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

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

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

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

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

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

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)

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

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

Begin at x=1

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

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

Example - Local error at any x See Excel sheet

Error can be reduced by smaller h - see Excel sheet

Effect of reducing step size Error vs h

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

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

Heun’s method is a predictor-corrector method

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