Can this type of integro-differential equation be solved somewhere?

Specific Domains Modelling & Simulations
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1

Hey,

I admit that I am a novice in differential equations solving, this problem came out of completely somthing else…

I want to solve an equation that has the following form :

S(t) = \exp(- H(t))
\partial H(t) = F(t, S(t), parameters...)

The function F(t) is highly non-linear, but :

  • it has jumps at previously known time values t_1,...t_n
  • between these times values it is continuous (but still non-linear).

I can provide starting values S(0) = 1.

Is this kind of stuff solvable ?

2 
using DifferentialEquations

function odefunc!(du,u,p,t)
    S = u[1]
    H = u[2]

    dS = exp(-H)
    dH = p * (S - t^2 + S^3)  #arbitrary nonlinear function, p is switched at predefined times in callback

    du[1] = dS
    du[2] = dH
    nothing
end

affect!(integrator) = (integrator.p > 0) ? integrator.p = -1 : integrator.p = 1
cb = PresetTimeCallback([2,5],affect!)

S0 = 1
H0 = 2
u0 = [S0,H0]

p = 1.0

prob = ODEProblem(odefunc!,u0,(0,10),p)
sol = solve(prob, callback = cb)

julia> sol.u
26-element Vector{Vector{Float64}}:
 [1.0, 2.0]
 [1.0130247068630718, 2.2164863270563178]
 [1.0360292186508027, 2.779573896877542]
 ⋮
 [4.875245836243441, 65.69652980394312]
 [4.875245836243441, 187.0985129901355]

image
image711×463 13.5 KB

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3

That’s the jumps part. The other part, using previous values, is just a DDE.

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