1. Chapter 6 Differential Equations
Chem Math 252
Chapter 6 Differential 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