1. Differential equations
There are ordinary differential equations - functions of one variable
And there are partial differential equations - functions of multiple variables

2. Order of differential equations
1st order
2nd order
etc.

3. Can always turn a higher order ode into a set of 1st order ode’s
Example:
Let
then
So solutions to 1st order are important

4. Linear and nonlinear ODEs
Linear: No multiplicative mixing of variables, no nonlinear functions
Nonlinear: anything else

5. Sometimes can linearize
Example: for small angles
then
which is linear

6. ODEs show up everywhere in engineering
dynamics (Newton’s 2nd law)
heat conduction (Fourier’s law)
diffusion (Fick’s law)

7. We’re going to cover
Euler and Heun's methods
Runge-Kutta methods
Adaptive Runge-Kutta
Multistep methods
Adams-Bashforth-Moulton methods
Boundary value problems
Goal is to get y(x) from dy/dx=f(x)

8. New value=old value+slope*step size
Runge Kutta methods - one step methods
Idea is that
New value=old value+slope*step size or
Slope is generally a function of x, hence y(x)
Different methods differ in how to estimate

9. Euler’s method
Use differential equation to estimate slope, by plugging in current values of x and y
Example: let
Integrate from 1 to 7. Let h=0.5. Initial condition is y(1)=1. Use f for

10. Begin at x=1

11. Ok, not so great
Truncation errors
Round off errors
There is
local truncation error - error from application at a single step
propagated truncation error - previous errors carried forward
sum is Global truncation error

12. Euler’s method uses Taylor series with only first order terms
Error is
Neglect higher order terms

13. Example -
Local error at any x
See Excel sheet

14. Error can be reduced by smaller h - see Excel sheet

15. Effect of reducing step size Error vs h

16. Improvements of Euler’s method - Heun’s method
derivative at beginning of interval is applied to entire interval
Heun’s method uses average derivative for entire interval

17. Graph of function with slope arrows explaining Heun’s method
Average the slopes

18. Heun’s method is a predictor-corrector method

19. Example of Heun’s method - see Excel
first few iterations -
yHeun(0)=2 (given)
y0=yHeun(0)+f(0)*h=2-500*0.5=-248
yHeun(0.5)=yHeun(0)+(f(0)+f(0.5))/2*0.5
=2+( )/2*0.5=
y0=yHeun(0.5)+f(0.5)*h
yHeun(1)=yHeun(0.5)+(f(0.5)+f(1))/2*h