Differential equations There are ordinary differential equations - functions of one variable

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. 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 • predictor • corrector

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+(-500-245.5)/2*0.5=-184.375 y0=yHeun(0.5)+f(0.5)*h yHeun(1)=yHeun(0.5)+(f(0.5)+f(1))/2*h