# MATH 685/CSI 700 Lecture Notes MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 10.

Ordinary differential equations. Initial value problems.

Differential equations

Differential equations involve derivatives of unknown solution function

Ordinary differential equation (ODE): all derivatives are with

respect to single independent variable, often representing time

Solution of differential equation is function in infinite-dimensional space of functions

Numerical solution of differential equations is based on finitedimensional approximation

Differential equation is replaced by algebraic equation whose solution approximates that of given differential equation

Order of ODE

Order of ODE is determined by highest-order derivative of solution function appearing in ODE

ODE with higher-order derivatives can be transformed into equivalent first-order system

We will discuss numerical solution methods only for

first-order ODEs

Most ODE software is designed to solve only firstorder equations

Higher-order ODEs Example:

Newtons second law u1 = solution y of the original

equation of 2nd order u2 = velocity y Can solve this by methods for 1st order equations

ODEs

Initial value problems Initial value problems

Example

Example (cont.) Stability of solutions

Solution of ODE is Stable if solutions resulting from perturbations of initial value remain close to original solution

Asymptotically stable if solutions resulting from

perturbations converge back to original solution

Unstable if solutions resulting from perturbations diverge away from original solution without bound

Example: stable solutions

Example: asymptotically stable solutions

Example: stability of solutions Example:

linear systems of ODEs Stability of solutions

Stability of solutions

Numerical solution of ODEs Numerical solution to ODEs

Eulers method

Example Example (cont.)

Example (cont.)

Example (cont.) Numerical errors in ODE

solution Global and local error

Global vs. local error

Global vs. local error Global vs. local error

Order of accuracy. Stability

Determining stability/accuracy Example: Eulers method

Example (cont.)

Example (cont.) Example (cont.)

Example (cont.)

Stability in ODE, in general Step size selection

Step size selection

Implicit methods Implicit methods, cont.

Backward Euler method

Implicit methods Backward Euler method

Backward Euler method

Unconditionally stable methods

Trapezoid method Trapezoid method

Implicit methods

Stiff differential equations Stiff ODEs

Stiff ODEs

Example Example, cont.

Example (cont.)

Example (cont.) Example (cont.)

Numerical methods for ODEs

Taylor series methods Taylor series methods

Runge-Kutta methods

Runge-Kutta methods Runge-Kutta methods

Runge-Kutta methods

Extrapolation methods Multistep methods

Multistep methods

Multistep methods Example of multistep methods

Example (cont.)

Multistep Adams methods Properties of multistep

methods Properties of multistep

methods Multivalue methods

Multivalue methods

Example Example (cont.)

Example (cont.)

Example (cont.) Multivalue methods, cont.

Variable-order/Variable-step methods