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 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 – • – 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