Computational Methods for Design Lecture 3 – Elementary Differential Equations John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary C enter for A pplied M athematics Virginia Polytechnic Institute and State University Blacksburg, Virginia A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N∞M∞T Series Two Course Canisius College, Buffalo, NY

Today’s Topics Lecture 3 – Elementary Differential Equations  A Review of the Basics  Equilibrium  Stability  Dependence on Parameters: Sensitivity  Numerical Methods

A Falling Object “Newton’s Second Law”. y(t) { { AIR RESISTANCE

Height: y(t)

Velocity: v(t)=y’(t)

System of Differential Equations

State Space STATE SPACE

System of Differential Equations THE PHYSICS – BIOLOGY – CHEMISTRY IS FINDING SELECTION OF THE “ CORRECT STATE SPACE ” IS A COMBINATION OF PHYSICS – BIOLOGY – CHEMISTRY AND MATHEMATICS

Parameters IN REAL PROBLEMS THERE ARE PARAMETERS SOLUTIONS DEPEND ON THESE PARAMETERS WE WILL BE INTERESTED IN COMPUTING SENSITIVITIES WITH RESPECT TO THESE PARAMETERS MORE LATER

Logistics Equation LE

Analytical Solution K

Initial p 0 : 1 < p 0 < 20,000 K

Equilibrium States LE EQUILIBRIUM STATES ARE CONSTANT SOLUTIONS

Equilibrium States K 0 UNSTABLESTABLE

A Falling Object {. y(t) NO EQUILIBRIUM STATES

Terminal Velocity

A Falling Object

Systems of DEs MORE EQUATIONS

Epidemic Models  SIR Models (Kermak – McKendrick, 1927) l S usceptible – I nfected – R ecovered/Removed

Epidemic Models  SIR Models (Kermak – McKendrick, 1927) l S usceptible – I nfected – R ecovered/Removed

Systems of DEs

Initial Value Problems MOST OF THE TIME WE FORGET THE ARROW AND f CAN DEPEND ON TIME t AND PARAMETERS q

Basic Results A solution to the ordinary differential equation (Σ) is a differentiable function (Σ)(Σ) defined on a connected interval (a,b) such that x(t) satisfies (Σ) for all t  (a,b). TWO SOLUTIONS

Solutions

Initial Condition

Basic Theorems Theorem 1. Let f: R n ---> R n be a continuous function on a domain D  R n, and x 0  D. Then there exists at least one solution to the initial value problem (IVP). (IVP) TO GET UNIQUENESS WE NEED MORE

Basic Theorems Theorem 2. If there is an open rectangle  about (t 0, x 0 ) such that is continuous at all points (t, x)  , then there a unique solution to the initial value problem (IVP). x0x0 t0t0

SIR Model

ALL ENTRIES ARE CONTINUOUS FOR ALL Theorem 1. IS OK

A Falling Object NO PROBLEM SO FAR

A Falling Object AGAIN … CONTINUOUS FOR ALL Theorem 1. IS OK

Parameter Dependence FOR THE FALLING OBJECT …

Examples: n=m=1 CONTINUOUS EVERYWHERE UNIQUE SOLUTION CONTINUOUS WHEN UNIQUE SOLUTION

Logistic Equation

Numerical Methods FORWARD EULER (IVP) t0t0 x0x0

Explicit Euler t0t0 x0x0

t0t0 x0x0

Example 1

Explicit Euler h=.2 h=.01 h=.1

Example 2

h=.2 h=.1

Typical MATLAB m files Eeuler_1.m Eeuler_2.m

Simple Example

Simple Example 3 IF 10 1 PROBLEM IS FINITE PRECISION ARITHMETIC MESH REFINEMENT MAKES THE PROBLEM WORSE