List of Figures

• 1.1. An example Laplace equation problem.
• 1.2. An example heat equation problem.
• 1.3. An example wave equation problem.
• 1.4. An improperly posed Laplace problem.
• 1.5. An improperly posed wave equation problem.
• 2.1. Chopping a one-dimensional function up into spikes.
• 2.2. Contribution of one spike to the solution.
• 2.3. Approximation of the spike by an infinitely narrow one.
• 2.4. Green’s function solution of an example one-dimensional Poisson equation.
• 2.5. Approximate Dirac delta function $\delta_\varepsilon(x-\xi)$ is shown left. The true delta function $\delta(x-\xi)$ is the limit when $\varepsilon$ becomes zero, and is an infinitely high, infinitely thin spike, whose bottom is shown right.
• 2.6. One of the spikes out of which an arbitrary two-dimensional function $f$ consists is shown in outline.
• 2.7. Sketch of the problem to be solved: heat is added only to the small dark rectangle around a point $(\xi,\eta)$.
• 2.8. Region of integration for the Green’s function integral. Excluded regions are left blank. The point $(x,y)$ at which the temperature $u$ is to be found is in the center of the excluded small circle.
• 2.9. Example finite domain $\Omega$ in which the Poisson or Laplace equation is to be solved.
• 2.10. Temperature distributions involved in the solution process: (i) $u$ is the desired temperature distribution satisfying the given boundary conditions; (ii) $u_{\rm out}$ is an external temperature distribution whose boundary conditions can be cleverly chosen to achieve various objectives; (iii) $u_{\rm inf}$ is the infinite-domain solution that satisfies no particular boundary conditions on $\delta\Omega$.
• 2.11. Region of integration of the integral for the infinite-space solution. Note that $\delta\Omega$ is a bounding surface of both dark grey domain $\Omega$ and of the light grey exterior region.
• 3.1. Characteristics of the partial differential equation of problem 5.30.
• 3.2. Given the value of $u$ at a single point on a characteristic line, $u$ can be found at every point on that line.
• 3.3. Region where $u$ is determined by an initial condition given on the line $x+y=1$.
• 3.4. Characteristics of Burgers’ equation for an example initial condition intersect for times greater than $t=1$.
• 3.5. Profiles at times $t = 0$, .5, 1, and 1.3 show wave steepening leading to a multiple-valued solution for times greater than $t=1$.
• 3.6. Correct solution of Burgers’ equation for the same initial condition as the previous subsection.
• 3.7. Correct profiles for Burgers’ equation for the same initial condition as the previous subsection.
• 3.8. Incorrect solution to Burgers’ equation for the initial pulse profile shown in the center graphic. The left shock violates the entropy condition.
• 3.9. Correct solution to Burgers’ equation for the initial pulse profile shown in the center graphic. The left shock has been replaced by an expansion fan.
• 5.1. Acoustics in a pipe.
• 5.2. Dependent variables.
• 5.3. Heat conduction in a bar.
• 5.4. Heat conduction in a bar.
• 5.5. Heat conduction in a bar.
• 5.6. Heat conduction in a bar.
• 7.1. Viscous flow next to a moving plate
• 7.2. Viscous flow next to a moving plate
• 7.3. Supersonic flow over a membrane.
• 7.4. Supersonic flow over a membrane.
• 7.5. Function $\bar f$.
• 7.5. Function $\bar f$.
• 7.7. Supersonic flow over a membrane.
• 7.8. Problem for v.