I am trying to reproduce the solution of heat 1D equation solution from https://databookuw.com/.

u_t = \alpha^2u_{xx}

where u(t,\ x) is the temperature distribution in time and space.

\mathcal{F}(u(t,\ x)) = \hat u(t, \ kappa)

u_x \xrightarrow{\mathcal{F}} i k \hat u

\hat u_t = - \alpha^2 k^2 \hat u

Here are the solutions in python and matlab.

and here is my take on it

```
using FFTW, DifferentialEquations
function heat()
α = 1.0
L = 100.0
N = 1000
dx = L / N
domain = range(-L/2, stop=L/2-dx, length=N)
u₀ = collect(0 * domain)
u₀[400:600] .= 1
u₀ = fft(u₀, 1)
kappa = (2π / L) * (-N/2:N/2-1)
kappa = fftshift(kappa, 1)
tspan = (0, 10)
t = tspan[1]:0.1:tspan[2]
params = (α, kappa)
function rhs(dû, û, p, t)
α, k = p
dû = -α^2 * (k.^2)' .* û
end
prob = ODEProblem(rhs, u₀, tspan, params)
û = solve(prob, saveat=t)
u = zeros(Complex, size(û)...)
for k in 1:length(t)
u[:, k] = ifft(û[:, k])
end
domain, real(u)
end
```

My results don’t match the other solutions. I have verified that up to calculating the kappa, I am consistent with the other solutions. So I am skeptical of the ODE solution or the ifft step.