Chem 302 - Math 252

1. Chem 302 - Math 252 Chapter 6Differential Equations

2. Differential Equations • Many problems in physical chemistry (eg. kinetics, dynamics, theoretical chemistry) require solution to a differential equation • Many can not be solved analytically • Deal only with first order ODE • Higher order equations can be reduced to a system of 1st order DE

3. Differential Equations • Simplest form • Can integrate analytically or numerically (using techniques of Chapter 4)

4. Differential Equations • General case • Many simpler problems can be solved analytically • Many involve ex • However, in chemistry (physics & engineering) many problems have to be solved numerically (or approximately)

5. Picard Method • Can not integrate exactly because integrand involves y • Approximate iteratively by using approximations for y • Continue to iterate until a desire level of accuracy is obtained in y • Often gives a power series solution

6. Picard Method – Example • Continue to iterate until a desire level of accuracy is obtained in y

7. Picard Method – Example 2

8. Euler Method • Assume linear between 2 consecutive points • Between initial point and 1st (calculated) point • User selects Dx • Need to be careful - too big or too small can cause problems

9. Euler Method – Example

10. Taylor Method • Based on Taylor expansion Euler method is Taylor method of order 1 Use chain rule

11. Taylor Method – Example

12. Improved Euler (Heun’s) Method • Euler Method • Use constant derivative between points i & i+1 • calculated at xi • Better to use average derivative across the interval • yi+1 is not known Predict – Correct(can repeat)

13. Improved Euler Method – Example

14. Modified Euler Method • Modified Euler Method • Use derivative halfway between points i & i+1

15. Modified Euler Method – Example

16. Runge-Kutta Methods • Improved and Modified Euler Methods are special cases • 2nd order Runge-Kutta • 4th order Runge-Kutta • Runge • Kutta • Runge-Kutta-Gill

17. Runge Methods

18. Kutta Methods

19. Runge-Kutta-Gill Methods

20. Systems of Equations • All the previous methods can be applied to systems of differential equations • Only illustrate the Runge method

21. Systems of Equations – Example 1

22. Systems of Equations – Example 2

23. Systems of Equations – Example 3

24. Systems of Equations – Example 4

25. Systems of Equations – Example 5