# Integro-differential equation and 2nd order differential equation
**URL:** https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799
**Category:** New to Julia
**Tags:** question, package, diffeq, differentialequation
**Created:** [March 31, 2022, 11:04am UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799 "2022-03-31T11:04:28Z")
**Posts on this page:** 9
**Page:** 1
### Author:  [@MartinMikkelsen](https://discourse.julialang.org/u/MartinMikkelsen)
#### Post date: [March 31, 2022, 11:04am UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/1 "2022-03-31T11:04:28Z")
I’m very new to Julia and want to convert from Python to Julia. I have a system of equations that I want to solve numerically in Julia. The system is
where f(r)=S\*exp(-r^2/b^2), S, b and m\_π are constants. In Python I used a general-purpose numerical integro-differential equation solver, [IDEsolver](https://idesolver.readthedocs.io/en/latest/index.html) – but this approach is very slow.
Any suggestions as to how this can be done using [DifferentialEquations.jl](https://diffeq.sciml.ai/dev/index.html)?
Thanks
---
### Author:  [@rveltz](https://discourse.julialang.org/u/rveltz)
#### Post date: [March 31, 2022, 11:53am UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/2 "2022-03-31T11:53:08Z")
I would recommand ApproxFun
---
### Author:  [@MartinMikkelsen](https://discourse.julialang.org/u/MartinMikkelsen)
#### Post date: [March 31, 2022, 12:06pm UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/3 "2022-03-31T12:06:31Z")
I will look into it. How would you use it?
---
### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [March 31, 2022, 1:16pm UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/4 "2022-03-31T13:16:56Z")
Basically, ApproxFun allows you to represent both the integral and derivative operators as matrices in your equation (see [ApproxFun: Operators](https://juliaapproximation.github.io/ApproxFun.jl/latest/usage/operators/)). Then you can just rearrange your equation into \phi = (\mbox{some operator}) \backslash f.
I’m not sure offhand [which function space](https://juliaapproximation.github.io/ApproxFun.jl/latest/usage/spaces/) you should use here, given the 1/r singularity at the origin or the fact that you want to go to r = \infty. The latter would seem to suggest a Laguerre space, but that won’t get rid of the origin singularity. What are your boundary conditions on \phi? (Maybe you can do a change of variables.)
---
### Author:  [@rveltz](https://discourse.julialang.org/u/rveltz)
#### Post date: [March 31, 2022, 1:50pm UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/5 "2022-03-31T13:50:35Z")
Sorry @ChrisRackauckas 🤣
I saved you from a technical answer though 😋
---
### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [March 31, 2022, 2:00pm UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/6 "2022-03-31T14:00:59Z")
> [@stevengj](#):
>
> Then you can just rearrange your equation into \phi = (\mbox{some operator}) \backslash f ϕ=(some operator)∖f\phi = (\mbox{some operator}) \backslash f .
On second glance, it’s not that simple, unfortunately, because your equation is nonlinear in \phi.
However, you can linearize it in \phi + \delta\phi, solve the linearized equations for \delta\phi given some \phi, and then perform Newton iterations.
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [March 31, 2022, 2:01pm UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/7 "2022-03-31T14:01:43Z")
Oh no, this is the right answer. Another answer could be NeuralPDE.jl, here’s a Physics-Informed Neural Network Integro-Differential Equation tutorial:
[https://neuralpde.sciml.ai/dev/pinn/integro\_diff/](https://neuralpde.sciml.ai/dev/pinn/integro_diff/)
that said, PINNs are a very slow method, especially at lower dimensions. This will be an easy but not efficient way of doing it.
---
### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [March 31, 2022, 2:05pm UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/8 "2022-03-31T14:05:09Z")
Another possible approach:
Given any value of E, you can solve your differential equation to find the corresponding \phi[E]. Conversely, given any \phi(r), you can perform the integral to compute E[\phi]. The combination of these is the fixed-point equation:
E[\phi[\mathcal{E}]] = \mathcal{E}
which is a nonlinear equation in a single variable that can be solved easily by a variety of root-finding algorithms.
The advantage of this is that you don’t need to implement an integro-differential solver directly — you just need an integration code for E[\phi] and a linear-ODE boundary-value solver for \phi[E], and then throw them at a root-finding algorithm.
---
### Author:  [@MartinMikkelsen](https://discourse.julialang.org/u/MartinMikkelsen)
#### Post date: [March 31, 2022, 2:26pm UTC](https://discourse.julialang.org/t/integro-differential-equation-and-2nd-order-differential-equation/78799/9 "2022-03-31T14:26:44Z")
That’s the approach I initially used in Python and it works out. I wanted to solve the same problem using DifferentialEquations.jl but turns out the problem is harder than I thought 😅