MathMods :: Joint MSc

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.

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: well-posed 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.

Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.

- 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.

The course is intended to introduce and develop an understanding of the concepts in nonlinear dynamical systems and bifurcation theory, and an ability to analyze nonlinear dynamic models of physical systems. The emphasis is to be on understanding the underlying basis of local bifurcation analysis techniques and their applications to structural and mechanical systems.

Review of: first-order nonlinear ODE, first-order linear systems of autonomous ODE. Local theory for nonlinear dynamical systems: linearization, stable manifold theorem, stability and Liapunov functions, planar non-hyperbolic critical points, center manifold theory, normal form theory. Global theory for nonlinear systems: limit sets and attractors, limit cycles and separatrix cycles, Poincaré map. Hamiltonian systems. Poincaré-Bendixson theory. Bifurcation theory for nonlinear systems: structural stability, bifurcation at non-hyperbolic equilibrium points, Hopf bifurcations, bifurcation at non hyperbolic periodic orbits. Applications.

Ordinary differential equations

Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001

Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form.

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions.  Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
• Introduction to the theory of unbounded operators. Linear differential operators. Applications.
• Introduction to infinite-dimensional differential calculus and variational methods.

Basic calculus and analysis in several variables, linear algebra.

• John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
• H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.