Applied partial differential equations
Additional Info
 ECTS credits: 6
 Code: I0183
 University: University of L'Aquila
 Semester: 1
 Lecturer 1: Corrado Lattanzio

Objectives:
LEARNING OBJECTIVES.
The course aims at providing basic properties and main techniques to solve basic partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematical Modelling was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know basic properties (existence, uniqueness, etc.) and main techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic partial differential equations (semilinear first order PDEs, heat, Laplace, wave equations);
2. understand and be able to explain thesis and proofs in the field of basic partial differential equations;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics. 
Topics:
First order partial differential equations. Definition of characteristic vectors and characteristic surfaces. Characteristics for (semi)linear partial differential equations of first order in two independent variables. Existence and uniqueness to initial value problems for first order semilinear partial differential equations in two independent variables Duhamel’s principle for non homogeneous first order partial differential equations.
Second order partial differential equations. Classification of second order semilinear partial differential equations in two independent variables. Canonical form for second order semilinear partial differential equations in two independent variables. Classification for second order semilinear partial differential equations in many independent variables.
Heat equation. Derivation of heat equation and well–posed problems in one space dimension. Solution of Cauchy–Dirichlet problem for one dimensional heat equation by means of Fourier method of separation of variables. Energy method and uniqueness. Maximum principle. Fundamental solution. Solution of global Cauchy problem. Non homogeneous problem: Duhamel’s principle.
Laplace equation. Laplace and Poisson equation: wellposed problems; uniqueness by means of energy method. Mean value property and maximum principles. Laplace equation in a disk by means of separation of variables. Poisson’s formula. Harnack’s inequality and Liouville’s Theorem. Fundamental solution of Laplace operator. Solution of Poisson’s equation in the whole space. Green’s functions and Green’s representation formula.
Wave equation. Transversal vibrations of a string. Well–posed problems in one space dimension. D’Alembert formula. Characteristic parallelogram. Domain of dependence and range of influence. Fundamental solution for one dimensional wave equation. Duhamel’s principle for non homogeneous one dimensional wave equations. Special solutions of multi–d wave equation: planar and spherical waves. Well–posedness for initial, boundary value problems: uniqueness by means of energy estimates. Separation of variables. Domain of dependence and range of influence in several space variables. Fundamental solution for multi–dimensional wave equation. Solution of 3–d wave equation: Kirchhoff’s formula and strong Huygens’ principle. Wave equation in two dimensions: method of descent. Fundamental solution in 2–d. Duhamel’s principle for non homogeneous wave equation in 3–d: delayed potentials. 
Prerequisites:
Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.

Books:
 L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010.
 S. Salsa. Partial Differential Equations in Actions: from Modelling to Theory. Springer–Verlag Italia, 2008.
 S. Salsa, G. Verzini. Equazioni a derivate parziali: complementi ed esercizi. Springer–Verlag Italia. 2005.
 W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons Inc., 2008.
 E.C. Zachmanoglou, D.W. Thoe. lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc., 1986.