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MATH 685/ CSI 700/ OR 682 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

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MATH 685/ CSI 700/ OR 682 Lecture Notes

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  1. MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 10. Ordinary differential equations. Initial value problems.

  2. 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 finite-dimensional approximation • Differential equation is replaced by algebraic equation whose solution approximates that of given differential equation

  3. 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 first-order equations

  4. Higher-order ODEs

  5. Example: Newton’s second law u1 = solution y of the original equation of 2nd order u2 = velocity y’ Can solve this by methods for 1st order equations

  6. ODEs

  7. Initial value problems

  8. Initial value problems

  9. Example

  10. Example (cont.)

  11. 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

  12. Example: stable solutions

  13. Example: asymptotically stable solutions

  14. Example: stability of solutions

  15. Example: linear systems of ODEs

  16. Stability of solutions

  17. Stability of solutions

  18. Numerical solution of ODEs

  19. Numerical solution to ODEs

  20. Euler’s method

  21. Example

  22. Example (cont.)

  23. Example (cont.)

  24. Example (cont.)

  25. Numerical errors in ODE solution

  26. Global and local error

  27. Global vs. local error

  28. Global vs. local error

  29. Global vs. local error

  30. Order of accuracy. Stability

  31. Determining stability/accuracy

  32. Example: Euler’s method

  33. Example (cont.)

  34. Example (cont.)

  35. Example (cont.)

  36. Example (cont.)

  37. Stability in ODE, in general

  38. Step size selection

  39. Step size selection

  40. Implicit methods

  41. Implicit methods, cont.

  42. Backward Euler method

  43. Implicit methods

  44. Backward Euler method

  45. Backward Euler method

  46. Unconditionally stable methods

  47. Trapezoid method

  48. Trapezoid method

  49. Implicit methods

  50. Stiff differential equations

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