ODE Problems
Mathematical Specification of an ODE Problem
To define an ODE Problem, you simply need to give the function $f$ and the initial condition $u₀$ which define an ODE:
\[\frac{du}{dt} = f(u,p,t)\]
f
should be specified as f(u,p,t)
(or in-place as f(du,u,p,t)
), and u₀
should be an AbstractArray (or number) whose geometry matches the desired geometry of u
. Note that we are not limited to numbers or vectors for u₀
; one is allowed to provide u₀
as arbitrary matrices / higher dimension tensors as well.
Problem Type
Constructors
ODEProblem(f::ODEFunction,u0,tspan,p=NullParameters();kwargs...)
ODEProblem{isinplace}(f,u0,tspan,p=NullParameters();kwargs...)
: Defines the ODE with the specified functions.isinplace
optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.
Parameters are optional, and if not given then a NullParameters()
singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a callback
in the problem, then that callback
will be added in every solve call.
For specifying Jacobians and mass matrices, see the DiffEqFunctions page.
Fields
f
: The function in the ODE.u0
: The initial condition.tspan
: The timespan for the problem.p
: The parameters.kwargs
: The keyword arguments passed onto the solves.
Example Problems
Example problems can be found in DiffEqProblemLibrary.jl.
To use a sample problem, such as prob_ode_linear
, you can do something like:
#] add DiffEqProblemLibrary
using DiffEqProblemLibrary.ODEProblemLibrary
# load problems
ODEProblemLibrary.importodeproblems()
prob = ODEProblemLibrary.prob_ode_linear
sol = solve(prob)
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_linear
— ConstantLinear ODE
\[\frac{du}{dt} = αu\]
with initial condition $u_0=\frac{1}{2}$, $α=1.01$, and solution
\[u(t) = u_0e^{αt}\]
with Float64s. The parameter is $α$
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_2Dlinear
— Constant4x2 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
\[u(t) = u_0e^{αt}\]
with Float64s
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_bigfloatlinear
— ConstantLinear ODE
\[\frac{du}{dt} = αu\]
with initial condition $u_0=\frac{1}{2}$, $α=1.01$, and solution
\[u(t) = u_0e^{αt}\]
with BigFloats
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_bigfloat2Dlinear
— Constant4x2 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
\[u(t) = u_0e^{αt}\]
with BigFloats
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_large2Dlinear
— Constant100x100 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
\[u(t) = u_0e^{αt}\]
with Float64s
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_2Dlinear_notinplace
— Constant4x2 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
\[u(t) = u_0e^{αt}\]
on Float64. Purposefully not in-place as a test.
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_lotkavoltera
— ConstantLotka-Voltera Equations (Non-stiff)
\[\frac{dx}{dt} = ax - bxy\]
\[\frac{dy}{dt} = -cy + dxy\]
with initial condition $x=y=1$
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_fitzhughnagumo
— ConstantFitzhugh-Nagumo (Non-stiff)
\[\frac{dv}{dt} = v - \frac{v^3}{3} - w + I_{est}\]
\[τ \frac{dw}{dt} = v + a -bw\]
with initial condition $v=w=1$
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_threebody
— ConstantThe ThreeBody problem as written by Hairer: (Non-stiff)
\[\frac{dy₁}{dt} = y₁ + 2\frac{dy₂}{dt} - \bar{μ}\frac{y₁+μ}{D₁} - μ\frac{y₁-\bar{μ}}{D₂}\]
\[\frac{dy₂}{dt} = y₂ - 2\frac{dy₁}{dt} - \bar{μ}\frac{y₂}{D₁} - μ\frac{y₂}{D₂}\]
\[D₁ = ((y₁+μ)^2 + y₂^2)^{3/2}\]
\[D₂ = ((y₁-\bar{μ})^2+y₂^2)^{3/2}\]
\[μ = 0.012277471\]
\[\bar{μ} =1-μ\]
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.
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_pleiades
— ConstantPleiades Problem (Non-stiff)
\[\\frac{d^2xᵢ}{dt^2} = \sum_{j≠i} mⱼ(xⱼ-xᵢ)/rᵢⱼ\]
\[\\frac{d^2yᵢ}{dt^2} = \sum_{j≠i} mⱼ(yⱼ-yᵢ)/rᵢⱼ\]
where
\[rᵢⱼ = ((xᵢ-xⱼ)^2 + (yᵢ-yⱼ)^2)^{3/2}\]
and initial conditions are
\[x₁(0) = 3\]
\[x₂(0) = 3\]
\[x₃(0) = -1\]
\[x₄(0) = -3\]
\[x₅(0) = 2\]
\[x₆(0) = -2\]
\[x₇(0) = 2\]
\[y₁(0) = 3\]
\[y₂(0) = -3\]
\[y₃(0) = 2\]
\[y₄(0) = 0\]
\[y₅(0) = 0\]
\[y₆(0) = -4\]
\[y₇(0) = 4\]
and with $\\frac{dxᵢ(0)}{dt}=\\frac{dyᵢ(0)}{dt}=0$ except for
\[\\frac{dx₆(0)}{dt} = 1.75\]
\[\\frac{dx₇(0)}{dt} = -1.5\]
\[\\frac{dy₄(0)}{dt} = -1.25\]
\[\\frac{dy₅(0)}{dt} = 1\]
From Hairer Norsett Wanner Solving Ordinary Differential Equations I - Nonstiff Problems Page 244
Usually solved from 0 to 3.
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_vanderpol
— ConstantVan der Pol Equations
\[\frac{dx}{dt} = y\]
\[\frac{dy}{dt} = μ((1-x^2)y -x)\]
with $μ=1.0$ and $u_0=[0,\sqrt{3}]$
Non-stiff parameters.
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_vanstiff
— ConstantVan der Pol Equations
\[\frac{dx}{dt} = y\]
\[\frac{dy}{dt} = μ(1-x^2)y -x\]
with $μ=10^6$ and $u_0=[0,\sqrt{3}]$
Stiff parameters.
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_rober
— ConstantThe Robertson biochemical reactions: (Stiff)
\[\frac{dy₁}{dt} = -k₁y₁+k₃y₂y₃\]
\[\frac{dy₂}{dt} = k₁y₁-k₂y₂^2-k₃y₂y₃\]
\[\frac{dy₃}{dt} = k₂y₂^2\]
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 $[0,1e11]$
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_rigidbody
— ConstantRigid Body Equations (Non-stiff)
\[\frac{dy₁}{dt} = I₁y₂y₃\]
\[\frac{dy₂}{dt} = I₂y₁y₃\]
\[\frac{dy₃}{dt} = I₃y₁y₂\]
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.
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_hires
— ConstantHires 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
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_orego
— ConstantOrego Problem (Stiff)
It is in the form of $\frac{dy}{dt}=f(y), \quad y(0)=y0,$ with
\[y \in ℝ^3, \quad 0 ≤ t ≤ 360\]
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 Notebook: Orego.ipynb
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_pollution
— ConstantPollution 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
\[y(0)=y0, \quad y \in ℝ^20, \quad 0 ≤ t ≤ 60\]
where $f$ is defined by
$f(y) = \begin{pmatrix} -\sum_{j∈{1,10,14,23,24}} r_j + \sum_{j∈{2,3,9,11,12,22,25}} r_j \\ -r_2 - r_3 - r_9 - r_12 + r_1 + r_{21} \\ -r_{15} + r_1 + r_{17} + r_{19} + r_{22} \\ -r_2 - r_{16} - r_{17} - r_{23} + r_{15} \\ -r_3 + 2r_4 + r_6 + r_7 + r_{13} + r_{20} \\ -r_6 - r_8 - r_{14} - r_{20} + r_3 + 2r_{18} \\ -r_4 - r_5 - r_6 + r_{13} \\ r_4 + r_5 + r_6 + r_7 \\ -r_7 - r_8 \\ -r_{12} + r_7 + r_9 \\ -r_9 - r_{10} + r_8 + r_{11} \\ r_9 \\ -r_{11} + r_{10} \\ -r_{13} + r_{12} \\ r_{14} \\ -r_{18} - r_{19} + r_{16} \\ -r_{20} \\ r_{20} \\ -r{21} - r_{22} - r_{24} + r_{23} + r_{25} \\ -r_{25} + r_{24} \end{pmatrix}$
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
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_nonlinchem
— ConstantNonlinear system of reactions with an analytical solution
\[\frac{dy_1}{dt} = -y_1\]
\[\frac{dy_2}{dt} = y_1 - y_2^2\]
\[\frac{dy_3}{dt} = y_2^2\]
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.
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_brusselator_1d
— Constant1D Brusselator
\[\frac{\partial u}{\partial t} = A + u^2v - (B+1)u + \alpha\frac{\partial^2 u}{\partial x^2}\]
\[\frac{\partial v}{\partial t} = Bu - u^2v + \alpha\frac{\partial^2 u}{\partial x^2}\]
and the initial conditions are
\[u(x,0) = 1+\sin(2π x)\]
\[v(x,0) = 3\]
with the boundary condition
\[u(0,t) = u(1,t) = 1\]
\[v(0,t) = v(1,t) = 3\]
From Hairer Norsett Wanner Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems Page 6
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_brusselator_2d
— Constant2D Brusselator
\[\frac{\partial u}{\partial t} = 1 + u^2v - 4.4u + \alpha(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}) + f(x, y, t)\]
\[\frac{\partial v}{\partial t} = 3.4u - u^2v + \alpha(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2})\]
where
$f(x, y, t) = \begin{cases} 5 & \quad \text{if } (x-0.3)^2+(y-0.6)^2 ≤ 0.1^2 \text{ and } t ≥ 1.1 \\ 0 & \quad \text{else} \end{cases}$
and the initial conditions are
\[u(x, y, 0) = 22\cdot y(1-y)^{3/2}\]
\[v(x, y, 0) = 27\cdot x(1-x)^{3/2}\]
with the periodic boundary condition
\[u(x+1,y,t) = u(x,y,t)\]
\[u(x,y+1,t) = u(x,y,t)\]
From Hairer Norsett Wanner Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems Page 152
DiffEqProblemLibrary.ODEProblemLibrary.prob_ode_filament
— ConstantFilament 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.