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Introduction to Numerical Methods Mathematical Procedures

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Introduction to Numerical Methods Mathematical Procedures. Mathematical Procedures. Nonlinear Equations Differentiation Simultaneous Linear Equations Curve Fitting Interpolation Regression Integration Ordinary Differential Equations Other Advanced Mathematical Procedures:

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Introduction to Numerical Methods Mathematical Procedures

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  1. Introduction to Numerical MethodsMathematical Procedures

  2. Mathematical Procedures • Nonlinear Equations • Differentiation • Simultaneous Linear Equations • Curve Fitting • Interpolation • Regression • Integration • Ordinary Differential Equations • Other Advanced Mathematical Procedures: • Partial Differential Equations • Optimization • Fast Fourier Transforms

  3. Nonlinear Equations How much of the floating ball is under water? Diameter=0.11m Specific Gravity=0.6

  4. Nonlinear Equations How much of the floating ball is under the water?

  5. Differentiation What is the acceleration at t=7 seconds?

  6. Differentiation What is the acceleration at t=7 seconds?

  7. Simultaneous Linear Equations Find the velocity profile, given Three simultaneous linear equations

  8. Interpolation What is the velocity of the rocket at t=7 seconds?

  9. What is Interpolation ? Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given. http://numericalmethods.eng.usf.edu

  10. Interpolants Polynomials are the most common choice of interpolants because they are easy to: • Evaluate • Differentiate, and • Integrate. http://numericalmethods.eng.usf.edu

  11. Newton’s Divided Difference Method Linear interpolation: Given pass a linear interpolant through the data where http://numericalmethods.eng.usf.edu

  12. Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for linear interpolation. Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time http://numericalmethods.eng.usf.edu

  13. http://numericalmethods.eng.usf.edu Linear Interpolation

  14. http://numericalmethods.eng.usf.edu Linear Interpolation (contd)

  15. http://numericalmethods.eng.usf.edu Quadratic Interpolation

  16. Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for quadratic interpolation. Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time http://numericalmethods.eng.usf.edu

  17. Quadratic Interpolation (contd) http://numericalmethods.eng.usf.edu

  18. http://numericalmethods.eng.usf.edu Quadratic Interpolation (contd)

  19. http://numericalmethods.eng.usf.edu Quadratic Interpolation (contd)

  20. General Form where Rewriting http://numericalmethods.eng.usf.edu

  21. http://numericalmethods.eng.usf.edu General Form

  22. http://numericalmethods.eng.usf.edu General form

  23. Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for cubic interpolation. Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time http://numericalmethods.eng.usf.edu

  24. http://numericalmethods.eng.usf.edu Example The velocity profile is chosen as we need to choose four data points that are closest to

  25. http://numericalmethods.eng.usf.edu Example

  26. http://numericalmethods.eng.usf.edu Example

  27. http://numericalmethods.eng.usf.edu Comparison Table

  28. http://numericalmethods.eng.usf.edu Distance from Velocity Profile Find the distance covered by the rocket from t=11s to t=16s ?

  29. http://numericalmethods.eng.usf.edu Acceleration from Velocity Profile Find the acceleration of the rocket at t=16s given that

  30. Regression Thermal expansion coefficient data for cast steel

  31. Regression (cont)

  32. Integration Finding the diametric contraction in a steel shaft when dipped in liquid nitrogen.

  33. Ordinary Differential Equations How long does it take a trunnion to cool down?

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