# ApproxFun: how to define initial conditions for a second-order ODE?
**URL:** https://discourse.julialang.org/t/approxfun-how-to-define-initial-conditions-for-a-second-order-ode/15295
**Category:** Modelling & Simulations
**Created:** [September 21, 2018, 4:29pm UTC](https://discourse.julialang.org/t/approxfun-how-to-define-initial-conditions-for-a-second-order-ode/15295 "2018-09-21T16:29:14Z")
**Posts on this page:** 5
**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-how-to-define-initial-conditions-for-a-second-order-ode/15295/1 "2018-09-21T16:29:14Z")
Hello,
I am struggling with the very basics of usage of [ApproxFun](https://github.com/JuliaApproximation/ApproxFun.jl) package. I would like to use it solve the very simplest second-order linear initial value problem \ddot u(t) + \dot u(t) + u(t) = 0, u(0) = 1, \dot u(0) = 0, on the interval [0,1]. The [examples on ApproxFun webpage](https://github.com/JuliaApproximation/ApproxFunExamples/tree/master/ODEs) do not give me much guidance - they either handle first-order problems or boundary value problems.
This is what I am trying in Julia:
```julia
using ApproxFun, LinearAlgebra
t₀ = 0.0; t₁ = 1.0; t = Interval(t₀,t₁);
B0 = Evaluation(Chebyshev(t),t₀)
B1 = Evaluation(Chebyshev(t),t₀,1)
B = [B0;B1]
x₀ = 1.0; v₀ = 0.0; u₀ = [x₀;v₀]
D = Derivative(t); D2 = Derivative(t,2);
N = D2 + D + I;
u = [B;N]\[u₀,0]
```
The error after the last line is
```julia
ERROR: LoadError: MethodError: ApproxFun.setdomain(::ConstantSpace{ApproxFun.AnyDomain,ApproxFun.UnsetNumber}, ::ApproxFun.AnyDomain) is ambiguous. Candidates:
setdomain(A::ConstantSpace{DD,R}, d) where {DD, R} in ApproxFun at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Fun/Space.jl:511
setdomain(sp::Space{D,R} where R, d::D) where D<:Domain in ApproxFun at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Fun/Space.jl:66
Possible fix, define
setdomain(::ConstantSpace{D<:Domain,R}, ::D<:Domain)
Stacktrace:
[1] ApproxFun.SumSpace(::ConstantSpace{ApproxFun.AnyDomain,ApproxFun.UnsetNumber}, ::ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Spaces/Modifier/SumSpace.jl:168
[2] ⊕(::ConstantSpace{ApproxFun.AnyDomain,ApproxFun.UnsetNumber}, ::ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Spaces/Modifier/Modifier.jl:9
[3] union_rule(::ConstantSpace{ApproxFun.Point{Float64},Float64}, ::ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Spaces/ConstantSpace.jl:111
[4] conversion_rule(::ConstantSpace{ApproxFun.Point{Float64},Float64}, ::ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Spaces/ConstantSpace.jl:102
[5] conversion_type(::ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64}, ::ConstantSpace{ApproxFun.Point{Float64},Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Fun/Space.jl:177
[6] defaultcoefficients(::Array{Float64,1}, ::ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64}, ::ConstantSpace{ApproxFun.Point{Float64},Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Fun/Space.jl:315
[7] coefficients(::Array{Float64,1}, ::ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64}, ::ConstantSpace{ApproxFun.Point{Float64},Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Fun/Space.jl:338
[8] coefficients at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Fun/Fun.jl:45 [inlined]
[9] (::getfield(Base, Symbol("##3#4")){typeof(coefficients)})(::Tuple{Fun{ApproxFun.ArraySpace{ConstantSpace{ApproxFun.AnyDomain,Float64},1,ApproxFun.AnyDomain,Float64},Float64,Array{Float64,1}},ConstantSpace{ApproxFun.Point{Float64},Float64}}) at ./generator.jl:36
[10] iterate at ./generator.jl:47 [inlined]
[11] collect(::Base.Generator{Base.Iterators.Zip2{Array{Fun,1},ApproxFun.ArraySpace{Space{D,Float64} where D,1,ApproxFun.Point{Float64},Float64}},getfield(Base, Symbol("##3#4")){typeof(coefficients)}}) at ./array.jl:619
[12] map at ./abstractarray.jl:2060 [inlined]
[13] coefficients(::Array{Fun,1}, ::ApproxFun.ArraySpace{Space{D,Float64} where D,1,ApproxFun.Point{Float64},Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Spaces/Modifier/ArraySpace.jl:208
[14] coefficients(::Array{Any,1}, ::ApproxFun.ArraySpace{Space{D,Float64} where D,1,ApproxFun.Point{Float64},Float64}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Spaces/Modifier/ArraySpace.jl:211
[15] #\#184(::Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}}, ::Function, ::ApproxFun.InterlaceOperator{Float64,1,Chebyshev{Segment{Float64},Float64},ApproxFun.ArraySpace{Space{D,Float64} where D,1,ApproxFun.Point{Float64},Float64},ApproxFun.CachedIterator{Tuple{Int64,Int64},ApproxFun.BlockInterlacer{Tuple{ApproxFun.Repeated{Bool}}}},ApproxFun.CachedIterator{Tuple{Int64,Int64},ApproxFun.BlockInterlacer{Tuple{ApproxFun.Take{ApproxFun.Repeated{Bool},Bool},ApproxFun.Take{ApproxFun.Repeated{Bool},Bool},ApproxFun.Repeated{Bool}}}},Tuple{ApproxFun.Infinity{Bool},ApproxFun.Infinity{Bool}}}, ::Array{Any,1}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Operators/ldiv.jl:11
[16] \(::ApproxFun.InterlaceOperator{Float64,1,Chebyshev{Segment{Float64},Float64},ApproxFun.ArraySpace{Space{D,Float64} where D,1,ApproxFun.Point{Float64},Float64},ApproxFun.CachedIterator{Tuple{Int64,Int64},ApproxFun.BlockInterlacer{Tuple{ApproxFun.Repeated{Bool}}}},ApproxFun.CachedIterator{Tuple{Int64,Int64},ApproxFun.BlockInterlacer{Tuple{ApproxFun.Take{ApproxFun.Repeated{Bool},Bool},ApproxFun.Take{ApproxFun.Repeated{Bool},Bool},ApproxFun.Repeated{Bool}}}},Tuple{ApproxFun.Infinity{Bool},ApproxFun.Infinity{Bool}}}, ::Array{Any,1}) at /home/hurak/.julia/packages/ApproxFun/wFAIT/src/Operators/ldiv.jl:4
[17] top-level scope at none:0
[18] include_string(::Module, ::String, ::String) at ./loading.jl:1002
[19] (::getfield(Atom, Symbol("##118#123")){String,String,Module})() at /home/hurak/.julia/packages/Atom/WSz3k/src/eval.jl:120
[20] withpath(::getfield(Atom, Symbol("##118#123")){String,String,Module}, ::String) at /home/hurak/.julia/packages/CodeTools/hB4Hy/src/utils.jl:30
[21] withpath at /home/hurak/.julia/packages/Atom/WSz3k/src/eval.jl:46 [inlined]
[22] #117 at /home/hurak/.julia/packages/Atom/WSz3k/src/eval.jl:117 [inlined]
[23] hideprompt(::getfield(Atom, Symbol("##117#122")){String,String,Module}) at /home/hurak/.julia/packages/Atom/WSz3k/src/repl.jl:76
[24] macro expansion at /home/hurak/.julia/packages/Atom/WSz3k/src/eval.jl:116 [inlined]
[25] (::getfield(Atom, Symbol("##116#121")){Dict{String,Any}})() at ./task.jl:85
in expression starting at /home/hurak/ownCloud/julia/Exploring ODEs/van_der_pol.jl:15
```
Obviously, I am failing and I am not able to decipher the error message. Any guidance?
---
### Author:  [@dpsanders](https://discourse.julialang.org/u/dpsanders)
#### Post date: [September 21, 2018, 10:07pm UTC](https://discourse.julialang.org/t/approxfun-how-to-define-initial-conditions-for-a-second-order-ode/15295/2 "2018-09-21T22:07:18Z")
Have you looked at the examples directory? There’s a 10th order one there.
[https://github.com/JuliaApproximation/ApproxFunExamples/blob/master/ODEs/tenthorderODE.jl](https://github.com/JuliaApproximation/ApproxFunExamples/blob/master/ODEs/tenthorderODE.jl)
---
### Author:  [@zdenek\_hurak](https://discourse.julialang.org/u/zdenek_hurak)
#### Post date: [September 21, 2018, 10:17pm UTC](https://discourse.julialang.org/t/approxfun-how-to-define-initial-conditions-for-a-second-order-ode/15295/3 "2018-09-21T22:17:26Z")
But that is a BVP, isn’t it? If fact, this is the example that I took the inspiration from for what I show above.
---
### Author:  [@dpsanders](https://discourse.julialang.org/u/dpsanders)
#### Post date: [September 21, 2018, 10:23pm UTC](https://discourse.julialang.org/t/approxfun-how-to-define-initial-conditions-for-a-second-order-ode/15295/4 "2018-09-21T22:23:06Z")
Ah sorry, my mistake.
---
### Author:  [@Domenico\_Lahaye](https://discourse.julialang.org/u/Domenico_Lahaye)
#### Post date: [August 25, 2024, 9:09pm UTC](https://discourse.julialang.org/t/approxfun-how-to-define-initial-conditions-for-a-second-order-ode/15295/5 "2024-08-25T21:09:49Z")
Has this issue been resolved?
Can this issue be extended from the scalar ODE of a mass-spring-damper to the PDE of a wave equation \ddot{u} = Laplacian(u) + boundary and initial conditions?