Hello, I am trying to solve a PDE where one of the terms is an integral of the solution variable. What would be the best way to calculate it?

Details:

We have multiple pdes of the form

$$ \frac{\partial^2 u_i}{\partial x^2} - u_i\int \frac{s(x’)}{\sqrt{(x-x’)^2+1}}dx’ = 0$$

Where $s$ is

$$ s(x) = \sum_i^n|u_i(x)|^2$$

The challenge here is that $s$ dependes on multiple $u_i$ and each solution $u_i$ depends on solving an integral of $s$ over the domain. What would be the best way to represent this in Modulus?

Some things that I have tried:

I tried using the `Integral`

function in sympy (Integrals — SymPy 1.10.1 documentation)

```
#assume dom is already set
# Equation set up in modulus
x = Symbol('x')
t = Symbol('t')
# make input variables
input_variables = {'x':x, 't':t}
# make u_i list
# List of sympy functions (u0, u1, ..., un)
u_list = [Function('u%d' % i)(*input_variables) for i in range(n)]
# make s
amp_list = [u_list[i]**2 for i in range(n)]
s = sum(amp_list)
# Integral term
xp = Symbol('xp')
s_p = s.subs(x,xp)
s_integral = integrate(s_p/(sqrt((x-xp)**2 + 1)),( xp,-dom,dom))
# set equations
self.equations = {}
for i in range(n):
u = u_list[i]
self.equations['u%d' %i] = u.diff(x,2) - u * s_integral
```

Here, I get an error saying that `Integral`

is not defined

```
...
...
File "/home/env/pinn/lib/python3.9/site-packages/modulus-22.3-py3.9.egg/modulus/sympy_utils/torch_printer.py", line 259, in forward
return {self.name: self.torch_expr(args)}
File "<lambdifygenerated-1>", line 5, in _lambdifygenerated
NameError: name 'Integral' is not defined
```

I checked the modulus sympy to torch converting functions and noticed that `sympy.integrate`

is not in the list of functions that are converted to torch by modulus. This file is `modulus/sympy_utils/torch_printer.py`

. I tried writing my own convertor to calculate an integral in pytorch from sympy but can’t figure that out.

I would be happy to explain this in more detail, and make necessary modifications in Modulus myself if you have some advice.