Mathematical models are used to understand, predict and optimize engineering systems. Many of these systems are deterministic and are modeled using differential equations. Others are random in nature and are analyzed using probability theory and statistics. This course provides an introduction to differential equations and their solutions and to probability and statistics, and relates the theory to physical systems and simple real world applications.

This course is designed to help students to understand the basic concepts and modelling of Ordinary differential equations, Laplace transforms, Matrices, Fourier series, Complex integration and Numeric linear algebra.

After completing this course, students will have professional knowledge to derive mathematical models of physical systems and solve differential equations using appropriate methods.

**WHO SHOULD COPMPLETE THIS COURSE?**

- Engineers of all disciplines
- Team leaders
- Subject matter experts
- Project managers
- Supervisors
- Analysts

**COURSE OUTLINE**

FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS (ODES)

- Basic concepts
- Separable ODEs
- Exact ODEs
- Linear ODEs
- Existence and Uniqueness of solutions

SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS (ODES)

- Homogeneous second order ODEs
- Modelling: Free Oscillations
- Euler – Cauchy Equations
- Modelling: Forced Oscillations
- Modelling: Electric Oscillators

HIGHER ORDER LINEAR ODES

- Homogeneous linear ODEs
- Superposition principle
- Initial value problem
- Non homogeneous linear ODEs

SYSTEM ODES

- Systems of ODE as models
- Conversion of Nth order ODE to a system

SERIES SOLUTIONS OF ODES

- Power series method
- Legendre's Equation
- Bessel functions
- Bessel function of the second kind
- Sturm-Liouville Problems
- Orthogonal Eigenvalue expansions

LAPLACE TRANSFORMS

- Transforms of Derivatives and Integrals
- ODEs
- Unit step functions
- Short Impulses
- Convolution
- Differentiation and Integration of Transforms

MATRICES, VECTORS AND DETERMINANTS

- Addition and Scalar multiplication
- Linear systems of Equations
- Gauss elimination
- Linear Independence and Rank of matrix
- Solutions of Linear systems
- Determinants
- Cramer's Rule
- Inverse of matrix
- Gauss Jordan elimination

MATRIX EIGENVALUES

- Applications of Eigenvalue problems
- Symmetric, skew-symmetric and orthogonal matrices
- Eigenbases, Diagonalization and Quadratic forms

FOURIER SERIES, INTEGRALS AND TRANSFORMS

- Even and odd series
- Half range expansions
- Approximation by trigonometric polynomials
- Fourier integrals
- Fourier cosine and sine transformers
- Discrete and Fast Fourier transforms

PARTIAL DIFFERENTIAL EQUATIONS PDES

- Basic concepts and modeling
- Solution by separating variables
- D'Alembart's Solution of wave equation characteristics
- Solution of heat equation
- Laplace's Equation in Cylindrical and Polar Coordinates
- Solution of PDEs by Laplace transforms

COMPLEX NUMBERS AND FUNCTIONS

- Polar form of complex numbers
- Cauchy-Riemann Equations
- Exponential functions
- Trigonometric and hyperbolic functions
- Logarithm and General power

COMPLEX INTEGRATION

- Line Integral in the complex plane
- Cauchy's Integral theorem
- Cauchy's Integral formula
- Derivatives of Analytic functions

POWER SERIES AND TAYLOR SERIES

- Sequence, series and convergence tests
- Power series
- Functions given by power series
- Taylor and Maclaurin series

LAURENT SERIES AND RESIDUE INTEGRATION

- Laurent series
- Singularities and Zeros
- Residue Integration method
- Residue Integration of Real Integrals

CONFORMAL MAPPING

- Geometry of analytic functions
- Linear fractional transformation
- Special linear fractional transformations
- Conformal mapping by other functions

COMPLEX ANALYSIS AND POTENTIAL THEORY

- Electrostatic fields
- Modelling and use of conformal mapping
- Heat problems
- Fluid flow
- Poisson's Integral formula
- General properties of harmonic functions

NUMERICS IN GENERAL

- Solutions of equation by iteration
- Interpolation
- Spline Interpolation
- Numeric Integration and Differentiation

NUMERIC LINEAR ALGEBRA

- Linear systems
- Least squares method
- Matrix eigenvalues
- Inclusion of Matrix eigenvalues
- Power method for Matrix eigenvalues
- Tridiagonalization and QR-factorization

NUMERIC FOR ODES AND PDES

- Methods for first order ODEs
- Multistep methods
- Methods for systems and higher order ODEs
- Methods for elliptic PDEs
- Methods for parabolic PDEs
- Methods for hyperbolic PDEs