Introduction to Applied Mathematics

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• Introduction to Applied Mathematics
• MA 472, Fall 2008
• Purdue University Calumet

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Some Information

• Instructor Nicolae Tarfulea
• Office CLO 370
• Office Hours MTWR 1100-1200
• Phone [phone]
• WEB http//ems.calumet.purdue.edu/tarfulea
• E-mail tarfulea_at_calumet.purdue.edu

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What is Applied Math Anyway?

• From Wikipedia Applied mathematics is a
  branch of mathematics that concerns itself with
  the mathematical techniques typically used in the
  application of mathematical knowledge to other
  domains.(Some) Divisions of Applied
  Mathematics Applied Analysis (Differential
  Equations, Approximation Theory, Applied
  Probability)Computational MathematicsNumber
  Theory (applications in cryptology)StatisticsMec
  hanicsFluid Dynamics, etc.

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Introduction to PDEs

• Many applications involve PDEs
• Fluid dynamics
• Electromagnetism
• Mechanics Quantum Mechanics
• Optics
• Reaction-diffusion processes

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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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Examples Heat Eqn.
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Interpretation of the Laplacian
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Interpretation of the Heat Eqn.
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Something Like This
The heat equation predicts that if a hot body is
placed in a box of cold water, the temperature of
the body will decrease, and eventually (after
infinite time, and subject to no external heat
sources) the temperature in the box will equalize.
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Examples Wave Eqn.
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Examples Wave Eqn.
A solution of the wave equation in two
dimensions with a zero-displacement boundary
condition along the entire outer edge.
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Examples Laplaces Eqn.
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Examples Laplaces Eqn.
Given Dirichlet boundary conditions on the
perimeter of a circle, Laplace's equation can be
solved for the value of the surface in the
interior of the circle.
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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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Solving a PDE

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

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Thank you!