Review MA 472

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Review MA 472
MA 472, Fall 2008
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What are PDEs?

• PDEs are equations that contain partial
  derivatives.
• Important issues1. formulate the PDE from the
  physical problem (the modeling process)2. solve
  the PDE (together with initial data and boundary
  conditions)

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Classifications

• Order of PDE
• Number of (Independent) Variables
• Linearity
• Homogeneity
• Kinds of Coefficients (e.g., constant or
  variable)
• Three basic types of linear equations parabolic,
  hyperbolic, and elliptic.

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Three Basic Types of Linear Equations
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Solving a PDE

• Separation of Variables
• Eigenfunction Expansion
• Integral Transforms
• Change of Coordinates
• Transformation of the Dependent Variable

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Separation of Variables
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Example
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Nonhomogeneous PDEs
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The Fourier Series and Transform
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The Fourier Series and Transform
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The 1D Wave Equation
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The DAlembert Solution
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The Finite Vibrating String
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The Method of Characteristics
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The Laplacian
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Interior Dirichlet Problem for a Circle
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The Dirichlet Problem in an Annulus
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Thank You!