# Solving 1D heat equation using FFT
**URL:** https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457
**Category:** Specific Domains
**Tags:** differentialequation
**Created:** [March 11, 2024, 12:15pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457 "2024-03-11T12:15:12Z")
**Posts on this page:** 11
**Page:** 1
### Author:  [@Abdulrhmn\_Ghanem](https://discourse.julialang.org/u/Abdulrhmn_Ghanem)
#### Post date: [March 11, 2024, 12:15pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/1 "2024-03-11T12:15:12Z")
I am trying to reproduce the solution of heat 1D equation solution from [https://databookuw.com/](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](https://github.com/dynamicslab/databook_python/blob/master/CH02/CH02_SEC03_1_FFTHeat.ipynb) and [matlab](https://github.com/dynamicslab/databook_matlab/blob/master/CH02/CH02_SEC03_1_FFTHeat.m).
and here is my take on it
```julia
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.
---
### Author:  [@John\_Gibson](https://discourse.julialang.org/u/John_Gibson)
#### Post date: [March 11, 2024, 12:29pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/2 "2024-03-11T12:29:32Z")
Compare your solution to a known analytic solution for the same \alpha and initial and boundary conditions. E.g. a simple sinusoid initial condition should decay exponentially.
---
### Author:  [@Abdulrhmn\_Ghanem](https://discourse.julialang.org/u/Abdulrhmn_Ghanem)
#### Post date: [March 11, 2024, 12:43pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/3 "2024-03-11T12:43:51Z")
My solution is wrong, I couldn’t make any sense of the output 😅
But the python and matlab code solutions match the analytical solution
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [March 11, 2024, 2:30pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/4 "2024-03-11T14:30:48Z")
> [@Abdulrhmn\_Ghanem](#):
>
> `u[k, :] = ifft(û[k, :])`
Why are you doing an ifft in time instead of space? It seems you messed up your indexing here.
---
### Author:  [@Abdulrhmn\_Ghanem](https://discourse.julialang.org/u/Abdulrhmn_Ghanem)
#### Post date: [March 11, 2024, 2:56pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/5 "2024-03-11T14:56:10Z")
> [@ChrisRackauckas](#):
>
> Why are you doing an ifft in time instead of space? It seems you messed up your indexing here.
Probably, I’ll flip it and try
---
### Author:  [@Abdulrhmn\_Ghanem](https://discourse.julialang.org/u/Abdulrhmn_Ghanem)
#### Post date: [March 11, 2024, 3:18pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/6 "2024-03-11T15:18:32Z")
Switching the Indies fixed the shape of the solution indeed
```julia
# fixed ifft indices
#...
û = solve(prob)
# ...
for k in 1:length(eachcol(u))
u[:, k] = ifft(û[:, k])
end
```
But the solution doesn’t get time-stepped (i.e., the final the distribution is the same as the initial condition)
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [March 11, 2024, 4:26pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/7 "2024-03-11T16:26:46Z")
> [@Abdulrhmn\_Ghanem](#):
>
> `dû = -α^2 * (k.^2)' .* û`
You’re not updating.
`dû .= -α^2 * (k.^2)' .* û`
---
### Author:  [@Abdulrhmn\_Ghanem](https://discourse.julialang.org/u/Abdulrhmn_Ghanem)
#### Post date: [March 11, 2024, 9:55pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/8 "2024-03-11T21:55:39Z")
The solution is unstable 😕 do I need a specific solver to handle `Complex` numbers?
```julia
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, Rodas4P(autodiff=false))
```
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [March 12, 2024, 12:02pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/9 "2024-03-12T12:02:59Z")
Aren’t you missing a `-`?
---
### Author:  [@photor](https://discourse.julialang.org/u/photor)
#### Post date: [March 12, 2024, 12:56pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/10 "2024-03-12T12:56:37Z")
exactly
---
### Author:  [@Abdulrhmn\_Ghanem](https://discourse.julialang.org/u/Abdulrhmn_Ghanem)
#### Post date: [March 12, 2024, 3:32pm UTC](https://discourse.julialang.org/t/solving-1d-heat-equation-using-fft/111457/11 "2024-03-12T15:32:34Z")
Thank you for y
> [@ChrisRackauckas](#):
>
> Aren’t you missing a `-`?
🤦♂️
Thank you for your precious time!