Aims and scope

Problem solving for ordinary and partial differential equations. Manipulation of analytic functions of complex variable, distributions, Fourier series, Fourier transform, Laplace transform, distribution signals. These topics will be presented in the perspective of their relevant applications to problems in Electrical Engineering.

The teaching activity of this course includes lecture hours and training sessions. Expectations of achievements and abilities resulting from a successful attendance of this course are those associated with the Dublin Descriptors DD1, DD2, DD3, DD4.

The course is focused on the acquisition and critical use of basic tools for: functions of complex variable, zeroes, poles, residues, methods for solving problems related to partial differential equations, distributions, Fourier series, Fourier transform, Laplace transform, separation of variables techniques, essentials of Functional Analysis. Main applications are: periodic solutions of ordinary differential equations; Cauchy problem for wave equation and heat equation; boundary value problems for Laplace equation; analysis of distribution signals; in-depth study of signal processing using time and frequency domain.

Lecture hours will allow students to know and understand the topics above (DD1) and to apply knowledge and understanding (DD2, DD3).

Training sessions will allow studentsto become familiar with the theoretical concepts presented during lecture hours (DD1, DD2, DD3),

help developing ability to make judgements, and improves communication skills (DD3, DD4).

PROGRAM

1 - Introduction - Mathematical modeling. Examples of partial differential equations. Well posed problems. Boundary value problems: Dirichlet and Neumann boundary conditions. Initial value problems. Classification of second order linear Partial Differential Equations with constant coefficients. Conservation Laws. Linear transport equation. Characteristics. Superposition principle for linear problems. Harmonic functions and their properties.

2 - Analytic functions - Complex derivative. Cauchy-Riemann conditions. Cauchy Theorem. Analiticity of holomorphic functions. Residues Theorem. Practical evaluation of residues. Jordan Lemma. Uniqueness of analytic extension. Euler Gamma function. Multivalued functions of complex variable. Logarithmic Index. Rouché Theorem. Evaluation of path integrals by complex variable techniques.

3 - Functional Analysis - Banach spaces. Hilbert spaces. $C^k$ spaces. Lebesgue integral. Dominated convergence Theorem. Absolutely continuous functions. $L^p$ spaces. H\"older and Minkowski inequalities. Test functions. Dirac delta. Distributions or generalized functions. Derivative of a distribution. Tempered distributions. Distributional convergence. Division problems. Convolution. The fundamental solution of Laplace equation in R^n. Newtonian potential.

4 - Heat equation - The diffusion equation. Heat conduction. Maximum principle and uniqueness. The fundamental solution. The Cauchy problem for the heat equation. Energy estimates. Separation of variables: domains with radial symmetry, domains with rectangular symmetry. Eigenvalue problems.

5 - Fourier transform - Fourier transform in L^1 and in L^2. Riemann-Lebesgue Lemma. Inverse transform. Plancherel Identity. Fourier transform of tempered distributions. Paley-Wiener Theorem. Transform-based techniques for solving differential equations with finite energy conditions. Analogic, discrete, causal signals. Discrete and continuous spectra. Sobolev spaces of periodic functions and regularity properties. Uncertainty principle. Fundamental Theorem of filters. Transfer function. Shannon sampling Theorem.

6 - Fourier series of periodic signals – Coefficients evaluation for locally integrable functions. Convergence properties of Fourier series. Parseval Identity. Periodic distributions. Interference, beats, chaotic superposition. Fourier series of periodic signal: relationship between Forier series and Fourier transform of a periodic signal. Fourier series and Fourier transform of the Dirac comb. Periodic solutions of ordinary differential equations. Separation of variables method for linear PDEs. Eigenvalue problems.

7 - Laplace transform – Laplace transformable functions. Laplace transformable distributionns. Analiticity of the Laplace transforms. Algebraic and functional transformation rules. Heaviside formula. Riemann-Fourier inversion formula. Relationship between Laplace transform and Fourier transform. Initial value Theorem. Final value Theorem. Transform-based methods for solving differential equations with conditions. Integro-differential equations. Differential equations with delayed argument.

8 - Wave equation - Transverse waves in a string. Vibrations of a membrane. The one-dimensional wave equation. The Cauchy problem and d’Alembert’s formula. Domain of dependence and range of influence. Huygens principle. Energy conservation. D'Alembert’s formula. Fundamental solution of the wave equation (N=1, 2, 3). Plane waves, Spherical waves. Huygens’principle. Helmoltz equation. Deduction of 3D wave equation from Maxwell’s system of equations. Scalar potential and vector potential. Separation of variables in rectangular and radial domains.

Prerequisites from LT courses (ANALISI MATEMATICA I, ANALISI MATEMATICA II):

differential calculus of several real variables, multiple integrals, integration by parts, divergence theorem, constant coefficients linear ordinary differential equations, power series, Taylor series expansion of elementary functions, Fourier series of periodic functions, Cauchy-Lipschitz Theorem, complex numbers.

The final exam is written, and consists of two parts. 

The first part is a multiple choice tests. Passing grade in the first part is a threshold for admittance to the second part.

The second part consists in solving exercises, providing detailed motivations and answering theoretical questions, all of them related with the topics of the course, in order to assess the knowledge of theorems and ability to exploit the mathematical tools presented in the frontal lessons. 

All exam tasks will have prescribed scores. 

The exam will check acquired skills, understanding of mathematical tools and ability to apply knowledge concerning: Linear Partial Differential Equations, Analytic functions of complex variable, Functional Analysis tools, Heat equation, Fourier transform of functions and distributions, Fourier series of periodic signals, Laplace transform, Wave equation and vibrations, in-depth study of signal processing using time and frequency domain.

Referring to the Dublin Descriptors, the first part of the exam is aimed at evaluating DD1, DD2;the second part is aimed at evaluating DD3, DD4.

The English version of the text will be also available as an E-book.