Numerical Methods for Solving ODEs Euler Method

Ordinary Differential Equations  A differential equation is an equation in which includes derivatives of a function. e.g.  An “Ordinary” Differential Equation is one that does not involve partial derivatives. or

 The “order” of a differential equation is the order of the highest derivative present. e.g.  A linear ODE is one in which the derivative terms are only to the 1 st power. 2 nd order, linear 1st order, linear 1st order, non-linear

Solving ODEs  “Solving” an ODE means finding a function f(x) that satisfies the ODE.  For example, if we measure the acceleration, a, of a body undergoing SHM as a function of its displacement from its equilibrium position with time, y(t), we find that  As acceleration is the second derivative of displacement with time. Either sin(ct) or cos(ct) will satisfy this ODE, provided the correct choice of the constant c is made.

Euler’s Method for Solving ODEs To use Euler’s method to solve an ODE we need to know  the rate of change of the function, f ’(x), (i.e. the first derivative), at any point, and  The value of the function f(x) at an initial point.

 We assume that over a small interval (h = dx) close to the known value for f(x 0 ) that both f(x) and f ’(x) are almost the same. y x Y = f(x) x0x0 y0y0 unknown yf ’(x 0 )

y x Y = f(x) x0x0 f(x 0 ) unknown f(x)f’(x 0 ) x1x1 f(x 1 ) h To find the next value of y, knowing y 0, we assume that y 1 will be

If h is small enough this will be true. Provided the steps h are small c.f. the curvature of the function we are trying to find and we can evaluate f ’(x) at any point, then Euler’s method gives us a good numerical approximation to y. where

Taylor Series Derivation  The Taylor series allows us to make a numerical (rather than geometrical) approximation to find successive values of y. n.b. h = x-a, where x-a is the step interval, and E is the error term

writing We can express the simple Euler method as the first two terms of the expansion to get The error will be of the order h 2.

Runge-Kutta Methods  For a better approximation to y(n+1) we can use the slope of the mid-point of the interval h, rather than using the slope at x itself.