{
"cells": [
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"---\n",
"title: \"Theory of IVPs\"\n",
"subtitle: \"Lecture 2: Theory of Initial Value Problems\"\n",
"date: 2026-01-26\n",
"abstract-title: Overview\n",
"abstract: | \n",
" (1) What are initial value problems?; \n",
" (2) Examples: what can happen? What can go wrong?; \n",
" (3) Existence and Uniqueness: when does it go right?; \n",
" (4) Well-posedness;\n",
"format:\n",
" html:\n",
" other-links:\n",
" - text: This notebook\n",
" href: L2.ipynb\n",
"---"
]
},
{
"cell_type": "markdown",
"id": "63cf0867",
"metadata": {},
"source": [
"::: {.callout-note}\n",
"\n",
"I encourage you to play around with the juptyer notebook for this lecture - you can copy the code with the ```this notebook``` button on the side of this page.\n",
"\n",
":::\n",
"\n",
"::: {.callout-warning}\n",
"\n",
"These notes are mainly a record of what we discussed and are not a substitute for attending the lectures and reading books! If anything is unclear/wrong, let me know and I will update the notes.\n",
"\n",
":::\n",
"\n",
"::: {.callout-tip}\n",
"\n",
"This lecture is mostly based on @Burden Section 5.1\n",
"\n",
":::"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "db6fe346",
"metadata": {},
"outputs": [],
"source": [
"#| echo: false\n",
"\n",
"using Plots, LaTeXStrings"
]
},
{
"cell_type": "markdown",
"id": "5ef0ea46",
"metadata": {},
"source": [
"Differential equations appear everywhere in science and engineering. Variables that change in time or space such as particles, planets, populations, .... can be modelled with differential equations. When there is one independent variable, these equations are known as *Ordinary Differential Equations (ODEs)*. In this chapter, we study scalar *initial value problems*: \n",
"\n",
"## Examples of Initial Value Problems\n",
"\n",
"Scalar initial value problems are of the form: Seek $u \\colon [0,T] \\to \\mathbb R$ such that \n",
"\n",
"\\begin{align}\n",
" u'(t) &= f\\big( t, u(t) \\big) \\nonumber\\\\\n",
" u(0) &= u_0 \\tag{IVP}\n",
"\\end{align}\n",
"\n",
"Remarks: \n",
"\n",
"* $t$ is the independent variable (often thought of as time), \n",
"* $u_0$ is the *initial condition*, \n",
"* $u(t)$ can be thought of as the solution at time $t$, \n",
"* If $f(t,u) = \\alpha(t) + u \\beta(t)$ is linear in $u$, we say the ODE is *linear*, otherwise it is *nonlinear*.\n",
"\n",
"::: {.callout-warning}\n",
"\n",
"There is lots of different notation for derivatives: \n",
"\n",
"\\begin{align}\n",
" u'(t) = \\frac{\\mathrm{d}u(t)}{\\mathrm{d}t} = \\frac{\\mathrm{d}}{\\mathrm{d}t}u(t) = \\dot{u}(t) = u_t = ...\n",
"\\end{align}\n",
"\n",
":::\n",
"\n",
"::: {#exm-exponential}\n",
"# Exponential decay/growth\n",
"\n",
"Consider the following linear IVP \n",
"\n",
"\\begin{align}\n",
" &u(0) = u_0, \\\\\n",
" &u'(t) = \\mu u(t) \\quad \\text{on } (0,T).\n",
"\\end{align}\n",
"\n",
"The rate of change of some quantity $u$ is a constant multiple of the quanity $u$ (e.g. growth of bacteria for small time scales, money in a fixed rate savings account, very simple model for the size of a population). One can see that there exists a unique solution given by $u(t) = u_0 e^{\\mu t}$. This equation therefore models expoential growth or decay.\n",
"\n",
":::\n",
"\n",
"::: {#exm-logistic}\n",
"# Logistic equation\n",
"\n",
"Population do not grow expentially. A more realistic model of population growth includes effects limiting the size of populations: for example\n",
"\n",
"\\begin{align}\n",
" &u(0) = u_0, \\\\\n",
" &u'(t) = \\mu u(t) - \\kappa u(t)^2 \\quad \\text{on } (0,T).\n",
"\\end{align}\n",
"\n",
"One can show that there exists a unique solution to this equation given by\n",
"\n",
"\\begin{align}\n",
" u(t) = \\frac{\\mu}{\\kappa u_0 (e^{\\mu t}-1) + \\mu } u_0 e^{\\mu t}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "15f52161",
"metadata": {},
"outputs": [
{
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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
""
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u0, μ, κ = 1, 1., .05\n",
"u(t) = ( μ/( κ*u0*(exp(μ*t) - 1) + μ) ) * u0 * exp( μ * t )\n",
"\n",
"plot(u, 0, 10, \n",
" title=L\"u'=\\mu u(t) - \\kappa u(t)^2\", legend=false,\n",
" xlabel=L\"t\", ylabel=L\"u(t)\")"
]
},
{
"cell_type": "markdown",
"id": "2c073b5b",
"metadata": {},
"source": [
":::\n",
"\n",
"::: {#exr-logistic}\n",
"# Logistic equation\n",
"\n",
"Let $u$ be the solution from @exm-logistic. Explain why $u(t) \\to \\frac{\\mu}{\\kappa}$ as $t\\to \\infty$.\n",
"\n",
":::\n",
"\n",
"In general, there is no analytical formula for solutions to $(\\text{IVP})$ and so, in practice, one needs to implement numerical methods to find solutions. For linear IVPs, one can find numerical solutions by approximating integrals:\n",
"\n",
"::: {#exr-IF}\n",
"# Integrating Factor\n",
"\n",
"For linear IVPs with $f(t,u) = \\alpha(t) + u \\beta(t)$, one can rewrite the solution as an integral involving the so-called *integrating factor* $I(t) := \\exp( - \\int_0^t \\beta(s) \\mathrm{d}s )$. Show that \n",
"\n",
"\\begin{align}\n",
" I(t) u(t) = u_0 + \\int_0^t I(s) \\alpha(s) \\mathrm{d}s.\n",
"\\end{align}\n",
"\n",
"In general, there is no analytical formula for this integral; one may approximate it using quadrature rules (see Math 5485).\n",
"\n",
":::\n",
"\n",
"In the following, we will use the package ```OrdinaryDiffEq``` to numerically solve some problems and see what solutions look like in practice:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "2166a563",
"metadata": {},
"outputs": [],
"source": [
"using OrdinaryDiffEq"
]
},
{
"cell_type": "markdown",
"id": "2aa31a2a",
"metadata": {},
"source": [
"::: {#exm-1}\n",
"\n",
"Consider the following linear IVP \n",
"\n",
"\\begin{align}\n",
" &u(0) = 0, \\\\\n",
" &u'(t) = t^2 - u \\sin(t) \\quad \\text{on } (0,10)\n",
"\\end{align}\n",
"\n",
"We numerically solve this equation:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "27937a4b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(u,p,t) = t^2 - u * sin(t)\n",
"# p is for a set of parameters that we don't need at the moment\n",
"u0, tspan = 0., (0., 10.)\n",
"ivp = ODEProblem(f, u0, tspan)\n",
"sol = solve(ivp, Tsit5()); #Tsit5() is a particular method\n",
"\n",
"plot(sol,\n",
" title=L\"u'=f(t,u)\", legend=false,\n",
" xlabel=L\"t\", ylabel=L\"u(t)\")\n",
"scatter!(sol.t, sol.u)"
]
},
{
"cell_type": "markdown",
"id": "fd9d8823",
"metadata": {},
"source": [
"\n",
"\n",
"The points $(t_j, u(t_j))$ in the plot are calculated using some approximate scheme and the graph above is an interpolation of these points.\n",
"\n",
"In this case, there exists a unique solution for all time.\n",
"\n",
"::: \n",
"\n",
"::: {#exr-1}\n",
"\n",
"Explain what happens to the solution to @exm-1 for large $t$. Verify your hypothesis numerically.\n",
"\n",
":::"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "f381e4d7",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"tspan = (0., 1000.)\n",
"ivp = ODEProblem(f, u0, tspan)\n",
"sol = solve(ivp, Tsit5()); #Tsit5() is a particular method\n",
"\n",
"plot(sol,\n",
" title=L\"u'=f(t,u)\", legend=false,\n",
" xlabel=L\"t\", ylabel=L\"u(t)\")"
]
},
{
"cell_type": "markdown",
"id": "d7fe66ec",
"metadata": {},
"source": [
"::: {#exm-2}\n",
"\n",
"Consider the following (nonlinear) IVP \n",
"\n",
"\\begin{align}\n",
" &u(0) = 1, \\\\\n",
" &u'(t) = u(t)^3 \\quad \\text{on } (0,1)\n",
"\\end{align}\n",
"\n",
"We try to solve this problem numerically:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "ddb37ef4",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"\u001b[33m\u001b[1m┌ \u001b[22m\u001b[39m\u001b[33m\u001b[1mWarning: \u001b[22m\u001b[39mAt t=0.4999988132731494, dt was forced below floating point epsilon 5.551115123125783e-17, and step error estimate = 6311.956553629581. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of Float64).\n",
"\u001b[33m\u001b[1m└ \u001b[22m\u001b[39m\u001b[90m@ SciMLBase C:\\Users\\math5\\.julia\\packages\\SciMLBase\\TZ9Rx\\src\\integrator_interface.jl:671\u001b[39m\n"
]
},
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(u,p,t) = u^3\n",
"u0, tspan = 1., (0., 1.)\n",
"ivp = ODEProblem(f, u0, tspan)\n",
"sol = solve(ivp, Tsit5())\n",
"\n",
"plot(sol, yaxis=:log,\n",
" title=L\"u'=u^3\", legend=false,\n",
" xlabel=L\"t\", ylabel=L\"u(t)\")"
]
},
{
"cell_type": "markdown",
"id": "66357572",
"metadata": {},
"source": [
"Notice that the solution appears to *blow-up in finite time*! This is **not** an artifact of the numerical scheme and, in fact, the solution that exists for small $t$ cannot be extended past $T = 0.5$.\n",
"\n",
":::\n",
"\n",
"::: {#exr-2}\n",
"\n",
"Show that the solution to @exm-2 is \n",
"\n",
"\\begin{align}\n",
" u(t) = \\frac{1}{\\sqrt{1-2t}}\n",
"\\end{align}\n",
"\n",
"which does indeed have a singularity at $T = \\frac12$.\n",
"\n",
":::\n",
"\n",
"::: {#exm-3}\n",
"\n",
"Consider the following IVP \n",
"\n",
"\\begin{align}\n",
" &u(0) = 0, \\\\\n",
" &u'(t) = \\sqrt{u(t)} \\quad \\text{on } (0,1)\n",
"\\end{align}\n",
"\n",
"We try to solve this problem numerically:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "d56d69f9",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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"\n",
"\n"
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]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(u,p,t) = sqrt(u)\n",
"u0, tspan = 0., (0., 1.)\n",
"ivp = ODEProblem(f, u0, tspan)\n",
"sol = solve(ivp, Tsit5())\n",
"\n",
"plot(sol,\n",
" title=L\"u'=√u\", legend=false,\n",
" xlabel=L\"t\", ylabel=L\"u(t)\", ylims=(-1,1))"
]
},
{
"cell_type": "markdown",
"id": "84bf5ff2",
"metadata": {},
"source": [
"Notice that $u(t) = 0$ is indeed a solution to this differential equation. However, this solution is **not** unique.\n",
"\n",
":::\n",
"\n",
"::: {#exr-3}\n",
"\n",
"Show that $\\tilde{u}(t) := \\frac14 t^{2}$ satisfies the IVP in @exm-3. (Notice that the numerical scheme implemented above did not find this solution).\n",
"\n",
"::: \n",
"\n",
"In these examples, we have seen IVPs that *(i)* has a unique solution for all time, *(ii)* has a unique solution for some time interval $[0,T)$ that cannot be extended to $[0,T]$ (finite time blow-up), and *(iii)* has non-unique solutions. It would be nice if we had some analytical results that tell us if/when there exists a unique solution, and over what kind of time interval; this is the content of the next section."
]
},
{
"cell_type": "markdown",
"id": "5fe9be2f",
"metadata": {},
"source": [
"## Existence and Uniqueness Theory\n",
"\n",
"Pages 259--262 of @Burden.\n",
"\n",
"::: {#thm-PL}\n",
"# Picard--Lindelöf \n",
"\n",
"Let $D := \\{ (t,u) : t \\in [0,T], u \\in \\mathbb R \\}$. Suppose that $f$ is continuous on $D$ and Lipschitz in its second argument (with constant independent of $t\\in [0,T]$). Then, $(\\text{IVP})$ has a unique solution $u : [0,T] \\to \\mathbb R$.\n",
"\n",
":::\n",
"\n",
"