First and second order linear Ordinary Differential Equations, with constant or continuous coefficients. Boundary value problems for ODEs, eigenvalues and eigenfunctions.
First order quasi-linear and second order linear PDEs with constant coefficients. Classification of second order equations. Boundary value problems for Laplace, heat and wave equations. Separation of the variables method.
Fourier Series and Transforms.
- Lecture notes written by the lecturer. See http://www.dma.unifi.it/~pera
- Mugelli F. -- Spadini M., Metodi matematici, Società Editrice Esculapio, 2013.
Recommended for consultation:
- Nakhlé H. Asmar, Partial Differential Equations, with Fourier Series and Boundary Value Problems, Pearson, 2004.
- Bramanti M., Metodi di analisi Matematica per l'Ingegneria, Società Editrice Esculapio, 2017.
- Tomarelli F., Mathematical Analysis Tools for Engineering, Società Editrice Esculapio, 2019.
Knowledge and understanding of both mathematical principles and the role of mathematical sciences as a tool for the analysis and the problem solving of mechanical engineering problems.
The training goal is to strengthen the disposition to the theoretical approach and to the logical-formal rigor acquired in the first year Math courses, by learning how to model various physical phenomena (potential, wave and heat transport) and by acquiring some solution methods and techniques at least in particular cases.
Courses of Mathematical Analysis 1 & 2, and the course of Geometry.
Theoretical lessons and tutorials
Type of Assessment
The exam consists in an oral discussion. The student must show to be able to apply the acquired tecniques and mathematical methods in modeling, analyzing and solving some problems coming from the applications, especially in the Mechanical Engineering context.
Ordinary Differential Equations.
First order equations. Maximal solutions. Cauchy problem.
First order linear equations; the parameter variation method.
Second order linear equations with constant or continuous coefficients, homogeneous or not.
The Wronskian determinant. The dimension theorem. Cauchy problem.
Methods for obtaining a solution of the non homogeneous equation.
Harmonic motion: forced and damped equations.
Legendre equation (of order 1) and the d'Alembert method.
Series solutions. Analytic solutions.
The Frobenius method.
Bessel functions and their properties.
Boundary value problems for second order equations. Eigenvalues and eigenfunctions.
Sturm Liouville problems. The Fredholm alternative theorem.
Norms and inner products.
Complete spaces. Banach spaces and Hilbert spaces.
The L^2 space: an orthonormal system of periodic functions.
Fourier series: pointwise convergence, uniform convergence.
Complex Fourier series. Bilateral series.
Partial Differential Equations.
Linear and quasi-linear equations.
Solving PDEs by means of Fourier series: the vibrating string equation and the one dimensional heat equation.
Cauchy problem for first order quasi-linear equations. Local existence and uniqueness theorem.
Conservation laws. Shocks.
Cars flow in a motorway.
Linear second'order equations.
Elliptic, parabolic and hyperbolic equations.
Dirichlet and Neumann problems for the Poisson equation.
Cauchy-Dirichlet and Cauchy-Neumann problems for parabolic and hyperbolic equations.
Two examples of not well posed problems: the heat equation for t
Radial solutions of the Laplace equation.
Laplacian in polar coordinates.
Solving the wave equation by means of the reflection method.
Fourier transform: properties and examples.
The Gaussian function transform.
Convolution and its transform.
Solving second order non homogeneous linear ordinary differential equations and Dirichlet problems for Laplace and heat equations in a half space, by means of Fourier transforms.
The fundamental solution of the heat equation.
The Laplace transform: properties and examples.
Convolution and its transform.
Abscissa of convergence.
Solving ordinary differential equations and integral-differential equations by means of Laplace transforms.