Lecture notes, available on-line at the URL: www.math.unifi.it/users/barletti (other books are cited in the References)
Learning Objectives
Use of Fourier and related methods fo the solution and/or the analysis of linear differential problems. Interpretation of such problems and methods as "linear algebra in infinite dimensions".
Prerequisites
Calculus I, Calculus II, Ordinary differential equations, Equations of Mathematical Physics
Teaching Methods
Class lectures
CFU: 9
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 153
Oral exam with an exercise and theoretical questions.
Course program
Recalls on Banach, Hilbert and L^p spaces.
Fourier series: punctual, uniform and L^2 convergence theorems.
Parseval identity. Application to initial value problems for the equation of the wave and heat.
Separation of variables and Sturm-Liouville problems. Elements of special functions:
Bessel, Hermite, Legendre, Laguerre functions. Application to problems in cylindrical symmetry.
Spherical harmonics and applications to problems in spherical symmetry. Calculation of energy levels of the hydrogen atom.
Fourier Transform, L^1 and L^2 definitions. Inversion theorem. Plancherel theorem. Application to transport and heat equations in R^n, and to waves in R equations.
Distributions, distributional derivative, tempered distributions. Fourier transform of tempered distributions. Applications to Poisson equation in R^n and to waves in R^2 and R^3 equation.
Semigroups with limited generator and elements on the non-limited case. Sources and perturbations. Dyson-Phillips series. Applications to problems with sources and/or not homogeneous data.