Question: Why Is Inhomogeneous Heat Equation Formula Not Satisfying IVP?

From Wikipedia,

However, when I plug the formula of u(x, t) into Maple, it doesn't seem to satisfy the PDE and is stuck evaluating.
 

restart

eq := diff(u(x, t), t)-k*(diff(u(x, t), x, x)) = f(x, t)

diff(u(x, t), t)-k*(diff(diff(u(x, t), x), x)) = f(x, t)

(1)

ic := u(x, 0) = 0

u(x, 0) = 0

(2)

"G(x,t):=1/(sqrt(4*pi*k*t))exp(-(x^(2))/(4*k*t))"

proc (x, t) options operator, arrow, function_assign; exp(-(1/4)*x^2/(k*t))/sqrt(4*pi*k*t) end proc

(3)

ans := u(x, t) = int(G(x-y, t-s)*f(y, s), y = -infinity .. infinity, s = 0 .. t)

u(x, t) = (1/2)*(int((int(exp(-(1/4)*(x-y)^2/(k*(t-s)))*f(y, s), y = -infinity .. infinity))/(pi*k*(t-s))^(1/2), s = 0 .. t))

(4)

`assuming`([simplify(pdetest(ans, [eq, ic]))], [t > 0])

``


 

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