{ "cells": [ { "cell_type": "markdown", "id": "05cedd92", "metadata": {}, "source": [ "## Ordinary Differential Equations (ODEs)\n", "\n", "Differential equations describe lots of things in pretty much every modern field including physics, chemistry, engineering, finance etc....\n", "\n", "They describe the rate of change generally of some quantity be it position, momentum, energy and so on. This is expressed as a derivative of some function of that quantity.\n", "\n", "The simplest type of this is when your equation is a function of one variable. This is called an ordinary differential equation.\n", "\n", "$$ \\frac{d^n}{dt^n}f(t) = g(t) $$" ] }, { "cell_type": "markdown", "id": "3a68f32f", "metadata": {}, "source": [ "The simplest type of ODE is when the derivative is a first derivative, for example\n", "\n", "$$ f'(t) = t $$ \n", "\n", "Does this equation have a unique solution? No, we need an initial condition\n", "\n", "$$ f(t_0) = f_0 $$\n", "\n", "In general though, a good question to ask is whether a solution actually exists for any right hand side $g(t)$? Furthermore, if a solution exists is it unique?\n", "\n", "Here are some cases:\n", "\n", "$$ f'(t) = \\sqrt{f(t)}, f(0) = 0 $$\n", "\n", "$$ f'(t) = f(t)^2, f(0) = 1 $$\n", "\n", "Do these have unique solutions? Turns out the first case does not have a unique solution, the second case does not have a global solution." ] }, { "cell_type": "markdown", "id": "27d28382", "metadata": {}, "source": [ "For the first case, can solve by just good old integration\n", "\n", "$$ \\int \\frac{1}{\\sqrt{f}}df = \\int t dt $$\n", "$$ 2\\sqrt{f(t)} = t $$\n", "$$ f(t) = \\frac{t^2}{4} $$\n", "\n", "Which is a solution. However so is the solution $f(t) = 0$. So which one is correct?\n", "\n", "For the second case\n", "$$ \\int \\frac{1}{f^2} df = \\int t dt $$\n", "$$ -\\frac{1}{f(t)} = t - 1 $$\n", "$$ f(t) = \\frac{1}{1-t} $$\n", "so the solution does not exist at $t=1$. So then what do we do?" ] }, { "cell_type": "markdown", "id": "0056c453", "metadata": {}, "source": [ "We are thus interested in knowing when we can expect the ODE to be uniquely solvable. This is called well-posedness. There is a very well known theorem for well-posedness of ODEs called the Picard-Lindelof theorem" ] }, { "cell_type": "markdown", "id": "cdf9af80", "metadata": {}, "source": [ "### Picard-Lindelof Theorem\n", "\n", "If the right hand side $g(t)$ is Lipschitz continuous, then a unique solution to the ODE exists.\n", "\n", "Lipschitz continuity: A function f is Lipschitz continuous if there exists some constant $L$ such that for all $x,y$, $|f(x)-f(y)| < K|x-y|$." ] }, { "cell_type": "markdown", "id": "74147c16", "metadata": {}, "source": [ "The fact that such a theorem exists is actually pretty amazing - for context there doesn't exist anything like this for PDEs (when f is a function of more than one variable). A lot of the problems in modern mathematics are on the topic of showing well-posedness of various PDEs (such as the Navier-Stokes equations)." ] }, { "cell_type": "markdown", "id": "ced1a8dd", "metadata": {}, "source": [ "For problems in this class, we will assume that they are well-posed so that we can actually even attempt to solve them computationally.\n", "\n", "The idea for solving these computationally lies in basic calculus, namely the FTC\n", "\n", "$$ f'(t) = g(t) $$\n", "$$ f(t) = f(t_0) + \\int_{t_0}^t f'(t) dt $$\n", "$$ f(t) = f(t_0) + \\int_{t_0}^t g(t) dt $$" ] }, { "cell_type": "markdown", "id": "f18e662f", "metadata": {}, "source": [ "If we knew the value of that integral, then we could calculate $f(t)$ exactly. Obviously we can't though in general which is where numerics come in. The goal then is to approximate that integral.\n", "\n", "The first thing you might say is that the integral is roughly approximated by $g(t_0) * (t-t_0)$ given our knowledge of left Riemann sums. This gives\n", "\n", "$$ f(t) \\approx f(t_0) + g(t_0)(t-t_0) $$\n", "\n", "This is known as the forward Euler method.\n", "\n", "The idea then is to use small timesteps $h$ such that if we know the solution at $t_0$, we can then use it to get the solution at $t_0 + h$, then use that to get the solution at $t_0+2h$, and so on." ] }, { "cell_type": "code", "execution_count": 3, "id": "955fce95", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "euler (generic function with 2 methods)" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using PyPlot\n", "\n", "function euler(f, y0, h, N, t0=0.0)\n", " t = t0 .+ h*(0:N)\n", " y = zeros(N+1, length(y0))\n", " \n", " y[1,:] .= y0\n", " for n = 1:N\n", " y[n+1,:] = y[n,:] + h * f(t[n], y[n,:])\n", " end\n", " \n", " return t,y\n", "end" ] }, { "cell_type": "code", "execution_count": 4, "id": "e50b02ed", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "text/plain": [ "Figure(PyObject
)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "f(t,y) = -y .+ sin(t)\n", "t,y = euler(f, 1, 0.2, 20)\n", "plot(t, y, \"-o\");" ] }, { "cell_type": "markdown", "id": "1b5c2647", "metadata": {}, "source": [ "How accurate is the approximation? We can check by trying different timestep sizes h:" ] }, { "cell_type": "code", "execution_count": 14, "id": "75f2ff03", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "text/plain": [ "Figure(PyObject
)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "yexact(t) = exp(-t) + (sin(t) - cos(t) + exp(-t)) / 2\n", "tt = 0:0.01:2\n", "plot(tt, yexact.(tt))\n", "for h = [0.4, 0.2, 0.1]\n", " t,y = euler(f, 1, h, round(Int, 2/h))\n", " plot(t, y, \"-o\")\n", "end\n", "legend((\"Exact\", \"h=0.4\", \"h=0.2\", \"h=0.1\"));" ] }, { "cell_type": "markdown", "id": "41a96f55", "metadata": {}, "source": [ "Unsurprisingly, it seems to get more accurate with smaller timesteps. How do we characterise how fast it gets more accurate? This is called convergence - we can check at the final time for instance how far from the real solution it is." ] }, { "cell_type": "code", "execution_count": 15, "id": "a66a19f4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3-element Vector{Any}:\n", " 0.0275012128275961\n", " 0.013522098820862971\n", " 0.006647820551018313" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "yexact(t) = exp(-t) + (sin(t) - cos(t) + exp(-t)) / 2\n", "tt = 0:0.01:2\n", "error = []\n", "for h = [0.4, 0.2, 0.1]\n", " t,y = euler(f, 1, h, round(Int, 2/h))\n", " push!(error, abs(y[end]-yexact(t[end])));\n", "end\n", "error" ] }, { "cell_type": "markdown", "id": "19bd5cee", "metadata": {}, "source": [ "It seems the error is roughly halving at each step as the timestep halves. This is called linear convergence." ] }, { "cell_type": "markdown", "id": "20fa0a2c", "metadata": {}, "source": [ "There are lots of methods to approximate that integral such as linear multistep methods, backwards differentiation formulae, etc. One famous type are the Runge-Kutta methods, none more so than RK4 which is sometimes known as the RK method. You can find its description on Wikipedia and we won't go into its derivation for this class." ] }, { "cell_type": "code", "execution_count": 16, "id": "9f9e348b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "rk4 (generic function with 2 methods)" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "function rk4(f, y0, h, N, t0=0)\n", " t = t0 .+ h*(0:N)\n", " y = zeros(N+1, length(y0))\n", " \n", " y[1,:] .= y0\n", " for n = 1:N\n", " k1 = h * f(t[n], y[n,:])\n", " k2 = h * f(t[n] + h/2, y[n,:] + k1/2)\n", " k3 = h * f(t[n] + h/2, y[n,:] + k2/2)\n", " k4 = h * f(t[n] + h, y[n,:] + k3)\n", " y[n+1,:] = y[n,:] + (k1 + 2k2 + 2k3 + k4) / 6\n", " end\n", " \n", " return t,y\n", "end" ] }, { "cell_type": "code", "execution_count": 23, "id": "0fff1072", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3-element Vector{Any}:\n", " 2.01614610818579e-5\n", " 1.34056601042154e-7\n", " 1.9447003896111426e-8" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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", "text/plain": [ "Figure(PyObject
)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "yexact(t) = exp(-t) + (sin(t) - cos(t) + exp(-t)) / 2\n", "tt = 0:0.01:2\n", "plot(tt, yexact.(tt))\n", "error = []\n", "for h = [0.4, 0.2, 0.1]\n", " t,y = rk4(f, 1, h, round(Int, 2/h))\n", " plot(t, y, \"-o\")\n", " push!(error, abs(y[end]-yexact(t[end])));\n", "end\n", "legend((\"Exact\", \"h=0.4\", \"h=0.2\", \"h=0.1\"));\n", "display(error)" ] }, { "cell_type": "markdown", "id": "2ec89900", "metadata": {}, "source": [ "It seems to be way more accurate. It's called RK4 as it is in fact 4th order accurate. This means as you halve the timestep your error should decrease by $\\frac{1}{2^4}$. In your HW you will code up the 5th order accurate version." ] }, { "cell_type": "markdown", "id": "f7e34787", "metadata": {}, "source": [ "We are brushing a lot of details under the rug here. In practice there are more nuances to this than simply accuracy, there is also something called stability that essentially asks even if a method is technically accurate can we still expect it to fail due to numerical error. This is somewhat covered in 128A but more so in 228A/B if you are interested.\n", "\n", "A final note, ODEs are something that we as a species have gotten quite good at solving. The real challenge nowadays is on solving PDEs, as they difficult and also what makes the world go round." ] }, { "cell_type": "markdown", "id": "b06f419d", "metadata": {}, "source": [ "### Solving higher derivatives\n", "\n", "A trick that actually works extremely well for solving ODEs with more than one derivative is to split the system by introducing a new variable for the derivative such that it becomes first order. For example\n", "\n", "$$ f''(t) = g(t) $$\n", "\n", "becomes\n", "\n", "$$ h'(t) = g(t) $$\n", "$$ f'(t) = h(t) $$\n", "\n", "which you can now solve as a system using the same techniques as above." ] }, { "cell_type": "markdown", "id": "2fbc3085", "metadata": {}, "source": [ "## Structs\n", "\n", "Structs are Julia's way of making new types. You have already encountered the built in types such as Integers, Floats, etc.... But what if you wanted to make your own?" ] }, { "cell_type": "code", "execution_count": 31, "id": "4bbd6b3c", "metadata": {}, "outputs": [], "source": [ "struct Point\n", " x\n", " y\n", " norm\n", " Point(x, y, norm=sqrt(x*x+y*y)) = new(x,y,norm)\n", "end" ] }, { "cell_type": "markdown", "id": "546dc7d0", "metadata": {}, "source": [ "This creates a struct called Point that has three fields, x,y,norm. The last line is the constructor, it says that you create a point with arguments x,y which are to be inputted, but the norm need not be as it has a default value indicated by the = sign." ] }, { "cell_type": "code", "execution_count": 32, "id": "525bd041", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Point(0.1, 0.2, 0.223606797749979)" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pt1 = Point( 0.1, 0.2 )" ] }, { "cell_type": "markdown", "id": "dec7f9a8", "metadata": {}, "source": [ "You cannot change values of a struct once they are set. They are so called immutable values. This is for speed of the code." ] }, { "cell_type": "code", "execution_count": 33, "id": "e86e35ae", "metadata": {}, "outputs": [ { "ename": "LoadError", "evalue": "setfield! immutable struct of type Point cannot be changed", "output_type": "error", "traceback": [ "setfield! immutable struct of type Point cannot be changed", "", "Stacktrace:", " [1] setproperty!(x::Point, f::Symbol, v::Float64)", " @ Base ./Base.jl:34", " [2] top-level scope", " @ In[33]:1", " [3] eval", " @ ./boot.jl:360 [inlined]", " [4] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)", " @ Base ./loading.jl:1116" ] } ], "source": [ "pt1.x = 0.2" ] }, { "cell_type": "markdown", "id": "a213aa02", "metadata": {}, "source": [ "To get the values though you just use the `.`" ] }, { "cell_type": "code", "execution_count": 34, "id": "38b561f4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.1" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pt1.x" ] }, { "cell_type": "markdown", "id": "8b75873e", "metadata": {}, "source": [ "Structs can act to simplify codes a lot to make them more readable. They exist in other languages too under different names such as classes. These form the basis of what is known as object-orientated programming." ] }, { "cell_type": "code", "execution_count": null, "id": "c9810933", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.6.5", "language": "julia", "name": "julia-1.6" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.6.5" } }, "nbformat": 4, "nbformat_minor": 5 }