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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

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Assessment

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