29145 - Integrational Elements of Mathematical Analysis L

Learning outcomes

The student consolidates his mathematical preparation, with particular regard to linear and nonlinear ordinary differential equations and linear partial differential equations of first and second order, applying the most significant boundary conditions for the various types of equations. He investigates some tools needed for this study and also having an autonomous interest, such as Fourier series and Fourier and Laplace transforms.

Course contents

Sequences and series of functions, various types of convergence, properties of the limit function. Power series and Taylor series.
Integrals depending on a parameter.
Boundary value problems for ordinary differential equations of second order.

Laplace transform: definition, abscissa of absolute convergence, formal properties, calculation of some transforms and inverse transforms, application to ordinary differential equations.

Fourier transform of integrable piecewise continuous functions. Properties  of the Fourier transform. Inversion formula for the Fourier transform.

Fourier series. Real and complex Fourier coefficients. Pointwise and uniform convergence of Fourier series. Properties of the Fourier coefficients. Bessel inequality and Parseval equality.

First order partial differential equations, the method of characteristics.

The Cauchy and Cauchy-Dirichlet problem for waves and heat equations in one space variable. The Laplace equation in two variables.

General properties of the second order equations: propagation velocity, energy conservation, maximum principle.

The Cauchy problem for for waves and heat equations in more space variables. The Dirichlet problem for the Laplace equation in a sphere.

G. C. Barozzi, G. Dore, E. Obrecht: Elementi di analisi matematica, vol. 2, Zanichelli.

P. Drabek, G. Holubova: Elements of Partial Differential Equations, de Gruyter.

Teacher's notes available at the page http://www.dm.unibo.it/~dore/CAM

Teaching methods

Lectures and exercises in the classroom.

Assessment methods

Oral exam aimed at verifying the learning and understanding of the topics covered by the course.