 0, k1 = 1/(2*sqrt(%pi))])
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