Introduction to Numerical Methods

Numerical Methods

• Numerical methods involve using simple arithmetic operations to solve complex mathematical problems by formulating models of physical situations.

History of Numerical Analysis

• The first numerical analysis is the Babylonian approximation.

Types of Numerical Methods

• Methods for finding the roots of an equation: bisection method, Regula-Falsi method, Secant method, Newton's method, and Fixed-point iteration method.
• Methods for solving systems of linear algebraic equations.
• Interpolation and regression analysis.
• Numerical differentiation.
• Numerical integration.
• Solution of differential equations.
• Solution of matrix problems.
• Solution of boundary value problems.

Need for Numerical Methods

• Mathematical models are central to science and engineering.
• Some models have closed-form solutions, but they can be too costly to be practical.
• All models can be solved computationally, providing useful results.

Why Numerical Methods are Used in Engineering

• Engineers use mathematical modeling, including equations and data, to describe and predict system behavior.
• Computers provide accurate results and are widely available and affordable.
• Many software packages are available for solving problems.

Steps Involved in Solving Numerical Methods

• Formulation of mathematical method.
• Construction of an appropriate numerical method.
• Implementation of the numerical method.
• Obtaining solutions by solving.

Applications of Numerical Methods

• Weather forecasting: uses Numerical Weather Prediction (NWP) techniques, based on atmospheric gases and fluid flow.
• Scientific computing: computes integrals and derivatives, solves differential equations, and performs root finding and numerical optimization.
• Estimation of ocean currents: models ocean circulation to estimate ocean currents.

Linear Equations

• A simple linear equation is of the form y = mx + c.
• Linear equations have a constant slope value and a degree of 1.
• Superposition principle applies to systems characterized by linear equations.
• The output of a linear system is directly proportional to its input.

Non-Linear Equations

• A simple non-linear equation is of the form ax2 + by2 = c.
• Non-linear equations have a variable slope value and a degree of at least 2.
• Superposition principle does not apply to systems characterized by non-linear equations.
• The input and output of a non-linear system are not directly related.

Applications of Numerical Methods (continued)

• Modelling of airplane: uses numerical simulations for aircraft wing design and predicts flying qualities using neural networks pilot model.
• Crime detection: uses numerical methods to predict crime by analyzing data.
• Heat equation: solves problems involving heat equations in science and engineering fields.