# ApproxFun: defining a nonlinear differential operator
**URL:** https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296
**Category:** Modelling & Simulations
**Created:** [September 21, 2018, 4:29pm UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296 "2018-09-21T16:29:15Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@zdenek\_hurak](https://discourse.julialang.org/u/zdenek_hurak)
#### Post date: [September 21, 2018, 4:29pm UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296/1 "2018-09-21T16:29:15Z")
Hello, I am struggling with the very basics of usage of ApproxFun package. I would like to define a nonlinear differential operator (the ultimate motivation is actually to define and solve a nonlinear ODE). The operator \mathcal{N} acts on u(t) as \mathcal{N}u = \ddot u + (1-u^2)\dot u + u.
How do I do it using ApproxFun? This is what I tried so far:
```julia
using ApproxFun, LinearAlgebra
t₀ = 0.0; t₁ = 1.0; t = Interval(t₀,t₁);
D = Derivative(t); D2 = Derivative(t,2);
```
```julia
N = D2 + (1-?)*D + I;
```
What do I put instead of the ? symbol on the last line? Or shall I do it in a completely different way? I feel my confusion might come from the interpretation of the \* symbol as both the multiplication and application of an operator.
---
### Author:  [@vtchen](https://discourse.julialang.org/u/vtchen)
#### Post date: [September 12, 2019, 10:57am UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296/2 "2019-09-12T10:57:57Z")
I have the same doubts. Have you solved this problem?
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### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [September 12, 2019, 11:04am UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296/3 "2019-09-12T11:04:42Z")
You have to translate back to position space to do it.
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### Author:  [@zdenek\_hurak](https://discourse.julialang.org/u/zdenek_hurak)
#### Post date: [September 12, 2019, 11:19am UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296/4 "2019-09-12T11:19:47Z")
Unfortunately no. I gave up, at least temporarily…
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### Author:  [@vtchen](https://discourse.julialang.org/u/vtchen)
#### Post date: [September 12, 2019, 11:54am UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296/5 "2019-09-12T11:54:16Z")
Dr chris and Dr Olver know the solution, and I hope the ApproxFun package will provide a solution to this problem in the near future.
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [September 12, 2019, 12:30pm UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296/6 "2019-09-12T12:30:04Z")
Here’s a few examples with finite discretizations at least:
[https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/allen\_cahn\_spectral\_wpd.jmd](https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/allen_cahn_spectral_wpd.jmd)
[https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/burgers\_spectral\_wpd.jmd](https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/burgers_spectral_wpd.jmd)
[https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/kdv\_spectral\_wpd.jmd](https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/kdv_spectral_wpd.jmd)
[https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/ks\_spectral\_wpd.jmd](https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/benchmarks/MOLPDE/ks_spectral_wpd.jmd)
---
### Author:  [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [September 12, 2019, 12:44pm UTC](https://discourse.julialang.org/t/approxfun-defining-a-nonlinear-differential-operator/15296/7 "2019-09-12T12:44:11Z")
An example of using ApproxFun to solve a nonlinear boundary-value problem is given here: [https://github.com/JuliaApproximation/ApproxFun.jl#nonlinear-boundary-value-problems](https://github.com/JuliaApproximation/ApproxFun.jl#nonlinear-boundary-value-problems)
The basic issue is that it is nonlinear, so you don’t define an operator and use `\`. Instead you define a function `N(u)` and run a Newton iteration (which internally performs a sequence of linearized solves).