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3: Boundary and Initial Conditions

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    As you all know, solutions to ordinary differential equations are usually not unique (integration constants appear in many places). This is of course equally a problem for PDE’s. PDE’s are usually specified through a set of boundary or initial conditions. A boundary condition expresses the behavior of a function on the boundary (border) of its area of definition. An initial condition is like a boundary condition, but then for the time-direction.

    • 3.1: Introduction to Boundary and Initial Conditions
      Not all boundary conditions allow for solutions, but usually the physics suggests what makes sense.
    • 3.2: Explicit Boundary Conditions
      For the problems of interest here we shall only consider linear boundary conditions, which express a linear relation between the function and its partial derivatives. As before the maximal order of the derivative in the boundary condition is one order lower than the order of the PDE. For a second order differential equation we have three possible types of boundary conditions: (1) Dirichlet boundary condition, (2) von Neumann boundary conditions and (3) Mixed (Robin’s) boundary conditions.
    • 3.3: Implicit Boundary Conditions
      In many physical problems we have implicit boundary conditions, which just mean that we have certain conditions we wish to be satisfied. This is usually the case for systems defined on an infinite definition area.
    • 3.4: More Realistic Examples of Boundary and Initial Conditions
      Realistic examples of boundary and initial conditions involving strings.

    This page titled 3: Boundary and Initial Conditions is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.