IVPs and BVPs#

In this class we first focus on two types of ODE problems: initial value problems (IVPs) and boundary value problems (BVPs). Back to our basic example, the harmonic oscillator

\[\frac{\dd^2 u}{\dd t^2} + u = 0 \quad \to \quad u(t) = A\cos(t) + B\sin(t)\]

By itself this ODE is not well-posed: any \( A, B \) give a solution. For the problem to be well-posed, we need to supply side information that constrains the general solution to a unique one. The general system is

\[\frac{\dd^n u}{\dd t^n} = f\left(t, u, \dv{u}{t}, \ldots, \frac{\dd^{n-1} u}{\dd t^{n-1}}\right)\]
  • Initial value problems supply the data at a fixed point in time, typically the initial time \( t_0 \) : the value of all arguments of \( f \) at that point. E.g. \( u(0) = u_0 \) , \( u'(0) = v_0 \) gives
\[u(t) = u_0 \cos(t) + v_0 \sin(t)\]
  • Boundary value problems supply data at the boundaries of the domain, typically two different points in time \( t_0 \) and \( t_1 \) . E.g. \( u(0) = u_0 \) , \( u(\pi/2) = u_1 \) gives
\[u(t) = u_0 \cos(t) + u_1 \sin(t)\]

Thinking about how this distinction affects us: for an IVP we are given enough information to push the initial solution infinitesimally forward in time, and the IVP above always has the unique solution \( A = u_0 \) , \( B = v_0 \) . For a BVP we need to find the entire solution at once, and it is more difficult to guarantee that it is well-posed. A BVP may have infinitely many solutions, or none. Consider

\[u(0) = c_1 \qquad u(\pi) = c_2\]

The coefficients must satisfy

\[\begin{aligned} A \cdot (1) + B \cdot (0) &= c_1 \\ A \cdot (-1) + B \cdot (0) &= c_2 \end{aligned} \quad \Leftrightarrow \quad \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} A \\ B \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}\]

The matrix is singular, so

  • If \( c_1 = c_2 = 0 \) , there are infinitely many solutions \( u(t) = B \sin(t) \): these boundary conditions are insufficient.
  • If \( c_2 \neq -c_1 \) , there is no solution at all.

Intuitively, IVPs are dynamic problems: start with initial data and march forward in time. BVPs are static problems: the solution must satisfy conditions at multiple points, and there’s no inherent direction of “marching”.

General Form of an IVP#

Definition: IVP#

The general form of a first-order IVP is a system of ODEs

\[\dv{\vec{u}}{t} = \vec{f}(t, \vec{u}) \qquad \vec{u}(t_0) = \vec{u}_0\]

where \( \vec{u} : [t_0, t_1] \to \mathbb{R}^d \) is the unknown solution, \( \vec{f} : [t_0, t_1] \times \mathbb{R}^d \to \mathbb{R}^d \) is the RHS, and \( \vec{u}_0 \in \mathbb{R}^d \) is the given initial condition.

All the examples from Lecture 1 are IVPs. It is sufficient to only consider first-order systems, because any higher-order system can be converted into a first-order one. Given

\[\frac{\dd^n u}{\dd t^n} = f\left(t, u, \dv{u}{t}, \ldots, \frac{\dd^{n-1} u}{\dd t^{n-1}}\right)\]

define auxiliary variables

\[\begin{aligned} u_1(t) &= u(t) \\ u_2(t) &= \dv{u}{t}(t) \\ &\;\;\vdots \\ u_n(t) &= \frac{\dd^{n-1} u}{\dd t^{n-1}}(t) \end{aligned} \qquad \qquad \vec{u} = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix}\]

Then the \( n^{\text{th}} \)-order ODE is equivalent to the first-order system

\[\dv{}{t} \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_{n-1} \\ u_n \end{bmatrix} = \begin{bmatrix} u_2 \\ u_3 \\ \vdots \\ u_n \\ f(t, u_1, u_2, \ldots, u_n) \end{bmatrix}\]

General Form of a BVP#

Definition: BVP#

The general form of a BVP is a system of ODEs

\[\dv{\vec{u}}{t} = \vec{f}(t, \vec{u}) \qquad \vec{g}\left(\vec{u}(t_0), \vec{u}(t_1)\right) = \vec{0}\]

where \( \vec{u} : [t_0, t_1] \to \mathbb{R}^d \) , \( \vec{f} : [t_0, t_1] \times \mathbb{R}^d \to \mathbb{R}^d \) , and \( \vec{g} : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^d \) specifies the boundary data at \( t_0 \) and \( t_1 \) .

Example: Poisson’s Equation#

Perhaps the most well-studied BVP is Poisson’s equation

\[-\frac{\dd^2 u}{\dd x^2} = f(x) \qquad x \in (0, 1) \qquad u(0) = u_0, \quad u(1) = u_1\]

a second-order ODE with Dirichlet boundary conditions. It describes the potential field \( u \) due to a charge distribution \( f(x) \), as well as heat conduction in a 1D rod and the deflection of an elastic beam under load.