pluto_ode.jl

### A Pluto.jl notebook ###
# v0.19.36

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

# ╔═╡ 9c4b446a-b42f-4292-91b6-b60b23b464da
begin
	using PlutoUI
	using Plots
	using Symbolics
    using ForwardDiff
	using LaTeXStrings
	using Images
	using TestImages
	using ImageView
	using LinearAlgebra
#	using FFTW
#	using DSP
	using DifferentialEquations

	nbsp = html"&nbsp"
end

# ╔═╡ a1dc1236-a3d4-456a-ab2b-5a3f2bbac38a
md"""
## Numerical Solutions to ODEs 
### 1. Basic Idea
Consider the initial value problem for first order ordinary differential equations of the form

$\frac{du}{dt}=f(t,u), \quad u(0)=u_0,$

where $u(t)$ is an unknown function of $t$, $f(t,u)$ is a given smooth function, and $u_0$ is a given initial data. This initial value problem is equivalent to the integral equation of the form 

$u(t)
=
u_0
+
\int_0^t
f\bigl(s,u(s)\bigr)
ds.$

Let $u(t)$ be a solution to the equation $u^\prime=f(t,u)$. Fix arbitrary $t\in\mathbb{R}$. We denote by $h>0$ the small step size of the approximation methed in $t$. Then we have 

$\begin{aligned}
  u(t+h)
& =
  u(t)
  +
  \int_0^h
  f\bigl(t+s,u(t+s)\bigr)
  ds
\\
& =
  u(t)
  +
  hf\bigl(t,u(t)\bigr)
  +
  \int_0^h
  \Bigl\{
  f\bigl(t+s,u(t+s)\bigr)
  -
  f\bigl(t,u(t)\bigr)
  \Bigr\}
  ds
\\
& =
  u(t)
  +
  hf\bigl(t,u(t)\bigr)
  +
  R
  +
  Q,
\\
  R
& =
  \int_0^h
  \Bigl\{
  f\Bigl(t+s,u(t)+sf\bigl(t,u(t)\bigr)\Bigr)
  -
  f\bigl(t,u(t)\bigr)
  \Bigr\}
  ds,
\\
  Q
& =
  \int_0^h
  \Bigl\{
  f\bigl(t+s,u(t+s)\bigr)
  -
  f\Bigl(t+s,u(t)+sf\bigl(t,u(t)\bigr)\Bigr)
  \Bigr\}
  ds.
\end{aligned}$

Taylor's theorem implies that 

$\begin{aligned}
  R
& =
  \int_0^h
  \Bigl\{
  sf_t\bigl(t,u(t)\bigr)
  +
  sf\bigl(t,u(t)\bigr)
  f_u\bigl(t,u(t)\bigr)
  +
  \mathcal{O}(s^2)
  \Bigr\}
  ds
 \\
& =
  \frac{h^2}{2}
  \Bigl\{
  f_t\bigl(t,u(t)\bigr)
  +
  f\bigl(t,u(t)\bigr)
  f_u\bigl(t,u(t)\bigr)
  \Bigr\}
  + 
  \mathcal{O}(h^3).
\end{aligned}$

Since $u(t)$ is a solution to $u^\prime=f(t,u)$, we deduce by Taylor's formula that 

$u(t+s)-u(t)-sf\bigl(t,u(t)\bigr)
=
u(t+s)-u(t)-su^\prime(t)
=
\mathcal{O}(s^2).$

Hence, by using the mean value theorem, we deduce that 

$\begin{aligned}
  Q
& =
  \int_0^h
  f_u(t+s,\dotsb)
  \Bigl\{
  u(t+s)-u(t)-sf\bigl(t,u(t)\bigr)
  \Bigr\}
  ds
\\
& =
  \int_0^s
  \mathcal{O}(s^2)
  ds
  =
  \mathcal{O}(h^3).
\end{aligned}$

Combining the above results, we obtain 

$\begin{aligned}
  u(t+h)
& =
  u(t)
  +
  hf\bigl(t,u(t)\bigr)
\\
& +
  \frac{h^2}{2}
  \Bigl\{
  f_t\bigl(t,u(t)\bigr)
  +
  f\bigl(t,u(t)\bigr)
  f_u\bigl(t,u(t)\bigr)
  \Bigr\}
  + 
  \mathcal{O}(h^3)
\\
& =
  u(t)
  +
  hf\bigl(t,u(t)\bigr)
  +
  \mathcal{O}(h^2).
\end{aligned}$

"""

# ╔═╡ 1e72b8d9-b161-4f31-92e1-b90a25a02a35
md""" 
### 2. The Euler Method
Let $h>0$ be a step size in $t$. The approximate solution is denoted by 

$U_0:=u_0. 
\quad
U_j \simeq u(jh),
\quad
j=1,2,3,\dotsc.$

We set 

$F_j:=f(jh,U_j),
\quad
F_{t,j}:=f_t(jh,U_j),
\quad
F_{u,j}:=f_u(jh,U_j),
\quad
j=0,1,2,\dotsc.$


#### The Euler Method
If we neglect $\mathcal{O}(h^2)$ and employ

$u(t+h)
\simeq
u(t)
+
hf\bigl(t,u(t)\bigr),$

then the approximation scheme is 

$U_{j+1}:=U_j+hF_j, \quad j=0,1,2,\dotsc.$

#### The Second Order Version
If we neglect $\mathcal{O}(h^3)$ and employ 

$u(t+h)
\simeq
u(t)
+
hf\bigl(t,u(t)\bigr)
+
\frac{h^2}{2}
\Bigl\{
f_t\bigl(t,u(t)\bigr)
+
f\bigl(t,u(t)\bigr)
f_u\bigl(t,u(t)\bigr)
\Bigr\},$

then the approximation scheme is 

$U_{j+1}:=U_j+hF_j+\frac{h^2}{2}(F_{t,j}+F_jF_{u,j}), \quad j=0,1,2,\dotsc.$

"""

# ╔═╡ 8f3a2089-c44c-450d-8d10-d355c9503ed7
md"""
### 3. $u^\prime=u$, $u(0)=1$, $t\in[0,2]$
The solution and approximation schemes are the following:
- The solution: $u(t)=e^t$.
- The Euler Method: $U_{j+1}=U_j+hU_j$.
- The second Order Version: $U_{j+1}=U_j+hU_j+\dfrac{h^2}{2}U_j$.
"""

# ╔═╡ 8b2273c4-8aa9-401e-a2ad-c3aefedc2ed9
md"""
 $N$ = $(@bind N1 Slider(8:1:100, show_value=true, default=10)) 
"""

# ╔═╡ 7f94440a-80e4-4b8e-b5b5-2efc47b5c152
begin
	f1(U0,p,t) = U0;
	tspan1=(0,2);
	U00=1;
    prob1 = ODEProblem(f1,U00,tspan1);
    sol1 = solve(prob1, maxiters=Int(1e6));
	
	h1=2/N1;
	U1=zeros(N1+1);
	U2=zeros(N1+1);
	U3=zeros(N1+1);
	U1[1]=1;
    U2[1]=1;
	U3[1]=1;
	for j=1:N1
		U1[j+1]=sol1(j*h1);
		U2[j+1]=(1+h1)*U2[j];
		U3[j+1]=(1+h1+h1^2/2)*U3[j];
	end
	
	plot([U1,U2,U3],
		 grid=false,
		 linewidth=2,
		 title="Solution to \$u'=u\$ and approximations with h=2/N",
		 ylim=(-0.1,9),
         xticks = ([1 1+N1/2 1+N1], ["0","1","2"]),
		 xlabel="\$t\$",
         yticks = ([0 1 exp(1) exp(2);], ["0","1","\$e\$","\$e^2\$"]),
		 ylabel="\$u(t)\$",
         label=["Exact Solution" "Euler Method" "Second Order Approx"],
	     legend=:topleft,
		 legendfont=font(12))
end

# ╔═╡ 33e36dbc-6682-4de6-a911-7eb2a0ec4f05
md"""
### 4. $u^\prime=u^2$, $u(0)=1$, $t\in[0,1)$
The approximation schemes are the following:
- The solution: $u(t)=\dfrac{1}{1-t}$
- The Euler Method: $U_{j+1}=U_j+hU_j^2$.
- Second Order Version: $U_{j+1}=U_j+hU_j^2+h^2U_j^3$.
"""

# ╔═╡ 5102229f-eb6e-4f07-80dd-cf2a7e151431
md"""
N = $(@bind N2 Slider(20:1:300, show_value=true, default=20)) 
"""

# ╔═╡ 95b1155d-442f-4c7a-a2a4-8fd34e854c62
begin
	f2(V0,p,t) = V0^2;
	tspan2=(0,95);
	V00=1;
    prob2 = ODEProblem(f2,V00,tspan2);
    sol2 = solve(prob2, maxiters=Int(1e6));
	
	h2=0.95/N2;
	V1=zeros(N2+1);
	V2=zeros(N2+1);
	V3=zeros(N2+1);
	V1[1]=1;
    V2[1]=1;
	V3[1]=1;
	for j=1:N2
		V1[j+1]=sol2(j*h2);
		V2[j+1]=V2[j]+h2*V2[j]*V2[j];
		V3[j+1]=V3[j]+h2*V3[j]^2+h2^2*V3[j]^3;
	end
	
	plot([V1,V2,V3],
		 grid=false,
		 linewidth=2,
		 title="Solution to \$u'=u^2\$ and approximations with h=0.95/N",
		 xlim=(1,N2/0.95),
		 ylim=(0,10),
         xticks = ([1 N2/0.95], ["0","1"]),
		 xlabel="\$t\$",
         yticks = ([0 1 5 10], ["0","1","5","10"]),
		 ylabel="\$u(t)\$",
         label=["Exact Solution" "Euler Method" "Second Order Approx"],
	     legend=:topleft,
		 legendfont=font(12))
end

# ╔═╡ a9f38308-8483-4b3a-9160-f2622027c7e2
md"""
### 5. Julia ODE Solvers
The Julia Programming Language has an ODE solver package: DifferentialEquations.jl. It solve the initial value problem, and integrate the equation or the system step by step.

We solve the initial value problem for the SIR model for infection disease of the form 

$\frac{dS}{dt}=-\beta IS,
\quad
\frac{dI}{dt}=\beta IS - \gamma I,
\quad
\frac{dR}{dt}=\gamma I,$

where $S(t)$, $I(t)$ and $R(t)$ are real-value unknown functions of $t$, and $\beta$ and $\gamma$ are positive constants. In mathematical epidemiology, $t$ is time, $S(t)$ is the number of susceptible people, $I(t)$ is the number of people infected,  $R(t)$ is the number of people who have recovered and developed immunity to the infection, $\beta$ is the infection rate, and $\gamma$ is the recovery rate. 

Since $\dfrac{d}{dt}(S+I+R)=0$, the total poputation $N:=S+I+R$ is preserved in this model. So we rewrite as 

$S:=S/N, \quad I:=I/N, \quad R:=R/N, \quad \beta:=\beta{N}, \quad \gamma:=\gamma$

Then we have apparently the same system 

$\frac{dS}{dt}=-\beta IS,
\quad
\frac{dI}{dt}=\beta IS - \gamma I,
\quad
\frac{dR}{dt}=\gamma I.$

In this setting 

$\rho_e(t):=\frac{\beta S(t)}{\gamma},
\quad
\rho_0:=\frac{\beta}{\gamma}$

are said to be the effective reproduction number and the basic reproduction number respectively. Note that the equation of $I(t)$ is

$\frac{dI}{dt}=\gamma\bigl(\rho_e(t)-1\bigr)I,$

and that the signature of $\rho_e(t)-1$ shows the increase/decrease of the number of active cases $I(t)$. For this reason $\rho_e(t)$ is the most important function in mathematical epidemiology.  


"""

# ╔═╡ b1e6cc97-a684-4cc2-8a53-5514544b453b
md"""
 $β$ = $(@bind b Slider(0:0.1:3, show_value=true, default=0.6)) 

 $γ$ = $(@bind g Slider(0:0.1:3, show_value=true, default=0.2))
"""

# ╔═╡ 09e2ec71-24e9-4c42-b073-bff7dac17697
begin
	f3(u,p,t) = [-b*u[1]*u[2];b*u[1]*u[2]-g*u[2]; g*u[2]];
	tspan3=(0,30);
	u3=[0.99;0.01;0];
    prob3 = ODEProblem(f3,u3,tspan3);
    sol3 = solve(prob3, maxiters=Int(1e6));

	plot(sol3, 
         grid=false,
         linewidth=2, 
		 xlim=(0,30),
         right_margin=Plots.Measures.Length(:mm, 10.0),
         left_margin=Plots.Measures.Length(:mm, 5.0),
         title="The SIR Model: \$S'=-βIS\$, \$I'=βIS-γI\$, \$R'=γI\$", 
         xlabel="time variable \$t\$",
         xticks = ([0 10 20 30], ["0","10","20","30"]), 
         yaxis="population rate",
         legendfont=font(10), 
         label=["\$S\$" "\$I\$" "\$R\$"], 
         palette = :seaborn_bright, 
         legend = :right)
end

# ╔═╡ 50acd500-9650-4b2d-bea8-1a60389a66b9
md"""
### 6. $u^{\prime\prime}+2\gamma{u^\prime}+k^2u=\sin(\omega{t})$, $\gamma,k,\omega\geqq0$

Note that the roots of the characteristic polynomial $z^2+2\gamma{z}+k^2z$ is 

$-\gamma \pm \sqrt{\gamma^2-k^2}.$


"""

# ╔═╡ 804ef4cd-4665-4f53-9034-64b96f0449b4
md"""
 $u(0)$ = $(@bind u40 Slider(0:0.1:0.5, show_value=true, default=0))

 $u'(0)$ = $(@bind u41 Slider(0:0.1:0.5, show_value=true, default=0))

 $γ$ = $(@bind γ Slider(0:0.1:5, show_value=true, default=0)) 

 $k$ = $(@bind k Slider(0:0.1:5, show_value=true, default=1))

 $ω$ = $(@bind ω Slider(0:0.1:5, show_value=true, default=0.1)) 
"""

# ╔═╡ 1169ae8c-6db8-430a-a11c-3252060ff4b6
begin
	f4(W,p,t) = [W[2];-2*γ*W[2]-k^2*W[1]+sin(ω*t)];
	tspan4=(0,10);
	u4=[u40;u41];
    prob4 = ODEProblem(f4,u4,tspan4);
    sol4 = solve(prob4, Tsit5(), reltol=1e-6, abstol=1e-6);
	
	m4,l4=size(sol4);
	for i=1:1
	end
end

# ╔═╡ 59bc7c84-4d39-48cb-8be8-852b66f86524
begin
	plot(sol4[1,:],
		 linewidth=2,
	     grid=false,
	     title="\$u''+2γu'+k^2u=\$sin \$(ωt)\$",
		ylim=(-2,2),
	     xlabel="\$t\$",
	     ylabel="\$u(t)\$",
	     xticks = ([1 l4/2 l4], ["0","5","10"]),
	     legend=false)

end

# ╔═╡ 42c00e90-77e9-4a23-9058-65a4b3fc4d8c


# ╔═╡ fb8ce483-506a-48ff-acb6-9bf100f2c7ff
size(sol4)

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[[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 = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[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 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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