# Integrate convolution symbolically to solve PDE

**URL:** <https://discourse.julialang.org/t/integrate-convolution-symbolically-to-solve-pde/123810>\
**Category:** Modelling & Simulations\
**Tags:** pde, integral\
**Created:** [December 13, 2024, 1:07pm UTC](https://discourse.julialang.org/t/integrate-convolution-symbolically-to-solve-pde/123810 "2024-12-13T13:07:16Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![B4rtel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/b4rtel/32/214171_2.png) [@B4rtel](https://discourse.julialang.org/u/B4rtel)\
**Post date:** [December 13, 2024, 1:07pm UTC](https://discourse.julialang.org/t/integrate-convolution-symbolically-to-solve-pde/123810/1 "2024-12-13T13:07:16Z")

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I would like to solve a PDE with MethodOfLines.jl and ModellingToolkit.jl similar to the example docs.sciml.ai/MethodOfLines/stable/tutorials/PIDE/. My modification is to change the integral term. Instead of integrating u(t,x) wrt x I would like to integrate (y-x)u(t,y) wrt y. Thus, I have to evaluate the function u which is originally defined as a function of x, with a paramter y in the integral. u still also needs to be able to be evaluated with parameter x since in the other parts of the PDE I still have to evaluate u(t,x). How can I do that? Because when I just define y as a paramter and then write u(t,y) I get the error message:  
ERROR: ArgumentError: invalid index: nothing of type Nothing  
Stacktrace:  
[1] to\_index(i::Nothing)  
@ Base ./indices.jl:300  
[2] to\_index(A::Vector{Symbolics.VarDomainPairing}, i::Nothing)  
@ Base ./indices.jl:277  
[3] \_to\_indices1(A::Vector{Symbolics.VarDomainPairing}, inds::Tuple{Base.OneTo{Int64}}, I1::Nothing)  
@ Base ./indices.jl:359  
[4] to\_indices  
@ Base ./indices.jl:354 [inlined]  
[5] to\_indices  
@ Base ./indices.jl:345 [inlined]  
[6] getindex  
@ Base ./abstractarray.jl:1288 [inlined]  
[7] (::PDEBase.var"#54#62"{Vector{Symbolics.VarDomainPairing}})(x::SymbolicUtils.BasicSymbolic{Real})  
@ PDEBase ~/.julia/packages/PDEBase/VVGPP/src/variable\_map.jl:34  
[8] iterate  
@ Base ./generator.jl:47 [inlined]  
[9] collect\_to!(dest::Vector{Pair{…}}, itr::Base.Generator{Vector{…}, PDEBase.var"#54#62"{…}}, offs::Int64, st::Int64)  
@ Base ./array.jl:892  
[10] collect\_to\_with\_first!(dest::Vector{…}, v1::Pair{…}, itr::Base.Generator{…}, st::Int64)  
@ Base ./array.jl:870  
[11] \_collect(c::Vector{…}, itr::Base.Generator{…}, ::Base.EltypeUnknown, isz::Base.HasShape{…})  
@ Base ./array.jl:864  
[12] collect\_similar(cont::Vector{Any}, itr::Base.Generator{Vector{Any}, PDEBase.var"#54#62"{Vector{…}}})  
@ Base ./array.jl:763  
[13] map(f::Function, A::Vector{Any})  
@ Base ./abstractarray.jl:3282  
[14] PDEBase.VariableMap(pdesys::PDESystem, disc::MOLFiniteDifference{…}; replaced\_vars::Dict{…})  
@ PDEBase ~/.julia/packages/PDEBase/VVGPP/src/variable\_map.jl:33  
[15] symbolic\_discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…})  
@ PDEBase ~/.julia/packages/PDEBase/VVGPP/src/symbolic\_discretize.jl:17  
[16] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…}; analytic::Nothing, kwargs::@Kwargs{})  
@ PDEBase ~/.julia/packages/PDEBase/VVGPP/src/discretization\_state.jl:57  
[17] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…})  
@ PDEBase ~/.julia/packages/PDEBase/VVGPP/src/discretization\_state.jl:54  
[18] top-level scope  
@ ~/Documents/implement/multid\_op\_form/example\_1.jl:27  
Some type information was truncated. Use `show(err)` to see complete types.

Here is the code I used:

```julia
using MethodOfLines, ModelingToolkit, OrdinaryDiffEq, DomainSets, Plots

@parameters t, x, y

@variables u(..) cumuSum(..) integrand(..)

Dt = Differential(t)

Dx = Differential(x)

xmin = 0.0

xmax = 2.0pi

# Integral limits are defined with DomainSets.ClosedInterval

Iy = Integral(y in DomainSets.ClosedInterval(xmin, xmax)) # basically cumulative sum from 0 to x

eq = [

integrand(t,x, y) ~ (y-x)*u(t, y), # Define the integrand as a separate function

cumuSum(t, x) ~ Iy(integrand(t,x,y)), # Note wrapping the argument to the derivative with an auxiliary variable

Dt(u(t, x)) + 2 * u(t, x) + 5 * Dx(cumuSum(t, x)) ~ 1

]

bcs = [u(0.0, x) ~ cos(x), Dx(u(t, xmin)) ~ 0.0, Dx(u(t, xmax)) ~ 0]

domains = [t ∈ Interval(0.0, 2.0), x ∈ Interval(xmin, xmax)]

@named pde_system = PDESystem(eq, bcs, domains, [t, x], [u(t, x), cumuSum(t, x)])

order = 2

discretization = MOLFiniteDifference([x => 30], t)

prob = MethodOfLines.discretize(pde_system, discretization)

sol = solve(prob, QNDF(), saveat = 0.1);

solu = sol[u(t, x)]

plot(sol[x], transpose(solu))

```

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [December 13, 2024, 6:21pm UTC](https://discourse.julialang.org/t/integrate-convolution-symbolically-to-solve-pde/123810/2 "2024-12-13T18:21:31Z")

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Right now we don’t have enough symbolic integral solving tools for this, but it is planned down the line.

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<div class="post-metadata">

**Author:** ![B4rtel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/b4rtel/32/214171_2.png) [@B4rtel](https://discourse.julialang.org/u/B4rtel)\
**Post date:** [December 13, 2024, 9:32pm UTC](https://discourse.julialang.org/t/integrate-convolution-symbolically-to-solve-pde/123810/3 "2024-12-13T21:32:41Z")

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Oh, that’s unfortunate. Thx for the answer tho and I’m looking forward to when it’s implemented 🙂