Differential equations There are ordinary differential equations functions
- Slides: 19
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 1 st order 2 nd order etc.
Can always turn a higher order ode into a set of 1 st order ode’s Example: Let then So solutions to 1 st 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 2 nd 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)
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 • predictor • corrector
Example of Heun’s method - see Excel first few iterations y. Heun(0)=2 (given) y 0=y. Heun(0)+f(0)*h=2 -500*0. 5=-248 y. Heun(0. 5)=y. Heun(0)+(f(0)+f(0. 5))/2*0. 5 =2+(-500 -245. 5)/2*0. 5=-184. 375 y 0=y. Heun(0. 5)+f(0. 5)*h y. Heun(1)=y. Heun(0. 5)+(f(0. 5)+f(1))/2*h
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