ODE Problems

Linear ODE

\[\frac{du}{dt} = αu\]

with initial condition $u_0=\frac{1}{2}$, $α=1.01$, and solution

\[u(t) = u_0 e^{αt}\]

with Float64s. The parameter is $α$

4×2 version of the Linear ODE

\[\frac{du}{dt} = αu\]

with initial condition $u_0$ as all uniformly distributed random numbers, $α=1.01$, and solution

\[u(t) = u_0 e^{αt}\]

with Float64s

Linear ODE

\[\frac{du}{dt} = αu\]

with initial condition $u_0=\frac{1}{2}$, $α=1.01$, and solution

\[u(t) = u_0 e^{αt}\]

with BigFloats

4×2 version of the Linear ODE

\[\frac{du}{dt} = αu\]

with initial condition $u_0$ as all uniformly distributed random numbers, $α=1.01$, and solution

\[u(t) = u_0 e^{αt}\]

with BigFloats

100×100 version of the Linear ODE

\[\frac{du}{dt} = αu\]

with initial condition $u_0$ as all uniformly distributed random numbers, $α=1.01$, and solution

\[u(t) = u_0 e^{αt}\]

with Float64s

4×2 version of the Linear ODE

\[\frac{du}{dt} = αu\]

with initial condition $u_0$ as all uniformly distributed random numbers, $α=1.01$, and solution

\[u(t) = u_0 e^{αt}\]

on Float64. Purposefully not in-place as a test.

Lotka-Volterra Equations (Non-stiff)

\[\begin{align*} \frac{dx}{dt} &= ax - bxy \\ \frac{dy}{dt} &= -cy + dxy \\ \end{align*}\]

with initial condition $x=y=1$

Fitzhugh-Nagumo (Non-stiff)

\[\begin{align*} \frac{dv}{dt} &= v - \frac{v^3}{3} - w + I_{est} \\ τ \frac{dw}{dt} &= v + a -bw \end{align*}\]

with initial condition $v=w=1$

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

\[\begin{align*} \frac{dy₁}{dt} &= y₁ + 2\frac{dy₂}{dt} - \bar{μ}\frac{y₁+μ}{D₁} - μ\frac{y₁-\bar{μ}}{D₂} \\ \frac{dy₂}{dt} &= y₂ - 2\frac{dy₁}{dt} - \bar{μ}\frac{y₂}{D₁} - μ\frac{y₂}{D₂} \end{align*}\]

\[\begin{align*} D₁ &= \left((y₁+μ)^2 + y₂^2\right)^{3/2} \\ D₂ &= \left((y₁-\bar{μ})^2 + y₂^2\right)^{3/2} \\ μ &= 0.012277471 \\ \bar{μ} &= 1-μ \end{align*}\]

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.

Pleiades Problem (Non-stiff)

\[\begin{align*} \frac{d^2xᵢ}{dt^2} &= \sum_{j≠i} mⱼ(xⱼ-xᵢ)/rᵢⱼ \\ \frac{d^2yᵢ}{dt^2} &= \sum_{j≠i} mⱼ(yⱼ-yᵢ)/rᵢⱼ \end{align*}\]

where

\[rᵢⱼ = \left((xᵢ-xⱼ)^2 + (yᵢ-yⱼ)^2\right)^{3/2}\]

and initial conditions are

\[\begin{align*} x₁(0) &= 3, & y₁(0) &= 3, \\ x₂(0) &= 3, & y₂(0) &= -3, \\ x₃(0) &= -1, & y₃(0) &= 2, \\ x₄(0) &= -3, & y₄(0) &= 0, \\ x₅(0) &= 2, & y₅(0) &= 0, \\ x₆(0) &= -2, & y₆(0) &= -4, \\ x₇(0) &= 2, & y₇(0) &= 4 \end{align*}\]

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

\[\begin{align*} \frac{dx₆(0)}{dt} &= 1.75, & \frac{dx₇(0)}{dt} &= -1.5, \\ \frac{dy₄(0)}{dt} &= -1.25, & \frac{dy₅(0)}{dt} &= 1 \end{align*}\]

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

Usually solved from 0 to 3.

Van der Pol Equations

\[\begin{align*} \frac{dx}{dt} &= y \\ \frac{dy}{dt} &= μ \left(\left(1-x^2\right) y - x\right) \end{align*}\]

with $μ=1.0$ and $u_0=[\sqrt{3}, 0]$ (where $u[1] = x$, $u[2] = y$)

Non-stiff parameters.

Van der Pol Equations

\[\begin{align*} \frac{dx}{dt} &= y \\ \frac{dy}{dt} &= μ \left(\left(1 - x^2\right) y - x\right) \end{align*}\]

with $μ=10^6$ and $u_0=[\sqrt{3}, 0]$ (where $u[1] = x$, $u[2] = y$)

Stiff parameters.

The Robertson biochemical reactions: (Stiff)

\[\begin{align*} \frac{dy₁}{dt} &= -k₁y₁ + k₃y₂y₃ \\ \frac{dy₂}{dt} &= k₁y₁ - k₂y₂^2 - k₃y₂y₃ \\ \frac{dy₃}{dt} &= k₂y₂^2 \end{align*}\]

where $k₁=0.04$, $k₂=3×10^7$, $k₃=10^4$. For details, see:

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

Usually solved on $[0,10^{11}]$

Rigid Body Equations (Non-stiff)

\[\begin{align*} \frac{dy₁}{dt} &= I₁y₂y₃ \\ \frac{dy₂}{dt} &= I₂y₁y₃ \\ \frac{dy₃}{dt} &= I₃y₁y₂ \end{align*}\]

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.

Hires Problem (Stiff)

It is in the form of

\[\frac{dy}{dt} = f(y)\]

with

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

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 Notebook: Hires.ipynb

Orego Problem (Stiff)

It is in the form of $\frac{dy}{dt}=f(y), \quad y(0)=y_0,$ with

\[y \in ℝ^3, \quad 0 ≤ t ≤ 360\]

where $f$ is defined by

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

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

Reference: demoorego.pdf Notebook: Orego.ipynb

Pollution Problem (Stiff)

This IVP is a stiff system of 20 non-linear Ordinary Differential Equations. It is in the form of

\[\frac{dy}{dt}=f(y)\]

with

\[y(0)=y_0, \quad y \in ℝ^{20}, \quad 0 ≤ t ≤ 60\]

where $f$ is defined by

\[f(y) = \begin{pmatrix} -\sum_{j\in{1,10,14,23,24}} r_j + \sum_{j\in{2,3,9,11,12,22,25}} r_j \\ -r_2 - r_3 - r_9 - r_12 + r_1 + r_{21} \\ -r_{15} + r_1 + r_{17} + r_{19} + r_{22} \\ -r_2 - r_{16} - r_{17} - r_{23} + r_{15} \\ -r_3 + 2r_4 + r_6 + r_7 + r_{13} + r_{20} \\ -r_6 - r_8 - r_{14} - r_{20} + r_3 + 2r_{18} \\ -r_4 - r_5 - r_6 + r_{13} \\ r_4 + r_5 + r_6 + r_7 \\ -r_7 - r_8 \\ -r_{12} + r_7 + r_9 \\ -r_9 - r_{10} + r_8 + r_{11} \\ r_9 \\ -r_{11} + r_{10} \\ -r_{13} + r_{12} \\ r_{14} \\ -r_{18} - r_{19} + r_{16} \\ -r_{20} \\ r_{20} \\ -r_{21} - r_{22} - r_{24} + r_{23} + r_{25} \\ -r_{25} + r_{24} \end{pmatrix}\]

with the initial condition of

\[y0 = (0, 0.2, 0, 0.04, 0, 0, 0.1, 0.3, 0.01, 0, 0, 0, 0 ,0, 0, 0, 0.007, 0, 0, 0)^T\]

Analytical Jacobian is included.

Reference: pollu.pdf Notebook: Pollution.ipynb

Nonlinear system of reactions with an analytical solution

\[\begin{align*} \frac{dy_1}{dt} &= -y_1 \\ \frac{dy_2}{dt} &= y_1 - y_2^2 \\ \frac{dy_3}{dt} &= y_2^2 \end{align*}\]

with initial condition $y=[1;0;0]$ on a time span of $t \in (0,20)$

From

Liu, L. C., Tian, B., Xue, Y. S., Wang, M., & Liu, W. J. (2012). Analytic solution for a nonlinear chemistry system of ordinary differential equations. Nonlinear Dynamics, 68(1-2), 17-21.

The analytical solution is implemented, allowing easy testing of ODE solvers.

1D Brusselator

\[\begin{align*} \frac{∂u}{∂t} &= A - (B+1) u + u^2 v + α \frac{∂^2 u}{∂x^2} \\ \frac{∂v}{∂t} &= B u - u^2 v + α \frac{∂^2 u}{∂x^2} \end{align*}\]

and the initial conditions are

\[\begin{align*} u(x,0) &= 1 + \sin(2πx) \\ v(x,0) &= 3 \end{align*}\]

with periodic boundary conditions

\[\begin{align*} u(0,t) &= u(1,t) \\ v(0,t) &= v(1,t) \end{align*}\]

From Hairer Norsett Wanner Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems Page 6

2D Brusselator

\[\begin{align*} \frac{∂u}{∂t} &= 1 + u^2v - 4.4u + α\left(\frac{∂^2 u}{∂x^2} + \frac{∂^2 u}{∂y^2}\right) + f(x, y, t) \\ \frac{∂v}{∂t} &= 3.4u - u^2v + α\left(\frac{∂^2 u}{∂x^2} + \frac{∂^2 u}{∂y^2}\right) \end{align*}\]

where

\[f(x, y, t) = \begin{cases} 5 & \text{if } (x-0.3)^2+(y-0.6)^2 ≤ 0.1^2 \text{ and } t ≥ 1.1 \\ 0 & \text{else} \end{cases}\]

and the initial conditions are

\[\begin{align*} u(x, y, 0) &= 22 ⋅ y(1-y)^{3/2} \\ v(x, y, 0) &= 27 ⋅ x(1-x)^{3/2} \end{align*}\]

with the periodic boundary condition

\[\begin{align*} u(x+1,y,t) &= u(x,y,t) \\ u(x,y+1,t) &= u(x,y,t) \end{align*}\]

From Hairer Norsett Wanner Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems Page 152

Filament PDE Discretization

Notebook: Filament.ipynb

In this problem is a real-world biological model from a paper entitled Magnetic dipole with a flexible tail as a self-propelling microdevice. It is a system of PDEs representing a Kirchhoff model of an elastic rod, where the equations of motion are given by the Rouse approximation with free boundary conditions.

Thomas' cyclically symmetric attractor equations

\[\begin{align*} \frac{dx}{dt} &= \sin(y) - bx \\ \frac{dy}{dt} &= \sin(z) - by \\ \frac{dz}{dt} &= \sin(x) - bz \end{align*}\]

with parameter $b = 0.208186$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference

Wikipedia

Lorenz equations

\[\begin{align*} \frac{dx}{dt} &= σ(y - x) \\ \frac{dy}{dt} &= x(ρ - z) - y \\ \frac{dz}{dt} &= xy - βz \end{align*}\]

with parameters $σ=10, ρ=28, β=8/3$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference

Wikipedia

Aizawa equations

\[\begin{align*} \frac{dx}{dt} &= (z - b)x - dy \\ \frac{dy}{dt} &= dx + (z - b)y \\ \frac{dz}{dt} &= c + az - \frac{z^3}{3} - (x^2 + y^2)(1 + ez) + fzx^3 \end{align*}\]

with parameters $a=0.95, b=0.7, c=0.6, d=3.5, e=0.25, f=0.1$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference

Dadras equations

\[\begin{align*} \frac{dx}{dt} &= y - ax + byz \\ \frac{dy}{dt} &= cy - xz + z \\ \frac{dz}{dt} &= dxy - ez \end{align*}\]

with parameters $a=3, b=2.7, c=1.7, d=2, e=9$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference

Chen equations

\[\begin{align*} \frac{dx}{dt} &= a(y - x) \\ \frac{dy}{dt} &= (c - a)x - xz + cy \\ \frac{dz}{dt} &= xy - bz \end{align*}\]

with parameters $a=35, b=3, c=28$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference

Rössler equations

\[\begin{align*} \frac{dx}{dt} &= -(y + z) \\ \frac{dy}{dt} &= x + ay \\ \frac{dz}{dt} &= b + z(x - c) \end{align*}\]

with parameters $a=0.2, b=0.2, c=5.7$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

ReferenceWikipedia

Rabinovich-Fabrikant equations

\[\begin{align*} \frac{dx}{dt} &= y(z - 1 + x^2) + bx \\ \frac{dy}{dt} &= x(3z + 1 - x^2) + by \\ \frac{dz}{dt} &= -2z(a + xy) \end{align*}\]

with parameters $a=0.14, b=0.10$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference

Sprott equations

\[\begin{align*} \frac{dx}{dt} &= y + axy + xz \\ \frac{dy}{dt} &= 1 - bx^2 + yz \\ \frac{dz}{dt} &= x - x^2 - y^2 \end{align*}\]

with parameters $a=2.07, b=1.79$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference

Hindmarsh-Rose equations

\[\begin{align*} \frac{dx}{dt} &= y - ax^3 + bx^2 - z + i \\ \frac{dy}{dt} &= c - dx^2 - y \\ \frac{dz}{dt} &= r(s(x - x_r) - z) \end{align*}\]

with parameters $a=1, b=3, c=1, d=5, r=10^{-2}, s=4, x_r=-8/5, i=5$ and initial conditions $x(0)=1, y(0)=0, z(0)=0$

Reference