Differential Equations: A Universal Language

```Differential Equations
A Universal Language
Bethany Caron
Spring 2008
Senior Honors Project
Math is not as scary as it seems!
• Explain complex mathematical concepts in “non-math”
language
• Used as a tool in modeling many different fields of study
• Pure Mathematics vs. Applied Mathematics
Derivatives
• Measures the instantaneous rate of change of a function
• Denoted by f ’ (x) or df/dx
Types of Differential Equations
• Ordinary Differential Equations
– Mathematical equation involving a function and its
derivatives
– Involve equations of one single variable
• Partial Differential Equations
– Involve equations of more than one variable and their partial
derivatives
– Much more difficult to study and solve
Uses of Differential Equations
• Study the relationship between a changing quantity
and its rate of change
• Help solve real life problems that cannot be solved
directly
• Model real life situations to further understand
natural and universal processes
• Model the behavior of complex systems
The Process
Real World
Situation
Formulation
Mathematical
Model
Interpretation
Mathematical
Analysis
Mathematical
Results
Practical Applications
• Physics
•
•
•
•
light and sound waves
Newton’s Laws of Motion
oscillation
• Economics and Finance
• equity markets
• net worth
• Biological Sciences
• predator / prey
• population growth
Practical Applications
• Engineering
• bridge design
• electrical circuits
• Astronomy
• celestial motion
• Chemistry
• interaction between neurons
• Newton’s Law of Cooling
– Forensics - time of death
– temperature of meat
Kermack-McKendrick Model:
Modeling Infectious Disease
•
Famous SIR model
– S: Susceptible People
– I: Infected People
– R: Removed People
•
Models contagious disease in a specific population over
time
•
Simplifies spread and recovery of disease
•
Created to model epidemics
–
–
–
–
–
Plague
Cholera
Flu
Measles
Tuberculosis
Kermack-McKendrick Model
• Assumes a fixed and closed population
– Model in 2-dimensional space
– Need for partial differential equations arises
• Much more difficult to understand and solve
• More effective and accurate model
Newton’s Law of Heating and Cooling
•
Rate of change of temperature
–
Proportional to difference between temperature of
object T and temperature of environment Ta
Law of Heating
–
•
–
Law of Cooling
•
•
Positively correlated
Negatively correlated
Forensics
–
–
Time of death
Temperature of body at different times
Newton’s Law of Heating and Cooling
• Cooling a cup of coffee
• Defrosting food
• Cooking time for meat
Conclusion
• Challenges
– Explaining complex mathematical ideas
– Finding and understanding solutions
“Mathematics knows no races or geographic
boundaries; for mathematics, the cultural
world is one country.” – David Hilbert
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