| 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. |