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## DIFFERENTIAL EQUATIONS AND TRANSFORM METHODS

Module code: EE206
Credits: 5
Semester: 1
Department: THEORETICAL PHYSICS
International: Coordinator: Jiri Vala (THEORETICAL PHYSICS) 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. Learning Outcomes Teaching & Learning methods Assessment Repeat options Pre-Requisites Timetable 