ode_simple_nonlinear_prob.jl

## Lotka-Volterra

function lotka(du, u, p, t)
    x = u[1]
    y = u[2]
    du[1] = p[1] * x - p[2] * x * y
    du[2] = -p[3] * y + p[4] * x * y
end

"""
Lotka-Volterra Equations (Non-stiff)

```math
\\frac{dx}{dt} = ax - bxy
```
```math
\\frac{dy}{dt} = -cy + dxy
```

with initial condition ``x=y=1``
"""
prob_ode_lotkavolterra = ODEProblem(lotka, [1.0, 1.0], (0.0, 1.0), [1.5, 1.0, 3.0, 1.0])

## Fitzhugh-Nagumo

function fitz(du, u, p, t)
    v = u[1]
    w = u[2]
    a = p[1]
    b = p[2]
    τinv = p[3]
    l = p[4]
    du[1] = v - v^3 / 3 - w + l
    du[2] = τinv * (v + a - b * w)
end
"""
Fitzhugh-Nagumo (Non-stiff)

```math
\\frac{dv}{dt} = v - \\frac{v^3}{3} - w + I_{est}
```
```math
τ \\frac{dw}{dt} = v + a -bw
```

with initial condition ``v=w=1``
"""
prob_ode_fitzhughnagumo = ODEProblem(fitz, [1.0; 1.0], (0.0, 1.0),
    (0.7, 0.8, 1 / 12.5, 0.5))

#Van der Pol Equations
@parameters μ
@variables x(t) y(t)

eqs = [D(y) ~ μ * ((1 - x^2) * y - x),
    D(x) ~ y]
van = ODESystem(eqs, t; name = :van_der_pol) |> structural_simplify |> complete

"""
Van der Pol Equations

```math
\\frac{dx}{dt} = y
```
```math
\\frac{dy}{dt} = μ((1-x^2)y -x)
```

with ``μ=1.0`` and ``u_0=[x => \\sqrt{3}, y => 0]``

Non-stiff parameters.
"""
prob_ode_vanderpol = ODEProblem(van, [y => 0, x => sqrt(3)], (0.0, 1.0), [μ => 1.0], jac=true, eval_module = @__MODULE__)

"""
Van der Pol Equations

```math
\\frac{dx}{dt} = y
```
```math
\\frac{dy}{dt} = μ((1-x^2)y -x)
```

with ``μ=10^6`` and ``u_0=[x => \\sqrt{3}, y => 0]``

Stiff parameters.
"""
prob_ode_vanderpol_stiff = ODEProblem(van, [y => 0, x => sqrt(3)], (0.0, 1.0), [μ => 1e6], jac=true, eval_module = @__MODULE__)

# ROBER
@parameters k₁ k₂ k₃
@variables y₁(t) y₂(t) y₃(t)

eqs = [D(y₁) ~ -k₁ * y₁ + k₃ * y₂ * y₃,
    D(y₂) ~ k₁ * y₁ - k₂ * y₂^2 - k₃ * y₂ * y₃,
    D(y₃) ~ k₂ * y₂^2]
rober = ODESystem(eqs, t; name = :rober) |> structural_simplify |> complete

"""
The Robertson biochemical reactions: (Stiff)

```math
\\frac{dy₁}{dt} = -k₁y₁+k₃y₂y₃
```
```math
\\frac{dy₂}{dt} =  k₁y₁-k₂y₂^2-k₃y₂y₃
```
```math
\\frac{dy₃}{dt} =  k₂y₂^2
```

where ``k₁=0.04``, ``k₂=3\\times10^7``, ``k₃=10^4``. For details, see:

Hairer Norsett Wanner Solving Ordinary Differential Equations I - Nonstiff Problems Page 129

Usually solved on ``[0,1e11]``
"""
prob_ode_rober = ODEProblem(rober, [y₁, y₂, y₃] .=> [1.0; 0.0; 0.0], (0.0, 1e11), [k₁, k₂, k₃] .=> (0.04, 3e7, 1e4), jac = true, eval_module = @__MODULE__)

# Three Body
const threebody_μ = big(0.012277471);
const threebody_μ′ = 1 - threebody_μ;

threebody = (du, u, p, t) -> begin
    # 1 = y₁
    # 2 = y₂
    # 3 = y₁'
    # 4 = y₂'
    D₁ = ((u[1] + threebody_μ)^2 + u[2]^2)^(3 / 2)
    D₂ = ((u[1] - threebody_μ′)^2 + u[2]^2)^(3 / 2)
    du[1] = u[3]
    du[2] = u[4]
    du[3] = u[1] + 2u[4] - threebody_μ′ * (u[1] + threebody_μ) / D₁ -
            threebody_μ * (u[1] - threebody_μ′) / D₂
    du[4] = u[2] - 2u[3] - threebody_μ′ * u[2] / D₁ - threebody_μ * u[2] / D₂
end

@doc doc"""
The ThreeBody problem as written by Hairer: (Non-stiff)

```math
\frac{dy₁}{dt} = y₁ + 2\frac{dy₂}{dt} - \bar{μ}\frac{y₁+μ}{D₁} - μ\frac{y₁-\bar{μ}}{D₂}
```
```math
\frac{dy₂}{dt} = y₂ - 2\frac{dy₁}{dt} - \bar{μ}\frac{y₂}{D₁} - μ\frac{y₂}{D₂}
```
```math
D₁ = ((y₁+μ)^2 + y₂^2)^{3/2}
```
```math
D₂ = ((y₁-\bar{μ})^2+y₂^2)^{3/2}
```
```math
μ = 0.012277471
```
```math
\bar{μ} =1-μ
```

From Hairer Norsett Wanner Solving Ordinary Differential Equations I - Nonstiff Problems Page 129

Usually solved on ``t₀ = 0.0`` and ``T = 17.0652165601579625588917206249``
Periodic with that setup.
"""
prob_ode_threebody = ODEProblem(threebody,
    [0.994, 0.0, 0.0, big(-2.00158510637908252240537862224)],
    (big(0.0), big(17.0652165601579625588917206249)))

# Rigid Body Equations

@parameters I₁ I₂ I₃
@variables y₁(t) y₂(t) y₃(t)

eqs = [D(y₁) ~ I₁ * y₂ * y₃,
    D(y₂) ~ I₂ * y₁ * y₃,
    D(y₃) ~ I₃ * y₁ * y₂]
rigid = ODESystem(eqs, t; name = :rigid_body) |> structural_simplify |> complete

"""
Rigid Body Equations (Non-stiff)

```math
\\frac{dy₁}{dt}  = I₁y₂y₃
```
```math
\\frac{dy₂}{dt}  = I₂y₁y₃
```
```math
\\frac{dy₃}{dt}  = I₃y₁y₂
```

with ``I₁=-2``, ``I₂=1.25``, and ``I₃=-1/2``.

The initial condition is ``y=[1.0;0.0;0.9]``.

From Solving Differential Equations in R by Karline Soetaert

or Hairer Norsett Wanner Solving Ordinary Differential Equations I - Nonstiff Problems Page 244

Usually solved from 0 to 20.
"""
prob_ode_rigidbody = ODEProblem(rigid, [y₁, y₂, y₃] .=> [1.0, 0.0, 0.9], (0.0, 20.0),
                                       [I₁, I₂, I₃] .=> (-2.0, 1.25, -0.5), jac = true,
                                       eval_module = @__MODULE__)

# Pleiades Problem

pleiades = (du, u, p, t) -> begin
    x = view(u, 1:7)   # x
    y = view(u, 8:14)  # y
    v = view(u, 15:21) # x′
    w = view(u, 22:28) # y′
    du[1:7] .= v
    du[8:14] .= w
    for i in 15:28
        du[i] = zero(eltype(u))
    end
    for i in 1:7, j in 1:7
        if i != j
            r = ((x[i] - x[j])^2 + (y[i] - y[j])^2)^(3 / 2)
            du[14 + i] += j * (x[j] - x[i]) / r
            du[21 + i] += j * (y[j] - y[i]) / r
        end
    end
end

@doc doc"""
Pleiades Problem (Non-stiff)

```math
\frac{d^2xᵢ}{dt^2} = \sum_{j≠i} mⱼ(xⱼ-xᵢ)/rᵢⱼ
```
```math
\frac{d^2yᵢ}{dt^2} = \sum_{j≠i} mⱼ(yⱼ-yᵢ)/rᵢⱼ
```

where

```math
rᵢⱼ = ((xᵢ-xⱼ)^2 + (yᵢ-yⱼ)^2)^{3/2}
```

and initial conditions are

```math
x₁(0) = 3
```
```math
x₂(0) = 3
```
```math
x₃(0) = -1
```
```math
x₄(0) = -3
```
```math
x₅(0) = 2
```
```math
x₆(0) = -2
```
```math
x₇(0) = 2
```
```math
y₁(0) = 3
```
```math
y₂(0) = -3
```
```math
y₃(0) = 2
```
```math
y₄(0) = 0
```
```math
y₅(0) = 0
```
```math
y₆(0) = -4
```
```math
y₇(0) = 4
```

and with ``\frac{dxᵢ(0)}{dt}=\frac{dyᵢ(0)}{dt}=0`` except for

```math
\frac{dx₆(0)}{dt} = 1.75
```
```math
\frac{dx₇(0)}{dt} = -1.5
```
```math
\frac{dy₄(0)}{dt} = -1.25
```
```math
\frac{dy₅(0)}{dt} = 1
```

From Hairer Norsett Wanner Solving Ordinary Differential Equations I - Nonstiff Problems Page 244

Usually solved from 0 to 3.
"""
prob_ode_pleiades = ODEProblem(pleiades,
    [
        3.0,
        3.0,
        -1.0,
        -3.0,
        2.0,
        -2.0,
        2.0,
        3.0,
        -3.0,
        2.0,
        0,
        0,
        -4.0,
        4.0,
        0,
        0,
        0,
        0,
        0,
        1.75,
        -1.5,
        0,
        0,
        0,
        -1.25,
        1,
        0,
        0
    ], (0.0, 3.0))

Random.seed!(100)
const mm_A = rand(4, 4)
mm_linear = function (du, u, p, t)
    mul!(du, mm_A, u)
end
const MM_linear = Matrix(Diagonal(0.5ones(4)))
mm_f = ODEFunction(mm_linear; analytic = (u0, p, t) -> exp(inv(MM_linear) * mm_A * t) * u0,
    mass_matrix = MM_linear)
prob_ode_mm_linear = ODEProblem(mm_f, rand(4), (0.0, 1.0))

@parameters p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12
@variables y1(t) y2(t) y3(t) y4(t) y5(t) y6(t) y7(t) y8(t)

eqs = [D(y1) ~ -p1 * y1 + p2 * y2 + p3 * y3 + p4,
    D(y2) ~ p1 * y1 - p5 * y2,
    D(y3) ~ -p6 * y3 + p2 * y4 + p7 * y5,
    D(y4) ~ p3 * y2 + p1 * y3 - p8 * y4,
    D(y5) ~ -p9 * y5 + p2 * y6 + p2 * y7,
    D(y6) ~ -p10 * y6 * y8 + p11 * y4 + p1 * y5 -
            p2 * y6 + p11 * y7,
    D(y7) ~ p10 * y6 * y8 - p12 * y7,
    D(y8) ~ -p10 * y6 * y8 + p12 * y7]
hires = ODESystem(eqs, t; name = :hires) |> structural_simplify |> complete

u0 = zeros(8)
u0[1] = 1
u0[8] = 0.0057

"""
Hires Problem (Stiff)

It is in the form of

```math
\\frac{dy}{dt} = f(y)
```

 with

```math
 y(0)=y_0, \\quad y \\in ℝ^8, \\quad 0 ≤ t ≤ 321.8122
```

where ``f`` is defined by

``f(y) = \\begin{pmatrix} −1.71y_1 & +0.43y_2 & +8.32y_3 & +0.0007y_4 & \\\\ 1.71y_1 & −8.75y_2 & & & \\\\ −10.03y_3 & +0.43y_4 & +0.035y_5 & & \\\\ 8.32y_2 & +1.71y_3 & −1.12y_4 & & \\\\ −1.745y_5 & +0.43y_6 & +0.43y_7 & & \\\\ −280y_6y_8 & +0.69y_4 & +1.71y_5 & −0.43y_6 & +0.69y_7 \\\\ 280y_6y_8 & −1.81y_7 & & & \\\\ −280y_6y_8 & +1.81y_7 & & & \\end{pmatrix}``

Reference: [demohires.pdf](http://www.radford.edu/~thompson/vodef90web/problems/demosnodislin/Demos_Pitagora/DemoHires/demohires.pdf)
Notebook: [Hires.ipynb](http://nbviewer.jupyter.org/github/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/StiffODE/Hires.ipynb)
"""
prob_ode_hires = ODEProblem(hires, [y1, y2, y3, y4, y5, y6, y7, y8] .=> u0, (0.0, 321.8122),
                            [p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12] .=> (1.71, 0.43, 8.32, 0.0007, 8.75,
                                    10.03, 0.035, 1.12, 1.745, 280.0,
                                    0.69, 1.81), jac = true, eval_module = @__MODULE__)

@parameters p1 p2 p3
@variables y1(t) y2(t) y3(t)

eqs = [D(y1) ~ p1 * (y2 + y1 * (1 - p2 * y1 - y2)),
    D(y2) ~ (y3 - (1 + y1) * y2) / p1,
    D(y3) ~ p3 * (y1 - y3)]
orego = ODESystem(eqs, t; name = :orego) |> structural_simplify |> complete

"""
Orego Problem (Stiff)

It is in the form of ``\\frac{dy}{dt}=f(y), \\quad y(0)=y0,`` with

```math
y \\in ℝ^3, \\quad 0 ≤ t ≤ 360
```

where ``f`` is defined by

``f(y) = \\begin{pmatrix} s(y_2 - y_1(1-qy_1-y_2)) \\\\ (y_3 - y_2(1+y_1))/s \\\\ w(y_1-y_3) \\end{pmatrix}``

where ``s=77.27``, ``w=0.161`` and ``q=8.375⋅10^{-6}``.

Reference: [demoorego.pdf](http://www.radford.edu/~thompson/vodef90web/problems/demosnodislin/Demos_Pitagora/DemoOrego/demoorego.pdf)
Notebook: [Orego.ipynb](http://nbviewer.jupyter.org/github/JuliaDiffEq/DiffEqBenchmarks.jl/blob/master/StiffODE/Orego.ipynb)
"""
prob_ode_orego = ODEProblem(orego, [y1, y2, y3] .=> [1.0, 2.0, 3.0], (0.0, 30.0), [p1, p2, p3] .=> [77.27, 8.375e-6, 0.161], jac = true, eval_module = @__MODULE__)