P.H. Trinh

MA20223 Lecture 22

This lecture is about terminology but it is important not to be bogged down by terminology. You will practice by doing! This lecture covered part of Chapter 15.

An example

We're now going to return to partial differential equations (which motivated Fourier series). We will mainly study boundary value problems (BVPs). These come with initial conditions (ICs) and boundary conditions (BCs). Here is an example: $$ \begin{gathered} u_t = \kappa u_{xx}, \quad x\in[0, L], t \geq 0 \\ u(0, t) = 0 = u(L, t) \\ u(x, 0) = f(x). \end{gathered} $$

This is attempting to model the distribution of heat in a bar between $x = 0$ and $x = L$. The ends of the bar are held at zero temperature (the BCs), and the bar is initially set to a temperature of $f(x)$ (ICs).

One question you might ask is: beyond physical sensibility, how could I have known how many boundary conditions are needed or how many initial conditions? This is not necessarily a question with a simple answer; in many real-world problems, the sufficiency of boundary conditions is sometimes unclear. But we will develop a better understanding through the study of simple toy problems.

Terminology

The notes, from Sec. 14.1 to 14.2 will set up systematic terminology for PDEs, but you only really only need to know the above (BVPs, ICs, BCs). It is worth remarking that the above equation is linear (definition 14.7) so if $u(x,t)$ and $v(x, t)$ are solutions, then $u + v$ is a solution (as well as any linear combination). This is an important property that allows us to construct solutions from establishing a basis (like Fourier series). In addition to this, let us cover three key types of boundary conditions:

  • Dirichlet BC: specify $u(\mathbf{x}, t) = h(\mathbf{x})$ on $\partial D$.
  • Neumann BC: specify $\frac{\partial u}{\partial n} = \nabla u \cdot \mathbf{n} = h(\mathbf{x})$ on $\partial D$.
  • Mixed/Robin BC: specify $\alpha(\mathbf{x}) u + \beta(\mathbf{x}) \frac{\partial u}{\partial n} = h(\mathbf{x})$ on $\partial D$.

All of these conditions can be made more complicated by requiring the boundary to change as a function of time.

Finally, the cases where the RHS = 0 in each (so $h = 0$) are known as homogeneous boundary conditions. If $h \neq 0$ then the boundary condition is inhomogeneous.

The 1D heat equation with zero Dirichlet conditions

Now let's return to the study of the heat equation with zero Dirichlet conditions stated at the top of this note. In the video, we will go through the procedure from start to end in solving the BVP. In the end, the solution will be given by a Fourier sine series,

$$ u(x, t) \sim \sum_1^\infty \left[b_n \sin\left(\frac{n\pi x}{L}\right)\right]. $$

where $b_n$ will be the Fourier sine coefficients of the odd $2\pi$ extension of $f(x)$ on $[0, \pi]$.

See the video for details of the calculation; we were not able to get to the end of the calculation by the lecture's end, but it will be simple for us to finish it all up in MA20223 Lecture 23