Partial Differential Equations

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12. Boundary Value Problems for Partial Differential Equations

We will consider the classical, physically important,  forms of second-order partial differential equations: the general heat, wave and potential equations with Dirichlet, Neumannn and Robin boundary conditions. Previous concepts from Part 2 are introduced in the context of PDEs and solution formulas are derived.

  • Zauderer: Section 7.1 in Chapter 7: Green's Functions
  • State the second-order elliptic, hyperbolic and parabolic PDEs as well as the three types of boundary conditions.
  • What are mixed boundary conditions?
  • How do the concepts familiar from ODEs carry over to the study of PDEs
  • Explain the following: formal adjoint, conjunct, Green’s formula, adjoint boundary value problem and Green’s function.
  • What role does the adjoint Green function play in the solution of the parabolic boundary value problem?
  • How is a causal fundamental solution for a time-dependent PDE defined?




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13. Eigenfunction Expansions

Finding Green functions is much more difficult for PDEs than for ODEs. Here, we introduce the method of full and partial eigenfunction expansions that can also yield approximations to actual Green functions.

  • Zauderer: Section 7.3 in Chapter 7: Green's Functions
  • Summarize the properties of the eigenvalues of the elliptic operator.
  • Explain what full and partial eigenfunction expansions are and the difference between them.




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14. The Method of Images

A beautiful approach for deriving Green functions in highly symmetrical situations is the method of images. Motivated by the idea of "virtual charges" in electrostatic, various examples are discussed, including a case of a line charge for a Robin problem.

  • Zauderer: Section 7.5 in Chapter 7: Green's Functions
  • How is Green's function for the elliptic Dirichlet problem constructed using image charges?
  • What approaches are used for the Neuman and Robin problem?
  • Is it possible to apply the method to evolution equations, e.g., the heat and wave equations? How?





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15. The Boundary Element Method

Fundamental solutions and Green functions are also useful in some non-analytic  approaches. The boundary element method (BEM) makes use of these for the numerical solution of problems.

  • Ang: A Beginner's Course in Boundary Element Methods