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 Interpretation Mathematical Analysis Mathematical Model Mathematical Results

Practical Applications Physics light and sound waves Newton’s Laws of Motion radioactive decay 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 Positively correlated Law of Cooling 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