EQUATIONS OF MATHEMATICAL PHYSICS
Academic year and teacher
If you can't find the course description that you're looking for in the above list,
please see the following instructions >>
- Versione italiana
- Academic year
- 2022/2023
- Teacher
- DIEGO GRANDI
- Credits
- 7
- Didactic period
- Primo Semestre
- SSD
- MAT/07
Training objectives
- The course provides basic concepts and methods in:
- Tensor Calculus: tensor algebra, hints of tensor analysis;
- introductory Continuum Mechanics: Kinematics, General Balance Laws, some constitutive models;
- introductory Partial differential equations (PDE), in connection with the applications to classical Continuum Mechanics: transport equation, wave-, heat- and Laplace/Poisson-equations.
At the end of the Course, the student will be able to:
- use the tensor formalism for Mathematical Physics applications;
- build PDE-based mathematical models to describe phenomena in Classical Mechanics;
- recognize well posed problems in PDE Theory and apply the most common solving techniques for particular PDEs. Prerequisites
- - (first year) Linear Algebra and Geometry;
- differential and integral Calculus of real functions in several variables; ordinary differential equations;
- Classical Mechanics of of particles and rigid bodies. Course programme
- 56 hours are scheduled about the following topics:
I) Elements of Tensor Algebra and Tensor Analysis (12 hours): reminders from Linear Algebra (duality, index notation, scalar products, volumes); tensor product; curvilinear coordinates in euclidean spaces, metric tensor and applications, derivation of tensor fields in general coordinates, divergence, curl, Laplace operator; Divergence Theorem.
II) Continuum Mechanics (18 hours): Kinematics (deformation, material derivatives, deformation measures, transport theorem); general Balance Laws and Cauchy Theorem (the tetrahedron argument): mass-, momentum-, angular momentum- and energy-balances; Clausius-Duhem inequality, free energy and dissipation inequality; constitutive laws and objectivity principle; constitutive models of elastic isotropic solid (in particular the linear one) and of the newtonian fluid.
III) Introduction to the classical theory of PDE (26 hours): general concepts; quasi-linear first order equation and the method of characteristics; wave equation (some classical solving techniques, separation of variables, Duhamel Principle, boundary/initial data problems, energy method, well posed problems); the Cauchy problem and the characteristic surfaces for general PDEs; Cauchy-Kowalevskaja theorem (statement and examples); heat equation (fundamental solution, well-posedness, maximum principle); Laplace/Poisson equation (boundary value problem and variational interpretation: Dirichlet Principle; fundamental solution, mean value property and consequences. Didactic methods
- The teaching usually consists of lectures on blackboard (occasionally with the projection of some slides), providing formal details and proper discussion of the subject. Interventions and questions from the students are welcomed and encouraged.
Learning assessment procedures
- The oral examination tests the knowledge of each of the parts of the course. The resolution of an exercise is required.
Reference texts
- Notes are available at the course Website.
Most topics can be found in:
Parts I-II: S. Forte, L. Preziosi, M. Vianello, Meccanica dei Continui, Springer-Verlag Italia, 2019
Part III: S. Salsa, Equazioni a derivate parziali: metodi, modelli e applicazioni. Springer Italia, 2004
Further readings:
L. C. Evans, Partial Differential Equations, AMS (2010), Chap. 1-4
