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Module DIFFERENTIAL EQUATIONS AND TRANSFORM METHODS

Module code: EE206
Credits: 5
Semester: 1
Department: THEORETICAL PHYSICS
International: Yes
Coordinator: Jiri Vala (THEORETICAL PHYSICS)
Overview Overview
 

Indicative Syllabus
Differential equations
Review of Basic Differentiation techniques and an introduction to modelling.
Second-order differential equations
Linear constant coefficient equations. Solution using method of variation of parameters and method of undetermined coefficients.
Application to RLC series circuit and spring-mass systems.
Higher order linear differential equations.
State-space
An introduction to state-space
State-space representation
Solving systems of linear ordinary differential equations in state-space form
Laplace Transform
Introduction to the Laplace transform. Idea of a linear operator. Rationale for solution method for differential equations.
Inverse Laplace Transform and its relation to the solution to a differential equation.
Laplace transforms of the elementary functions.
Shifting theorems: s-shifting and t-shifting.
Applying the Laplace transform to solving first and second-order differential equations with elementary forcing functions.
Introduction to the unit step function and its use in modelling switching functions.
Introduction to the Dirac delta function and its use in modelling impulsive functions.
Applying the Laplace transform to solving first and second-order differential equations with non-trivial forcing functions (switching, impulsive, periodic).
Laplace transform of periodic functions (square waves, triangular waves) and solving differential equations with forcing functions of this type.
Laplace transform of integrals.
Convolution theorem. Solving integral equations using the convolution theorem and their relation to time-delay systems.
Fourier Series
Introduction to Fourier series and its importance in applications.
Derivation of the Fourier coefficients.
Calculation of Fourier series for several types of periodic function.
Development and motivation of the Fourier transform.
Calculation of the Fourier Transform for various and special functions
Relation of the Fourier transform to signal analysis.
Z-Transform
Introduction to simple difference equations. Idea that solution is a sequence.
Motivational examples of how difference equations arise in practice.
Solution techniques for simple linear first and second order difference equations.
Introduction to discrete transforms and specifically the Z-transform.
Calculation of the Z-transform of certain special sequences. (Unit Step sequence, Unit Impulse sequence)
Using the Z-transform to solve difference equations and the analogy between this method and the use of the Laplace Transform.

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