1.1: Solving ordinary differential equations
- Page ID
- 218596
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Solving ordinary differential equations (ODEs) is a major focus of numerical computing. We start by considering first-order ODEs of the form.
\[ y' = f(t, y) \label{eq:1.1}\]
Where \(y(t)\) is the unknown (i.e. the thing we want to find), and \(t\) is the independent variable. When dealing with such an ODE, we usually think of \(t\) as time, and \(y\) as something depending upon time, like the population of an animal species or the voltage of a node in an electronic circuit. Equation \ref{eq:1.1} says that the way \(y\) changes in time depends upon some function \(f\). In this booklet, we assume the function \(f\) is smooth and well-behaved, and that it is easily calculable in a computer program.
One may write down solutions to \ref{eq:1.1}, but to get a unique solution an additional piece of information is required: An initial condition. That is, we need to know something about \(y(t)\) at one point in time so we can select one individual solution from all possible solutions which satisfy \ref{eq:1.1}. One traditionally uses a value for \(y(t)\) at time \(t=0\) as the initial condition, but other choices are also possible (but uncommon).
Here’s a very basic example. Consider the ODE
\[ y' = a y \label{eq:1.2}\]
You likely encountered this ODE early in your mathematical career, and you know the solution by heart:
\[ y(t) = C e^{a t} \label{eq:1.3}\]
where \(C\) is some constant. The solution \ref{eq:1.3} obviously satisfies equation \ref{eq:1.2}, but it is not useful in practice as the solution to the equation. Why? The reason is that there are an infinite number of possible solutions depending upon the value of \(C\) chosen. That is, \ref{eq:1.3} defines a family of solutions, but in a real, practical problem we only want one of those solutions. For example, how would you make a plot of \ref{eq:1.3}? The answer is, you can’t, until you chose a specific value of \(C\).
In a practical problem, you usually know the value of your function at a start-point in time, almost always at \(t = 0\). Call that value \(y_0\). In this case, the specific solution which satisfies \ref{eq:1.2} and also satisfies \(y(t=0) = y_0\) is
\[ y(t) = y_0 e^{a t} \label{eq:1.4}\]
The point is that numerically solving an ODE requires two pieces of information: 1. The ODE itself, and 2. An initial condition. The initial value picks one solution out of the entire family of solutions which satisfy \ref{eq:1.2}. The concept is shown in 1.1
Such a problem is called an "initial value problem" (IVP) to distinguish it from other types of differential equations (which we will encounter later in the class). "Initial value problem" simply means we know the beginning (initial) value of \(y\), and want to find its behavior as we move forward in time. Once the equation \ref{eq:1.1} and an initial condition for \(y\) are specified, then we have all the information required to find a unique solution \(y(t)\) numerically.
Methods to find numerical solutions IVPs are the focus of this booklet. As a matter of nomenclature, we generally say "solve the ODE", or "solve the IVP". This means we compute the function \(y(t)\) starting from time \(t=0\) to some point in the future, \(t_{end}\). Another way to say the same thing is we "integrate the ODE", or "integrate the IVP". When speaking of ODEs, "integrate" means the same thing as "solve".