- 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 is shown left. The true delta function is the limit when 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 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 .
- 2.8. Region of integration for the Green’s function integral. Excluded regions are left blank. The point at which the temperature is to be found is in the center of the excluded small circle.
- 2.9. Example finite domain in which the Poisson or Laplace equation is to be solved.
- 2.10. Temperature distributions involved in the solution process: (i) is the desired temperature distribution satisfying the given boundary conditions; (ii) is an external temperature distribution whose boundary conditions can be cleverly chosen to achieve various objectives; (iii) is the infinite-domain solution that satisfies no particular boundary conditions on .
- 2.11. Region of integration of the integral for the infinite-space solution. Note that is a bounding surface of both dark grey domain 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 at a single point on a characteristic line, can be found at every point on that line.
- 3.3. Region where is determined by an initial condition given on the line .
- 3.4. Characteristics of Burgers’ equation for an example initial condition intersect for times greater than .
- 3.5. Profiles at times , .5, 1, and 1.3 show wave steepening leading to a multiple-valued solution for times greater than .
- 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 .
- 7.5. Function .
- 7.7. Supersonic flow over a membrane.
- 7.8. Problem for v.