Welcome to the course, meaning and need for numerical analysis.

Introduction

Meaning and formulation of mathematical models for actual physical problems.

Mathematical Modelling

Meaning and evaluation of truncation errors.

Truncation Errors

Meaning and evaluation of round-off errors.

Round-Off Errors

Meaning and use of algorithms and pseudocodes.

Algorithms

An introduction to numerical analysis - mathematical modelling, error analysis, numerical algorithms and pseudocodes.

Introduction to Numerical Analysis

Some MATLAB built-in functions and their use.

Built-In Functions

Plotting graphs of various dimensions with MATLAB.

Graphics

Taking inputs from users and presenting outputs.

Input and Output

Welcome to the course, installation of software and overview of the MATLAB environment.

Variable declaration, value assignment and overview of different data types.

Variables and Data Types

Mathematical operators, including operations on vectors and matrices.

Mathematical Operations

Definition and use of relational and logical operators.

Relational and Logical Operators

Definition and use of MATLAB functions and script files.

Functions and Script Files

Definition and use of conditional statements - if else and switch structures.

Conditional Statements

Definition and use of loops - for and while structures.

Loops

Problem Solving

MATLAB fundamentals and its applications to numerical analysis.

Introduction to Programming with MATLAB

Introduction to numerical analysis, solutions of equations in one variable, existence of solution in an interval.

Principle, procedure and worked examples on the bisection method of solution of equations in one variable.

Bisection Method (1)

Error evaluation, advantages and disadvantages of the bisection method of solution of equations in one variable.

Bisection Method (2)

Definition, formulation and use of first-order fixed point iteration for the solution of equations in one variable.

Fixed Point Iteration

Convergence of first-order fixed point recurrence relations.

Fixed Point Convergence

Aitken's accelerated convergence scheme for the solution of equations in one variable.

Accelerated Convergence (1)

g-factor accelerated convergence scheme for the solution of equations in one variable.

Accelerated Convergence (2)

Newton-Raphson iteration as an optimized accelerated convergence scheme for the solution of equations in one variable.

Newton-Raphson Iteration

Numerical solutions of equations in one variable.

Solutions of Equations in One Variable - Numerical Analysis

Welcome and outline of course. Meaning of of polynomials and an introduction to interpolations.

Definition and approximation with Lagrange polynomials.

Lagrange Polynomials (1)

Error evaluation for Lagrange polynomial approximations.

Lagrange Polynomials (2)

Recursively-generated Lagrange polynomials - Neville's method.

Lagrange Polynomials (3)

Lagrange polynomial interpolations and approximations.

Interpolation and Polynomial Approximation - Numerical Analysis

Welcome and course outline. Review of elementary calculus concepts, including Taylor's series.

Definition and use of divided differences.

Divided Differences

Definition and use of second and higher forward and backward differences of error order h.

Higher Forward and Backward Differences

Fitting data at a series of equally-spaced points with a polynomial, using difference expressions.

Differences and Polynomials

Definition, geometric interpretation and use of the first forward  and backward differences of error order h.

First Forward and Backward Differences

Definition and use of first and higher central differences of error order h.

Central Differences

Definition and use of forward and backward differences of error order h-squared and higher.

Higher-Error-Order Forward and Backward Differences

Numerical differentiation (finite difference calculus) and its applications.

Differentiation - Numerical Analysis

Welcome and course outline. The difference operator and definition of difference equations.

Classification of difference equations based on various criteria.

Classification

Meaning and forms of the solution of a difference equation. Formation of difference equations from their solutions. 

Solutions

Solutions of homogeneous difference equations.

Homogeneous Equations

Solutions of non-homogeneous difference equations.

Non-Homogeneous Difference Equations

Solutions and applications of difference equations.

Difference Equations - Numerical Analysis

Welcome and course outline. Review of elementary integral calculus concepts. Motivation for numerical integration.

Definition and use of the trapezoidal rule.

Trapezoidal Rule

Definition and use of Simpson's 1/3 and 3/8 rules.

Simpson's Rules

Richardson's extrapolation and its general use for accelerating convergence in numerical analysis.

Richardson's Extrapolation

Romberg's integration technique - a combination of trapezoidal rule and Richardson's extrapolation.

Romberg's Integration

Numerical integration techniques and their applications.

Integration - Numerical Analysis

Welcome and course outline. Review of ordinary differential equation concepts and definitions.

Meaning of initial value problems in contrast to boundary value problems. Existence and uniqueness of solutions of initial value problems.

Initial Value Problems

Definition and use of Euler's method for the solution of first-order initial value problems.

Euler's Method

Definition and use of Runge-Kutta formulas for the solution of first-order initial value problems.

Runge-Kutta Formulas

Numerical solutions of initial value problems for ordinary differential equations.

Ordinary Differential Equations - Numerical Analysis

Meaning of differential equations, order, degree, solutions, etc., of differential equations.

Meaning of the solution of differential equations.

Solution of differential equations

Worked examples on boundary-value and initial-value problems.

Worked examples (2)

Meaning of the order of differential equations.

Order of differential equations

Definition

Classification of differential equations into ordinary and partial differential equations.

Ordinary and partial differential equations

Identifying linear and non-linear differential equations.

Linear and non-linear differential equations

Worked examples on identifying differential equations and their solutions.

Worked examples (1)

An overview of common notations for derivatives.

Notations

Identifying the degree of differential equations.

Degree of differential equations

Meaning of initial-value problems and boundary-value problems.

Initial-value and boundary-value problems

Applications of first-order ordinary differential equations.

Modelling temperature change problems with first-order ordinary differential equations.

Newton's law of cooling

Worked examples on Newton's second law of motion.

Newton's second law of motion

Determining the oblique trajectories of a family of curves.

Oblique trajectories

Determining the orthogonal trajectories of a family of curves.

Orthogonal trajectories

Modelling radioactive decay with first-order ordinary differential equations.

Radioactive decay

Modelling electric circuit problems with first-order ordinary differential equations.

Electric circuits

Modelling exponential growth and decay with first-order ordinary differential equations.

Exponential growth and decay

Modelling population growth with first-order ordinary differential equations.

Population growth

Modelling rate of chemical reactions with first-order ordinary differential equations.

Rate of chemical reactions

More examples on applications of first-order ordinary differential equations.

More worked examples (1)

Applications of First-Order Ordinary Differential Equations

Analytical solutions of homogeneous linear second-order ordinary differential equations with constant coefficients.

How to obtain the auxiliary equation of homogeneous equations with constant coefficients.

Solving homogeneous equations with constant coefficients (1)

More worked examples on solving homogeneous linear ordinary differential equations with constant coefficients.

Understanding linear dependence of functions and the Wronskian.

Linear dependence

Meaning of homogeneous and non-homogeneous linear differential equations.

Linearly-independent solutions and the general solution of homogeneous linear second-order differential equations.

General solution of homogeneous equations

How to obtain the general solution of homogeneous equations with constant coefficients from the roots of the auxiliary equation.

Solving homogeneous equations with constant coefficients (2)

General solution of non-homogeneous linear second-order differential equations.

General solution of non-homogeneous equations

Solutions of Second-Order Ordinary Differential Equations (1)

Analytical solutions of first-order ordinary differential equations, such as variable-separable equations, homogeneous equations, non-homogeneous equations convertible to homogeneous forms, exact differential equations, inexact differential equations, linear differential equations, Bernoulli equation, and Riccati equation.

Solution of Bernoulli's ordinary differential equation.

Simple non-linear equations (1)

Solution of homogeneous differential equations.

Homogeneous equations

Solution linear differential equations with integrating factors.

Linear differential equations

Solution of non-homogeneous differential equations reducible to homogeneous form.

Non-homogeneous equations

Solution of Riccati's ordinary differential equation.

Simple non-linear equations (2)

Solution of variable-separable differential equations.

Variable-separable equations

Inexact differential equations and their solutions with integrating factors.

Inexact differential equations

Solution of exact differential equations.

Exact differential equations

An introduction to first-order differential equations and their solutions.

More worked examples on solution of first-order ordinary differential equations.

More worked examples (2)

More worked examples (3)

Solutions of First-Order Ordinary Differential Equations

Analytical solutions of non-homogeneous linear second-order ordinary differential equations by the methods of undetermined coefficients and variation of parameters.

An overview of methods of determining the particular integral.

Solution of non-homogeneous second-order linear ordinary differential equations by the method of undetermined coefficients.

Undetermined coefficients

More worked examples on the method of undetermined coefficients.

Variation of parameters

Worked examples on the method of undetermined coefficients.

Worked examples (3)

Worked examples on the method of variation of parameters.

Worked examples (4)

Solutions of Second-Order Ordinary Differential Equations (2)

A comprehensive treatise of ordinary differential equations covering solutions of first-order and second-order ordinary differential equations, and applications of first-order ordinary differential equations.

Ordinary Differential Equations - Advanced Calculus

Welcome and course outline. Review of elementary integral concepts. Motivation for integral transforms.

Definition and existence of Laplace transforms of real-valued or complex-valued functions.

Laplace Transforms (1)

Definition and existence of Mellin transforms and their inverses.

Mellin Transforms

Integral transforms (Laplace, Fourier, Mellin, Hankel and other transforms ), their inverses and applications to the solutions of ordinary differential equations.

Integral Transforms and Their Applications

Advanced engineering mathematics covering numerical and analytical methods of engineering analysis.

Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

CHE 306: Engineering Analysis II