NUMERICAL METHOD 1 Introduction Definition The study of

NUMERICAL METHOD 1

Introduction Definition : The study of approximation techniques for solving mathematical problems, taking into account the extent of possible errors. 2

Why use Numerical Methods? • To solve problems that cannot be solved exactly

Why use Numerical Methods? • To solve problems that are intractable!

Steps in Solving a Mathematical Problem 5

How do we solve a mathematical problem? Problem Description Mathematical Model Prosedure/ Numerical Solution of Mathematical Model Using the Solution 6

Mathematical Procedures 7

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 8

Nonlinear Equations How much of the floating ball is under water? Diameter=0. 11 m Specific Gravity=0. 6 9

Nonlinear Equations How much of the floating ball is under the water? 10

Differentiation What is the acceleration at t=7 seconds? 11

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

Simultaneous Linear Equations Find the velocity profile, given Time (s) 5 8 12 Vel (m/s) 106 177 600 Three simultaneous linear equations 13

Interpolation What is the velocity of the rocket at t=7 seconds? Time (s) Vel (m/s) 5 106 8 177 12 600 14

Regression Thermal expansion coefficient data for cast steel 15

Regression (cont) 16

Integration Finding the diametric contraction in a steel shaft when dipped in liquid nitrogen. 17

Ordinary Differential Equations How long does it take a trunnion to cool down? 18