This course represents an introduction to partial differential equations. Particular emphasis is given to the linear case (e.g. heat equation, Helmholtz and Laplace equations), where a solution can be constructed using kernels. A part of the class is dedicated to nonlinear partial differential equations such as Korteweg-De Vries or sine-Gordon, and some tools to solve them, like Baecklund transformations, are introduced.
Expected learning outcomes
At the end of the course the students are expected to have the following skills: 1. construction of the kernel for the most important partial differential equations like the heat equation or the Helmholtz and Laplace equations; 2. knows the method of separation of variables; 3. knows some important special functions like Euler's Gamma function or the Bessel functions; 4. ability to classify quasilinear partial differential equations, knows the Cauchy problem and the Cauchy-Kowalewsky theorem; 5. knows some techniques to solve nonlinear differential equations, like e.g. the method of characteristics or the Baecklund transformations.
Partial differential equations. Revisitation of old facts from Physicis and Mathematical Analysis about differential equations (e. g. Frobenius theorem , Helmholtz decomposition and the likes). Comments on the deduction of the vibrating string. The wave equation, homogeneous and inhomogeneous. Uniqueness of the solutions with the energy integral. Construction of the solutions with the separation of variables technique. The eigen-value problem. Series expansion in orthogonal functions. Poisson equation, uniqueness of the solutions with the minimum-maximum theorems, construction of the solutions by sepearation of variables. Heat equation, uniqueness and construction of the solutions, convolution of functions, heat and Schroedinger kernels. Fourier transform applied to the solution of said equations; integral as principal value and Jordan Lemma. Distributions. Test functions, space K, distributions, elementary operations, derivation, regularization of non-locally summable functions, Plemelj formulae. Convolution of distributions and the elementary solutions of the Cauchy problem. Fourier transform of distributions and applications to the solutions of partial differential equations with initial data. Advaced and retarded Green's functions for the wave equation