Dr. Mubashir Alam King Saud University

Outline Ordinary Differential Equations (ODE) ODE: An Introduction (8.1) ODE Solution: Euler’s Method (8.2) ODE Solution: Runge-Kutta Method (Order 2) (8.5)

Ordinary Differential Equations (ODE)

ODE: General Solution

ODE: Stability

Assume the solution Y(x) is being sought in the interval x 0 ≤ x ≤ b, and for an initial value Y 0 Change the initial value from Y 0 to Y 0 +ε, and lets call the resulting solution Y ε (x), i.e. Then a solution is stable if for small value of ε Thus a small change in the initial solution Y 0 will only lead to small change in the solution Y(x) of the initial value problem.

Numerical Methods for ODE

Euler’s Method

Proof of Euler’s Method Ch#5

Geometric Approach

Example: Find the solution: Y`(x) = -Y(x), Y(0)=1 True Solution: Y(x)=e -x Euler Method Solution: y n+1 = y n -hy n, n ≥ 0 Y 0 =1 and x n = nh

Example: hxy h (x)ErrorRelative Error E-14.02E E-12.80E E-21.46E E-26.79E E-32.96E

Example

Runge-Kutta Method of Order 2

How to choose constants γ 2 = 1/2, or ¾ or 1

Runge-Kutta Method of Order 2

Example:8.5.2

hxy h (x)Error E E E E E-003

Example: hxy h (x)Error E E E