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ODEs and parameterized types.jl
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### A Pluto.jl notebook ### | |
# v0.19.20 | |
#> [frontmatter] | |
#> chapter = 3 | |
#> video = "https://www.youtube.com/watch?v=S71YIZ8e7MQ" | |
#> image = "https://user-images.githubusercontent.com/6933510/136199713-25eb2e90-c5cd-4e11-8463-6f6069a81a51.png" | |
#> section = 2 | |
#> order = 2 | |
#> title = "ODEs and parameterized types" | |
#> layout = "layout.jlhtml" | |
#> youtube_id = "S71YIZ8e7MQ" | |
#> description = "" | |
#> tags = ["lecture", "module3"] | |
using Markdown | |
using InteractiveUtils | |
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). | |
macro bind(def, element) | |
quote | |
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end | |
local el = $(esc(element)) | |
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) | |
el | |
end | |
end | |
# ╔═╡ 5997bb40-b508-4647-8b12-a7cc5152e550 | |
using DifferentialEquations, Plots, PlutoUI, MappedArrays | |
# ╔═╡ ad58bd8c-be22-4203-96b9-04b0407c32c4 | |
md""" | |
!!! info | |
This notebook is from [`https://computationalthinking.mit.edu/Fall22/climate_science/odes_and_parameterized_types/`](https://computationalthinking.mit.edu/Fall22/climate_science/odes_and_parameterized_types/). | |
""" | |
# ╔═╡ 38c74e63-13e9-49a9-8eae-4a898197647a | |
TableOfContents() | |
# ╔═╡ 7a1d476b-1302-4b98-b9fa-01b990979985 | |
md""" | |
# Julia concepts | |
- Using libraries | |
- Libraries are built around types and objects | |
- Plot recipes: Defining how to plot our objects | |
- Making objects into callable functions | |
- Parameterized types | |
""" | |
# ╔═╡ 3004336a-0250-483f-be00-60db592c9fc8 | |
md""" | |
# ODEs and parameterized types | |
""" | |
# ╔═╡ a1cfd00b-825f-494e-91a9-192815f06ad6 | |
md""" | |
In the previous lecture we started to look at Ordinary Differential Equations (ODEs). These are equations relating a function to its derivatives, telling us how to move at each moment in time. | |
For example, exponential decay is modelled by | |
$$\dot{u} = -p \, u,$$ | |
in other words | |
$$\dot{u}(t) = - p \, u(t)$$ | |
for all real $t \ge 0$, | |
with initial condition | |
$$u(0) = u_0$$ | |
where $u(t)$ is the number of light bulbs still functioning at time $t$, the number of radioactive nuclei that have not yet decayed, or the number of people who are still infectious and have not yet recovered. The parameter $p$ is the decay rate. | |
We saw that **time stepping** methods provide a class of numerical methods for solving such initial-value problems, of which the Euler method is the simplest example. | |
Unfortunately the Euler method is *not* a good choice, since it is not accurate enough. In numerical analysis classes (like [18.330](https://github.com/mitmath/18330)) various better methods are derived. | |
Fortunately those methods are available in a Julia **library** or package, `DifferentialEquations.jl`. | |
""" | |
# ╔═╡ b2d2b8f0-c81d-437a-83cf-726b5caa77ab | |
md""" | |
# The `DifferentialEquations.jl` library | |
""" | |
# ╔═╡ d27ec502-ea0c-4d15-a223-7cbf6b784d1c | |
md""" | |
[DifferentialEquations.jl](https://diffeq.sciml.ai/v2.0/) (often abbreviated as DiffEq) is a Julia library for solving many types of differential equations. It has become the best (most complete and fastest) general library for **solving** ordinary differential equations (and many other types) in any programming environment. It is also a very good example of the way that Julia libraries are often structured. | |
We will see some basic usage examples and how Julia **types** are used both to control the library and to store the input and output to the **solvers**. | |
""" | |
# ╔═╡ 6f87059a-dbdc-46dd-933b-7bb36980868e | |
md""" | |
## Basic usage of DifferentialEquations.jl | |
""" | |
# ╔═╡ 557cedbc-6e44-40a8-b0ab-37ae11b63f55 | |
md""" | |
Let's try to solve the above equation that we derived in the previous lecture, $\dot{u} = -p \, u$. (We are using $u$ as the name of the dependent variable for consistency with the `DifferentialEquations.jl` documentation and internal structure.) | |
We write a general ODE in 1 variable as | |
$\dot{u} = f_p(t, u).$ | |
The right-hand side of the ODE is given by a function $f$, which may depend on a **parameter** $p$, i.e. a constant which does not change for any given run of the ODE. | |
$f$ may also depend explicitly on time. | |
In our example we do indeed have a parameter $p$ that we want to change, but $f$ does not have an explicit time dependence, so we write | |
""" | |
# ╔═╡ a6ae905b-db28-4e75-b567-54840dcda9a5 | |
f(u, p, t) = -p * u | |
# ╔═╡ f16644d0-e2d2-44ac-988c-aa5b178844f5 | |
md""" | |
Note that you always need the three input arguments `u`, `p` and `t`, in that order, even if you don't use them. | |
""" | |
# ╔═╡ 1ea2acfe-6ba1-44d3-9a24-a904a5fac0c4 | |
md""" | |
Now we define our initial condition $u_0$ and the time span to integrate over, given as a tuple (initial\_time, final\_time): | |
""" | |
# ╔═╡ 264a27cd-9ace-48cc-9556-7eb798298f17 | |
begin | |
u0 = 100.0 | |
time_span = (0.0, 10.0) | |
end | |
# ╔═╡ 70ebe76d-6107-492d-a3d2-84c1dd7f59fb | |
md""" | |
To set up the **problem** instance we use a type `ODEProblem` that is defined in the `DifferentialEquations.jl` package, into which we pass all the information necessary to define the problem. The parameters must go in the following order: | |
""" | |
# ╔═╡ b5379dc2-d97f-47ed-8737-35e3fe59285c | |
md""" | |
(For more advanced use there are also some additional, optional, keyword arguments.) | |
Note that the displayed output does not currently include information about the function, nor the parameters. | |
""" | |
# ╔═╡ 0ed8e007-2137-43e0-93b8-6f5dfb9496f5 | |
md""" | |
To solve the ODE we call the `solve` function: | |
""" | |
# ╔═╡ 64b26404-1ca0-4aa4-bea0-3c9075b08298 | |
md""" | |
What happened here? A suitable solver (i.e. an algorithm to calculate the solution) was chosen *automatically*, and it chose certain moments in time at which to output information about the (approximate, but very accurate) solution. | |
In this particular case it chose to output data at only eight points in time between $t=0$ and $t=10$. | |
Let's try to plot the `solution` object: | |
""" | |
# ╔═╡ 0b1855de-a8c9-492e-a249-3238e41fe84c | |
md""" | |
### Plot recipes | |
""" | |
# ╔═╡ 0190bdb0-97bf-4c35-b3c8-e97c1940b702 | |
md""" | |
Two surprising things happen. Firstly, there is no reason to expect this to have worked: `solution` is some kind of Julia object, but somehow `Plots.jl` knows how to plot it. This is because `DifferentialEquations.jl` defines a **plot recipe**. This specifies a way to turn solution objects into plots! Any package can do this relatively easily. | |
""" | |
# ╔═╡ 555a6328-8aff-4f9f-87f5-51a67450002d | |
md""" | |
The second surprise is that the output looks like a smooth curve, rather than just 8 points. Let's see those points on top of the curve. We can extract the relevant data from the `solution` object: | |
""" | |
# ╔═╡ b274c61b-4fab-4f4a-99df-5d39e0f56aa1 | |
md""" | |
We see that the package in fact gives not only the value at those points, but it is in fact also capable of calculating an (approximate) solution at *any* intermediate point, using **interpolation**. In fact, we can access this by treating `solution` as if it were a function: | |
""" | |
# ╔═╡ 13b38f88-2ead-46b3-bc96-eae2ea10204d | |
md""" | |
For this particular ODE we know the analytical solution. Let's compare them as we vary the parameter $p$: | |
""" | |
# ╔═╡ 475ef0e5-7f3a-4bbb-b513-8be29abfd3f9 | |
md""" | |
p = $(@bind p Slider(0.0:0.1:2.0, show_value=true)) | |
""" | |
# ╔═╡ 6e7c8e9e-bef0-4bf8-b1b1-d50c82aa203e | |
problem = ODEProblem(f, u0, time_span, p) | |
# ╔═╡ ff91515a-89a4-4423-a186-5572c712493d | |
solution = solve(problem) | |
# ╔═╡ b07de2ce-640b-42d6-8b60-39fcbcd116e7 | |
plot(solution, size=(500, 300), label="solution") | |
# ╔═╡ d65372df-e6ca-4165-8d9b-b21cc5c9f796 | |
scatter!(solution.t, solution.u, label="discrete output") | |
# ╔═╡ e3d9e0be-6cf4-4ae1-8c7f-b444a153a9f5 | |
begin | |
tt = 3.5 | |
solution(tt) | |
end | |
# ╔═╡ 180bcdc1-5f51-4b32-8b9c-5000605cdf32 | |
scatter!([tt], [solution(tt)], label="t = $(tt)", ms=5, m=:square) | |
# ╔═╡ 0ed6d65c-707d-4666-b203-2ad0ea822687 | |
let | |
problem = ODEProblem(f, u0, time_span, p) | |
solution = solve(problem) | |
plot(solution, | |
linewidth=3, xlabel="t", yaxis="x(t)", label="numerical", size=(500, 300)) | |
plot!(t -> u0 * exp(-p*t), lw=3, ls=:dash, label="exact") | |
ylims!(0, 100) | |
title!("p = $p") | |
end | |
# ╔═╡ cf9a293c-aa0f-40ce-960b-d4c3b82b8346 | |
md""" | |
We see that the numerical and exact solutions are (to the eye) indistinguishable, and that the package is fast enough to calculate the solution basically in real time. | |
""" | |
# ╔═╡ efcfdc1a-5a80-4780-ba56-43c5e2d6ea36 | |
md""" | |
## Systems of ODEs | |
""" | |
# ╔═╡ 5d7a81d7-8080-4390-924a-72c34c0a5e23 | |
md""" | |
Now let's try to solve the SIR equations: | |
$$\begin{aligned} | |
\dot{s} &= -\beta s i \\ | |
\dot{i} &= +\beta s i - \gamma i\\ | |
\dot{r} &= + \gamma i | |
\end{aligned}$$ | |
""" | |
# ╔═╡ 6c607126-dd0b-438d-8724-c1ef5e79c71d | |
md""" | |
We need to think of the system in a vector form, | |
$$\dot{\mathbf{x}} = \mathbf{f}_\mathbf{p}(t, \mathbf{x}),$$ | |
where: | |
- the vector $\mathbf{x}$ is a vector of all the variables, $\mathbf{x} := (x_1, x_2, x_3) := (s, i, r)$ | |
- the function $\mathbf{f}$ is a vector-valued function $\mathbb{R}^3 \to \mathbb{R}^3$, where $f_k$ gives the right-hand side of the equation for $\dot{x_k}$; and | |
- the vector $\mathbf{p} = (\beta, \gamma)$ is a vector of the parameters. | |
""" | |
# ╔═╡ d16e06f4-55c4-4c0e-bde5-c0ce4b36b043 | |
x0 = [0.99, 0.01, 0.0] | |
# ╔═╡ aa0e4260-6268-4eff-a662-f3fd55e7d450 | |
function SIR(x, p, t) | |
s, i, r = x # unpack the vectors into scalar values | |
β, γ = p | |
# build a new vector to return: | |
return [-β * s * i, | |
+β * s * i - γ * i, | |
+ γ * i] | |
end | |
# ╔═╡ 7aa45efe-865d-4e0e-9c71-0c032c72c40d | |
md""" | |
Now we see that the solverr has recognised that everything is a vector, and it returns a vector at each time stamp. | |
Again we can plot: | |
""" | |
# ╔═╡ bca8112f-cb61-43dc-ae87-383915c8a89b | |
gr() | |
# ╔═╡ 76cbc37d-54c8-4626-8bfe-58b63a602c38 | |
md""" | |
β = $(@bind β Slider(-0.5:0.01:2.0, default=1.0, show_value=true)) | |
γ = $(@bind γ Slider(-0.5:0.01:2.0, default=0.1, show_value=true)) | |
""" | |
# ╔═╡ d0f40681-73df-4cd3-bbd5-3edb8193153e | |
params = [β, γ] | |
# ╔═╡ fd122a25-5655-410b-aa7f-34936fc97b53 | |
SIR_problem = ODEProblem(SIR, x0, (0.0, 50.0), params) | |
# ╔═╡ 8ac7da6b-46e0-470b-91b3-1a61c226fa4a | |
sol = solve(SIR_problem) | |
# ╔═╡ a766a141-5d7b-499f-9e18-48bf926ee7ea | |
plot(sol) | |
# ╔═╡ c2765282-bcc7-4110-9822-10557326461e | |
md""" | |
It knows to plot each variable separately. | |
We can instead plot combinations of variables in *phase space* or *state space*: | |
""" | |
# ╔═╡ 7671506d-d4b7-4792-9571-e003097235e1 | |
gr() | |
# ╔═╡ 203e90b0-4d0c-4999-8513-d4eb43f53aac | |
plot(sol, vars=(1, 2), xlabel="s", ylabel="i", arrow=true, xlims=(-0.1, 1.0), size=(500, 300)) | |
# ╔═╡ 326890ae-0675-4779-b5e8-9b3e0412a52b | |
md""" | |
And even in 3D: | |
""" | |
# ╔═╡ d7025d1f-c3b8-461a-94dc-2414fbdfd373 | |
plotly() | |
# ╔═╡ 7f5c41f9-96b9-40d6-87f8-3e81c372b48e | |
plot(sol, vars=(1, 2, 3), xlabel="s", ylabel="i", zlabel="r") | |
# ╔═╡ d8f15e9b-aef1-4795-8522-85ea3564e351 | |
md""" | |
The [ModelingToolkit.jl](https://mtk.sciml.ai/stable/tutorials/ode_modeling/) library provides ways to make creating ODE models more intuitive, using symbolic equation objects. | |
""" | |
# ╔═╡ e723400e-3f8f-47d6-8944-28355a54c698 | |
md""" | |
## Making objects into callable functions | |
""" | |
# ╔═╡ 15fdb430-4157-4401-85b9-665dde795a22 | |
md""" | |
One of the interesting things we have seen is the way that the `solution` object can be **called** using the normal syntax for calling functions: | |
""" | |
# ╔═╡ a410e98d-dd7f-4f55-8812-8b0339a56c86 | |
solution(3.5) | |
# ╔═╡ 239f39b2-53c6-4da9-ac25-b5e7d19deb5f | |
md""" | |
Not all types allow this. For example, if we define a matrix | |
""" | |
# ╔═╡ 6630cbc8-1b2f-4ea9-bcd6-e4f4fcb2e844 | |
A = rand(2, 2) | |
# ╔═╡ 5c1c7016-69aa-4009-8dc8-0c68c925ee66 | |
md""" | |
we cannot usually call it with a vector: | |
""" | |
# ╔═╡ 83719ef1-e8d6-44d0-98ab-a851b1082fa5 | |
A(rand(2)) | |
# ╔═╡ e4d36db5-352b-4fb2-9169-da5c6a3458b7 | |
md""" | |
If we wanted to do so, we could, however, using the following syntax: | |
""" | |
# ╔═╡ 167015ff-1caf-42a8-85f1-070da3e5d5c5 | |
# (A::Matrix)(x) = A * x | |
# ╔═╡ f8bcd82d-089c-49a2-99d0-b17a5ac5c509 | |
md""" | |
Note that this makes objects (instances) of the type callable, rather than the type itself, which we can already call as a **constructor** to create objects of that type: | |
""" | |
# ╔═╡ cd17e738-089b-4934-af63-d27706c5d1c7 | |
Matrix([1 2; 3 4]) | |
# ╔═╡ fc525f77-ad62-43cb-92e4-0fc1354be99c | |
md""" | |
For example, if we wanted to write our own type to store the output of the Euler method, we could do something like | |
""" | |
# ╔═╡ ec90842a-a20c-4847-a803-e070d343b98d | |
begin | |
struct SimpleEulerOutput | |
times::Vector{Float64} | |
values::Vector{Float64} | |
end | |
function (soln::SimpleEulerOutput)(t) | |
# use interpolation to calculate the value at time t | |
# using `soln.times` and `soln.values` | |
end | |
end | |
# ╔═╡ 9fad6f01-a3b7-4dbb-a8f5-29f2ea1f545a | |
md""" | |
The definition of the function makes it callable. | |
This says that "the way to call an object called `soln`, of type `EulerOutput`, is the following:". Note that inside the function you will need too access the `times` and `values` from *inside* the `soln` object, using `soln.times` and `soln.values`. | |
Note that at the time of writing, Pluto requires that you put this call definition in the same cell as the type definition. | |
""" | |
# ╔═╡ 658d8d15-5b3e-4d35-88b1-4e2fa0f3b17a | |
md""" | |
## Digging into objects | |
""" | |
# ╔═╡ aafeeb55-943f-4375-9cc0-e0dce659d161 | |
md""" | |
A key feature of the `DifferentialEquations.jl` library, and of most other libraries in Julia, is the way they define and use new **types**. | |
As we have seen previously, we can define new types using `struct` which store data, and then define functions which act on those types. | |
""" | |
# ╔═╡ 3c45ad21-d18b-41fa-a0fd-64d8988621f3 | |
md""" | |
Let's start by looking at the `ODEProblem` type: | |
""" | |
# ╔═╡ 96e871a8-48b9-459b-a1a0-94fc1deb941e | |
problem | |
# ╔═╡ 2622801c-4e37-42f1-ab77-939d461b8d20 | |
md""" | |
To see what fields, or attributes, the object contains we can use | |
""" | |
# ╔═╡ 43e7b270-6452-4a4c-ba6f-c286a337a423 | |
fieldnames(typeof(problem)) | |
# ╔═╡ dbdb9998-ca3b-46fb-87cf-5b7c6e387ab9 | |
md""" | |
In Pluto and other interactive environments we can get the same information using `problem.<TAB>`, i.e. by typing the TAB key after `problem.` | |
""" | |
# ╔═╡ b8fbff51-72fe-4432-9f6f-06f4628320d4 | |
md""" | |
As usual we can extract the data contained in the object as usual using `.`: | |
""" | |
# ╔═╡ ef677abe-c96d-41cc-9b4a-192ff29cdb86 | |
problem.u0 | |
# ╔═╡ 78933b33-8a3c-492e-8a58-19ad7187b45e | |
problem.tspan | |
# ╔═╡ 9ef4aa78-2ecd-4d53-87ae-f0909bb46915 | |
md""" | |
To see everything contained in the object, we can use `Dump` in Pluto, or `dump` if we are not using Pluto: | |
""" | |
# ╔═╡ 798bc482-7556-4904-8946-0d9898ce2d33 | |
Dump(problem) | |
# ╔═╡ 32e9f6e5-8df6-4d40-8092-efaf2bce28d3 | |
md""" | |
This seems (much) more complicated than you might expect. The library is clearly doing more than just storing the data that we provide in a type! The main thing that has happened is that the function we passed in has been processed by wrapping it into yet another type, `ODEFunction`. This contains information such as how to calculate the derivative of the function that can be provided for more advanced usage of the package. | |
""" | |
# ╔═╡ 6dfc2699-fe5e-4b55-8437-0866f715170a | |
md""" | |
Similarly we can look inside the solution object: | |
""" | |
# ╔═╡ 17d6021b-0d21-4d04-a8ae-64743cc01404 | |
fieldnames(typeof(solution)) | |
# ╔═╡ f4c3b174-57ac-461b-a816-869460d8896d | |
Dump(solution) | |
# ╔═╡ 500973f5-faa7-4a31-b812-d0b20dbb9d82 | |
md""" | |
The solution is even more complicated, containing not only the data that was calculated, but also all information about which algorithms were used to solve the problem and tables of coefficients for interpolation. | |
""" | |
# ╔═╡ 4a575627-32c2-4842-a540-52253f79ff89 | |
md""" | |
## Parameterized types | |
""" | |
# ╔═╡ bea223ef-f961-457c-8962-d0ed396f4717 | |
md""" | |
Let's look now at the *type* of the DiffEq objects: | |
""" | |
# ╔═╡ 92bd82dc-eaab-4869-9d95-cb4647092352 | |
typeof(problem) | |
# ╔═╡ d518828a-1972-4bfa-b96f-0ebbf86b6193 | |
md""" | |
We see that indeed `problem` is an object of type `ODEProblem`, but that it also has several **type parameters**, which are listed inside the curly braces (`{` and `}`): | |
""" | |
# ╔═╡ c3570557-55bd-4189-a332-aa2e54693665 | |
[typeof(problem).parameters...] | |
# ╔═╡ 1f684b1b-8e43-4e0e-8a47-1f2474cc5e4f | |
md""" | |
For example, we see that the second type parameter is of type `Tuple{Float64, Float64}`, which is the type of the variable `time_span`: | |
""" | |
# ╔═╡ 1bf84cfe-9ddc-4644-b7ea-8ac43b01d477 | |
typeof(time_span) | |
# ╔═╡ d2a1f97e-2e3e-4ae9-8287-2de62065d708 | |
md""" | |
One of the reasons for using parameterized types like this is for efficiency: Julia is fastest when it knows the exact types of every variable, since then it can generate efficient machine code. | |
""" | |
# ╔═╡ a71aaf0b-8ce9-4242-b31f-8f05730ce2e7 | |
md""" | |
# Making parameterized types | |
""" | |
# ╔═╡ 09ddc00b-2361-44c5-b194-7b4b6d24bef4 | |
md""" | |
Let's see how and why we would make a parameterized type. | |
Let's think again about the `EulerOutput` type that we started to write above. | |
What happens if, as in the `DifferentialEquations.jl` library, we would like to handle both *scalar* ODEs (one single real dependent variable), and *vector* ODEs? | |
We can no longer specify that the output type is a `Vector{Float64}`, so we would need to leave it unspecified: | |
""" | |
# ╔═╡ c7c16e89-3ccf-4d70-9cdb-d5ae71a5ea49 | |
struct SimpleEulerOutput2 | |
times | |
values | |
end | |
# ╔═╡ 1872c30d-aece-44a5-b3b6-36e82c072609 | |
md""" | |
However, it turns out that we then lose efficiency (speed). | |
What we would need is two different types: one in which we specify the type of `values` as `Vector{Float64}`, and the other as `Vector{Vector{Float64}}`. | |
Julia provides a mechanism to define *both at once*: this is what parameterized types do! | |
""" | |
# ╔═╡ 5d06734e-ffbd-421e-820e-609e373ccc7b | |
md""" | |
Let's define our own `MyODEProblem` type as a simple example. We specify type parameters for each variable: | |
""" | |
# ╔═╡ 62d55814-c863-4d4d-8582-908721d37326 | |
struct MyODEProblem{T,U} | |
t0::T | |
tfinal::T | |
u0::U | |
end | |
# ╔═╡ 470b7808-5ff6-4cda-82cc-2ac250d51909 | |
md""" | |
Note that the starting and final times have been specified to require the *same* type `T`. | |
""" | |
# ╔═╡ d392de63-137e-43fb-86ca-4f344769bc71 | |
md""" | |
Now if we create objects of this type, we see that Julia matches the types we pass in to the correct type parameters: | |
""" | |
# ╔═╡ e07830be-3aab-4410-a830-786a30eb162a | |
newprob = MyODEProblem(3, 4, 100.0) | |
# ╔═╡ 6d809ff0-20c5-4910-aca0-612496955812 | |
newprob.t0 | |
# ╔═╡ 816a64ed-1f88-43a8-9df7-8070372e84ea | |
newprob.tfinal | |
# ╔═╡ cdd6a0dd-e188-4d37-ab6d-12e7fecd3e90 | |
newprob.u0 | |
# ╔═╡ c29d4796-9e52-4234-8325-5bf31099c79d | |
md""" | |
If we change the types of the inputs, the type parameters change: | |
""" | |
# ╔═╡ 7e5b0073-1bc7-41c7-ae74-c9c1bdf6c6ea | |
MyODEProblem(3.0, 4.0, 100.0) | |
# ╔═╡ 67e36f7c-0cb9-4ead-82ad-4c47639802a8 | |
md""" | |
If we mix types for the first two arguments then we get an error, since Julia can't match to a single type `T`: | |
""" | |
# ╔═╡ cdc25238-ed48-42e5-9db4-2367577b215d | |
MyODEProblem(3.0, 5, 100.0) | |
# ╔═╡ f66b6078-bab5-48ef-9241-8d5fb0dc78ea | |
md""" | |
We can use vectors: | |
""" | |
# ╔═╡ 73016af9-5efe-4af2-9e99-ce1a7de6fa96 | |
MyODEProblem(3, 4, [17.0, 18.0]) | |
# ╔═╡ 00000000-0000-0000-0000-000000000001 | |
PLUTO_PROJECT_TOML_CONTENTS = """ | |
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""" | |
# ╔═╡ 00000000-0000-0000-0000-000000000002 | |
PLUTO_MANIFEST_TOML_CONTENTS = """ | |
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version = "1.1.3+4" | |
[[deps.Xorg_libXext_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3" | |
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3" | |
version = "1.3.4+4" | |
[[deps.Xorg_libXfixes_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4" | |
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8" | |
version = "5.0.3+4" | |
[[deps.Xorg_libXi_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"] | |
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246" | |
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492" | |
version = "1.7.10+4" | |
[[deps.Xorg_libXinerama_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"] | |
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123" | |
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3" | |
version = "1.1.4+4" | |
[[deps.Xorg_libXrandr_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"] | |
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631" | |
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484" | |
version = "1.5.2+4" | |
[[deps.Xorg_libXrender_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96" | |
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" | |
version = "0.9.10+4" | |
[[deps.Xorg_libpthread_stubs_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb" | |
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" | |
version = "0.1.0+3" | |
[[deps.Xorg_libxcb_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] | |
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6" | |
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" | |
version = "1.13.0+3" | |
[[deps.Xorg_libxkbfile_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2" | |
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a" | |
version = "1.1.0+4" | |
[[deps.Xorg_xcb_util_image_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97" | |
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b" | |
version = "0.4.0+1" | |
[[deps.Xorg_xcb_util_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"] | |
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1" | |
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5" | |
version = "0.4.0+1" | |
[[deps.Xorg_xcb_util_keysyms_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00" | |
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7" | |
version = "0.4.0+1" | |
[[deps.Xorg_xcb_util_renderutil_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e" | |
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e" | |
version = "0.3.9+1" | |
[[deps.Xorg_xcb_util_wm_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67" | |
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361" | |
version = "0.4.1+1" | |
[[deps.Xorg_xkbcomp_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"] | |
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b" | |
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4" | |
version = "1.4.2+4" | |
[[deps.Xorg_xkeyboard_config_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"] | |
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d" | |
uuid = "33bec58e-1273-512f-9401-5d533626f822" | |
version = "2.27.0+4" | |
[[deps.Xorg_xtrans_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" | |
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" | |
version = "1.4.0+3" | |
[[deps.Zlib_jll]] | |
deps = ["Libdl"] | |
uuid = "83775a58-1f1d-513f-b197-d71354ab007a" | |
version = "1.2.13+0" | |
[[deps.Zstd_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927" | |
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" | |
version = "1.5.2+0" | |
[[deps.ZygoteRules]] | |
deps = ["MacroTools"] | |
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0" | |
uuid = "700de1a5-db45-46bc-99cf-38207098b444" | |
version = "0.2.2" | |
[[deps.fzf_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56" | |
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09" | |
version = "0.29.0+0" | |
[[deps.libaom_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" | |
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" | |
version = "3.4.0+0" | |
[[deps.libass_jll]] | |
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" | |
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" | |
version = "0.15.1+0" | |
[[deps.libblastrampoline_jll]] | |
deps = ["Artifacts", "Libdl"] | |
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" | |
version = "5.2.0+0" | |
[[deps.libfdk_aac_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" | |
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" | |
version = "2.0.2+0" | |
[[deps.libpng_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" | |
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" | |
version = "1.6.38+0" | |
[[deps.libvorbis_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] | |
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" | |
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" | |
version = "1.3.7+1" | |
[[deps.nghttp2_jll]] | |
deps = ["Artifacts", "Libdl"] | |
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" | |
version = "1.48.0+0" | |
[[deps.p7zip_jll]] | |
deps = ["Artifacts", "Libdl"] | |
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" | |
version = "17.4.0+0" | |
[[deps.x264_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" | |
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" | |
version = "2021.5.5+0" | |
[[deps.x265_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" | |
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" | |
version = "3.5.0+0" | |
[[deps.xkbcommon_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] | |
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" | |
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" | |
version = "1.4.1+0" | |
""" | |
# ╔═╡ Cell order: | |
# ╟─ad58bd8c-be22-4203-96b9-04b0407c32c4 | |
# ╠═5997bb40-b508-4647-8b12-a7cc5152e550 | |
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# ╟─3004336a-0250-483f-be00-60db592c9fc8 | |
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# ╟─6f87059a-dbdc-46dd-933b-7bb36980868e | |
# ╟─557cedbc-6e44-40a8-b0ab-37ae11b63f55 | |
# ╠═a6ae905b-db28-4e75-b567-54840dcda9a5 | |
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# ╟─70ebe76d-6107-492d-a3d2-84c1dd7f59fb | |
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# ╟─b5379dc2-d97f-47ed-8737-35e3fe59285c | |
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# ╠═0ed6d65c-707d-4666-b203-2ad0ea822687 | |
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# ╠═bca8112f-cb61-43dc-ae87-383915c8a89b | |
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# ╟─326890ae-0675-4779-b5e8-9b3e0412a52b | |
# ╠═d7025d1f-c3b8-461a-94dc-2414fbdfd373 | |
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# ╟─d8f15e9b-aef1-4795-8522-85ea3564e351 | |
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# ╟─15fdb430-4157-4401-85b9-665dde795a22 | |
# ╠═a410e98d-dd7f-4f55-8812-8b0339a56c86 | |
# ╟─239f39b2-53c6-4da9-ac25-b5e7d19deb5f | |
# ╠═6630cbc8-1b2f-4ea9-bcd6-e4f4fcb2e844 | |
# ╟─5c1c7016-69aa-4009-8dc8-0c68c925ee66 | |
# ╠═83719ef1-e8d6-44d0-98ab-a851b1082fa5 | |
# ╟─e4d36db5-352b-4fb2-9169-da5c6a3458b7 | |
# ╠═167015ff-1caf-42a8-85f1-070da3e5d5c5 | |
# ╟─f8bcd82d-089c-49a2-99d0-b17a5ac5c509 | |
# ╠═cd17e738-089b-4934-af63-d27706c5d1c7 | |
# ╟─fc525f77-ad62-43cb-92e4-0fc1354be99c | |
# ╠═ec90842a-a20c-4847-a803-e070d343b98d | |
# ╟─9fad6f01-a3b7-4dbb-a8f5-29f2ea1f545a | |
# ╟─658d8d15-5b3e-4d35-88b1-4e2fa0f3b17a | |
# ╟─aafeeb55-943f-4375-9cc0-e0dce659d161 | |
# ╟─3c45ad21-d18b-41fa-a0fd-64d8988621f3 | |
# ╠═96e871a8-48b9-459b-a1a0-94fc1deb941e | |
# ╟─2622801c-4e37-42f1-ab77-939d461b8d20 | |
# ╠═43e7b270-6452-4a4c-ba6f-c286a337a423 | |
# ╟─dbdb9998-ca3b-46fb-87cf-5b7c6e387ab9 | |
# ╟─b8fbff51-72fe-4432-9f6f-06f4628320d4 | |
# ╠═ef677abe-c96d-41cc-9b4a-192ff29cdb86 | |
# ╠═78933b33-8a3c-492e-8a58-19ad7187b45e | |
# ╟─9ef4aa78-2ecd-4d53-87ae-f0909bb46915 | |
# ╠═798bc482-7556-4904-8946-0d9898ce2d33 | |
# ╟─32e9f6e5-8df6-4d40-8092-efaf2bce28d3 | |
# ╟─6dfc2699-fe5e-4b55-8437-0866f715170a | |
# ╠═17d6021b-0d21-4d04-a8ae-64743cc01404 | |
# ╠═f4c3b174-57ac-461b-a816-869460d8896d | |
# ╟─500973f5-faa7-4a31-b812-d0b20dbb9d82 | |
# ╟─4a575627-32c2-4842-a540-52253f79ff89 | |
# ╟─bea223ef-f961-457c-8962-d0ed396f4717 | |
# ╠═92bd82dc-eaab-4869-9d95-cb4647092352 | |
# ╟─d518828a-1972-4bfa-b96f-0ebbf86b6193 | |
# ╠═c3570557-55bd-4189-a332-aa2e54693665 | |
# ╟─1f684b1b-8e43-4e0e-8a47-1f2474cc5e4f | |
# ╠═1bf84cfe-9ddc-4644-b7ea-8ac43b01d477 | |
# ╟─d2a1f97e-2e3e-4ae9-8287-2de62065d708 | |
# ╟─a71aaf0b-8ce9-4242-b31f-8f05730ce2e7 | |
# ╟─09ddc00b-2361-44c5-b194-7b4b6d24bef4 | |
# ╠═c7c16e89-3ccf-4d70-9cdb-d5ae71a5ea49 | |
# ╟─1872c30d-aece-44a5-b3b6-36e82c072609 | |
# ╟─5d06734e-ffbd-421e-820e-609e373ccc7b | |
# ╠═62d55814-c863-4d4d-8582-908721d37326 | |
# ╟─470b7808-5ff6-4cda-82cc-2ac250d51909 | |
# ╟─d392de63-137e-43fb-86ca-4f344769bc71 | |
# ╠═e07830be-3aab-4410-a830-786a30eb162a | |
# ╠═6d809ff0-20c5-4910-aca0-612496955812 | |
# ╠═816a64ed-1f88-43a8-9df7-8070372e84ea | |
# ╠═cdd6a0dd-e188-4d37-ab6d-12e7fecd3e90 | |
# ╟─c29d4796-9e52-4234-8325-5bf31099c79d | |
# ╠═7e5b0073-1bc7-41c7-ae74-c9c1bdf6c6ea | |
# ╟─67e36f7c-0cb9-4ead-82ad-4c47639802a8 | |
# ╠═cdc25238-ed48-42e5-9db4-2367577b215d | |
# ╟─f66b6078-bab5-48ef-9241-8d5fb0dc78ea | |
# ╠═73016af9-5efe-4af2-9e99-ce1a7de6fa96 | |
# ╟─00000000-0000-0000-0000-000000000001 | |
# ╟─00000000-0000-0000-0000-000000000002 |
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