Introduction to Numerical Methods Mathematical Procedures 1 Mathematical

Introduction to Numerical Methods Mathematical Procedures 1

Mathematical Procedures n n n n 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 2

Nonlinear Equations How much of the floating ball is under water? Diameter=0. 11 m Specific Gravity=0. 6 3

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

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

Differentiation What is the acceleration at t=7 seconds? Time (s) Vel (m/s) 5 106 8 177 12 600 6

Simultaneous Linear Equations Find the velocity profile, given Time (s) 5 8 12 Vel (m/s) 106 177 600 Three simultaneous linear equations 7

Interpolation What is the velocity of the rocket at t=7 seconds? Time (s) Vel (m/s) 5 106 8 177 12 600 8

What is Interpolation ? Given (x 0, y 0), (x 1, y 1), …… (xn, yn), find the value of ‘y’ at a value of ‘x’ that is not given. 9

Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate. 10

Newton’s Divided Difference Method Linear interpolation: Given linear interpolant through the data where 11 pass a

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. t v(t) s m/s 0 0 10 227. 04 15 362. 78 20 517. 35 22. 5 602. 97 30 901. 67 Table 1: Velocity as a function of time 12 Figure 2: Velocity vs. time data for the rocket example

Linear Interpolation 13

Linear Interpolation (contd) 14

Quadratic Interpolation 15

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. t v(t) s m/s 0 0 10 227. 04 15 362. 78 20 517. 35 22. 5 602. 97 30 901. 67 Table 1: Velocity as a function of time 16 Figure 2: Velocity vs. time data for the rocket example

Quadratic Interpolation (contd) 17

Quadratic Interpolation (contd) 18

Quadratic Interpolation (contd) 19

General Form where Rewriting 20

General Form 21

General form 22

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. t v(t) s m/s 0 0 10 227. 04 15 362. 78 20 517. 35 22. 5 602. 97 30 901. 67 Table 1: Velocity as a Figure 2: Velocity vs. time data for the rocket example function of time 23

Example The velocity profile is chosen as we need to choose four data points that are closest to 24

Example 25

Example 26

Comparison Table 27

Distance from Velocity Profile Find the distance covered by the rocket from t=11 s to t=16 s ? 28

Acceleration from Velocity Profile Find the acceleration of the rocket at t=16 s given that 29

Regression Thermal expansion coefficient data for cast steel 30

Regression (cont) 31

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

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