29161 - Mathematical Methods M

Learning outcomes

To know and to be able to use some mathematical techniques for the information engineering. Competencies: to know the theory of linear differential equations and systems; to be able to solve constant coefficient linear differential equations and systems; to know the Laplace transform and its use in solving linear differential equations; to have a basic knowledge of dynamical systems. Detailed contents: linear ordinary differential equations, Cauchy problem, existence and uniqueness of solutions. First-order linear equations. Discussion of existence and uniqueness of solutions of first-order differential equations and applications. Higher-order linear differential equations. Numerical solutions of differential equations. Introduction to nonlinear systems. Laplace transform: definition, convergence abscissa; formal properties of the Laplace transform; Laplace transforms of standard functions. Step functions and their transforms. Laplace transforms of some further special functions: the saw-tooth function, the Dirac delta. Applications of Laplace transform to ordinary differential equations: theory and application in solving simple ordinary differential equations with constant coefficients and given boundary conditions. Basic facts about linear transformations; eigenvalues, eigenvectors. Systems of linear differential equations; matrix exponential; dynamical systems, stability; numerical solutions of differential equations. General form of solutions. Transfer function. Stabilization problem.

Course contents

Module 1

Graphs and subgraphs. Trees. Connectivity. Euler tours and Hamilton cycles. Matchings. Edge colourings. Independent sets and cliques. Vertex colourings. Planar graphs. Directed graphs. Hints at networks. Detailed information in http://www.dm.unibo.it/~ferri/hm/progmame.htm

Module 2

Normed spaces. Hilbert spaces. Fourier series and applications. Functions of a complex variable. Harmonic functions. Dirichlet's problem for Laplace equations. Fourier transform. Applications to the equations of heat transfer and waves. Hints of spectral theory.

Readings/Bibliography

Module 1

J.A. Bondy and U.S.R. Murty, "Graph theory with applications",
North Holland, 1976.
Freely downloadable at http://book.huihoo.com/pdf/graph-theory-With-applications/

Module 2

Lecture notes of the teacher. The notes (pdf) will be made available through the institutional site AMS-Campus. Students can also make use of the following textbooks:

- Davide Guidetti: Notes of the course Mathematical Methods (Pdf file available on AMS-Campus:
http://campus.unibo.it/id/ eprint/157317) : Chapters 2 (normed spaces, Fourier series) and Chapter 4 (Fourier transform)

- Erwin Kreyszig: Advanced Engineering Mathematics, 10th Edition J. Wiley (2014) Chapters 6
(Laplace transform), Chapter 11 (Fouries series and Fourier transform ) and Chapter 12 (PDEs)

Teaching methods

Lectures and exercises.

Assessment methods

Module 1

A mid-term test with exercises. An oral exam.

Module 2

Written test with exercises and theory questions.

Teaching tools

Module 1

Textbook available at http://book.huihoo.com/pdf/graph-theory-With-applications/

Additional material at http://www.dm.unibo.it/~ferri/hm/progmame.htm

Module 2

Notes and exercises will be made available at AMS-Campus.

Office hours

See the website of Massimo Ferri

See the website of Simonetta Abenda