Differential Equations

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• Differential Equations
• MA 264, Spring 2009

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Some Information

• Instructor Nicolae Tarfulea
• Office CLO 370
• Office Hours MW 200-500 pm
• Phone [phone]
• WEB http//ems.calumet.purdue.edu/tarfulea
• E-mail tarfulea_at_calumet.purdue.edu

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Introduction

• Mathematical Modeling is the process of
  mimicking reality by using the language of
  mathematics.
• It has three stages
• formulate the problem
• develop the mathematical model
• test the model

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Introduction

• Many mathematical models involve DEs
• Population Models
• Electric Circuits
• Economics
• Medicine
• Psychology

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Introduction

• Many mathematical models involve DEs
• Fluid dynamics
• Electromagnetism
• Mechanics Quantum Mechanics
• Optics
• Reaction-diffusion processes

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What are DEs?

• Differential Equations are equations that contain
  derivatives.
• Important issues1. formulate the DE from the
  physical problem (the modeling process)2. solve
  the DE

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Example
Computer Solution
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Example
Computer Solution
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Example
Computer Solution
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Example
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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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Example
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Something Like This
A solution of the wave equation in two
dimensions with a zero-displacement boundary
condition along the entire outer edge.
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Classifications

• Order of DE
• Number of (Independent) Variables
• Linearity
• Homogeneity
• Kinds of Coefficients (e.g., constant or variable)

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Solution
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Initial Value Problem