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: Partial Differential Equations Optimization Fast Fourier Transforms

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

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

Differentiation What is the acceleration at t=7 seconds?

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

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

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

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

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

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

Figure 2: Velocity vs. time data 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 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time http://numericalmethods.eng.usf.edu

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

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

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

Figure 2: Velocity vs. time data 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 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time http://numericalmethods.eng.usf.edu

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

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

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

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

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

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

Figure 2: Velocity vs. time data 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 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time http://numericalmethods.eng.usf.edu

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

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

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

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

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

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

Regression Thermal expansion coefficient data for cast steel

Regression (cont)

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

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