1. 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 24061-0531 A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N∞M∞T Series Two Course Canisius College, Buffalo, NY

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

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

4. Height: y(t)

5. Velocity: v(t)=y’(t)

6. System of Differential Equations

7. State Space STATE SPACE

8. 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

9. 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

10. Logistics Equation LE

11. Analytical Solution K

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

13. Equilibrium States LE EQUILIBRIUM STATES ARE CONSTANT SOLUTIONS

14. Equilibrium States K 0 UNSTABLESTABLE

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

16. Terminal Velocity

17. A Falling Object

18. Systems of DEs MORE EQUATIONS

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

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

21. Systems of DEs

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

23. 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

24. Solutions

25. Initial Condition

26. 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

27. 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

28. SIR Model

29. ALL ENTRIES ARE CONTINUOUS FOR ALL Theorem 1. IS OK

30. A Falling Object NO PROBLEM SO FAR

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

32. Parameter Dependence FOR THE FALLING OBJECT …

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

34. Logistic Equation

35. Numerical Methods FORWARD EULER (IVP) t0t0 x0x0

36. Explicit Euler t0t0 x0x0

37. t0t0 x0x0

38. Example 1

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

40. Example 2

41. h=.2 h=.1

42. Typical MATLAB m files Eeuler_1.m Eeuler_2.m

43. Simple Example 3 10 1

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