# Solving integral equation using DifferentialEquations

**URL:** https://discourse.julialang.org/t/solving-integral-equation-using-differentialequations/28992
**Category:** Numerics
**Tags:** question
**Created:** [September 20, 2019, 9:39am UTC](https://discourse.julialang.org/t/solving-integral-equation-using-differentialequations/28992 "2019-09-20T09:39:01Z")
**Posts on this page:** 1
**Showing post:** 5

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### Author: ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [September 20, 2019, 11:26am UTC](https://discourse.julialang.org/t/solving-integral-equation-using-differentialequations/28992/5 "2019-09-20T11:26:24Z")

</div>

Use Newton and a quadrature routine. That is, you are trying to find a root of

f(x) = x - \int\_x^\infty A(z) dz

The derivative of this is f'(x) = 1 + A(x). So, you can just repeatedly take Newton steps

x \leftarrow x - \frac{f(x)}{f'(x)}

where f(x) is evaluated with a quadrature routine like QuadGK:

```julia
using QuadGK
f(x) = x - quadgk(A, x, Inf, rtol=1e-4)[1]

```

for example. (QuadGK handles the semi-infinite integral for you by a coordinate transformation.)

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