# Course Details

#### Name:

**Equation of Mathematical Physics / Equazioni della fisica matematica**

##### Type:

### Basic information

##### Code:

##### Sector:

##### Credits:

*Bachelor Degree in Mathematics:*6 Ects (b)+3 Ects (c)

##### Term:

*Module Partial differential equations:*1° semester

*Module Modelli Matematici:*1° semester

##### Degree(s):

^{rd}year Bachelor Degree in Mathematics curriculum Generale

##### Language:

##### Teacher(s):

##### Schedule:

### Course Objectives

**Module Partial differential equations:** EQUATIONS OF MATHEMATICAL PHYSICS
Students will know basic of properties (existence, uniqueness, etc.) and techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic PDEs (conservation laws, heat, Laplace, wave equations).**Module Modelli Matematici:** This course aims to enable the students to understand basic Quantum Mechanics and to handle the Schrödinger
Equation.

### Course Content

##### Module Partial differential equations

- Integral curves and surfaces of vector fields. First order partial differential equations. Linear and quasi linear partial differential equations (PDEs) of first order. Method of characteristics. The initial value problem: existence and uniqueness. Development of shocks.
- The Cauchy-Kovalevsky theorem. Linear partial differential operators and their characteristic curves and surfaces. Methods for finding characteristic curves and surfaces. The initial value problem for linear first order equations in two independent variables. Holmgren's uniqueness theorem. Canonical form of first order equations. Classification and canonical forms of second order equations in two independent variables. Second order equations in two or more independent variables. The principle of superposition.
- The divergence theorem and the Green's identities. Equations of Mathematical Physics.
- LAPLACE'S EQUATION AND HARMONIC FUNCTIONS Elementary harmonic functions. Separation of variables. Inversion with respect to circles and spheres. Boundary value problems associated with Laplace's equation. Representation theorem. Mean value property. Maximum principle. Harnack’s inequality and Liouville’s theorem. Well-posedness of the Dirichlet problem. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson's integral. Analytic functions of a complex variable and Laplace's equation in two dimensions. The Neumann problem.
- GREEN'S FUNCTIONS. Solution to the Dirichlet problem for a ball in three dimensions. Further properties of harmonic functions. The Dirichlet problem in unbounded domains. Method of electrostatic images.
- THE WAVE EQUATION. Cauchy problem. Energy method and uniqueness. Domain of dependence and range of influence. Conservation of energy. One-dimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multi-dimensional wave equation. Well posed problems. Fundamental solution (n=3) and strong Huygens’ principle. Kirchhoff formula. Method of descent. Poisson?s formula (n=2). Wave propagation in regions with boundaries. Uniqueness of solution of the initial-boundary value problem. Separation of variables. Reflection of waves.
- THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initial-boundary value problem for the one dimensional heat equation. Method of separation of variables. The initial value problem for the one dimensional heat equation. Fundamental solution. Non homogeneous case and Duhamel’s method. Heat conduction in more than one space dimension.

##### Module Modelli Matematici

- The behavior of quantum systems and a little of hystory.
- Postulates, principles and mathematical tools of Quantum Mechanics.
- Position and momentum: the Heisenberg’s uncertainty principle.
- The dynamics: the Shrödinger equation.
- The Shrödinger equation in one dimension.
- The Shrödinger equation for the harmonic oscillator.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

##### Module Partial differential equations

- have advanced knowledge of classical theory for first and second order PDEs of Mathematical Physics;
- have basic notions of fluid mechanics from an engineering point of view;
- be able to understand when an elementary problem from Mathematical Physics is well posed;
- be able to solve classical problems coming from Mathematical Physics such as initial, boundary, initial-boundary value problems;
- be able to learn autonomously additional results for PDEs of Mathematical Physics.

##### Module Modelli Matematici

- • have acquired the basic notions of Quantum Mechanics, • be able to handle the Shrödinger Equation in simple cases, • be able of reading and understanding more advanced topics in Quantum Mechanics, • have acquired a deeper comprehension of the physical world, • be able to face novel problems with a similar mathematical modeling.

### Prerequisites and Learning Activities

**Module Partial differential equations:** The student must know the basic notions of mathematical analysis, including Fourier series and
ordinary differential equations and the basic notions of continuum mechanics.**Module Modelli Matematici:** Classical Mechanics, Elementary Probability Theory, Linear Algebra.

### Teaching Methods

**Language**: English
**Module Partial differential equations:** Lectures and exercise sessions.**Module Modelli Matematici:** Lectures and exercises.

### Assessment Methods and Criteria

**Module Partial differential equations:** Written and oral.**Module Modelli Matematici:** Written and, if necessary, oral examination.

### Textbooks

##### Module Partial differential equations

- S. Salsa, G. Verzini,
**Equazioni a derivate parziali: complementi ed esercizi**. Springer-Verlag Italia. 2005.*In Italian, ISBN 10 88-470-0260-5* - E. C. Zachmanoglou and Dale W. Thoe,
**lntroduction to Partial Differential Equations with Applications**. Dover Publications, Inc.. 1986.*ISBN 0-486-65251-3* - L.C. Evans,
**Partial Differential Equations**. American Mathematical Society. 2010.*Second edition, ISBN-13: 978-0821849743* - S. Salsa,
**Partial Differential Equations in Actions: from Modelling to Theory**. Springer-Verlag Italia. 2008.*ISBN 978-88-470-0751-2* - W. A. Strauss,
**Partial Differential Equations, Student Solutions Manual: An Introduction**. John Wiley & Sons, LTD. 2008.*Second edition, ISBN-13: 978-0470260715* - W. A. Strauss,
**Partial Differential Equations: an introduction**. John Wiley & Sons, LTD. 2007.*Second edition, ISBN-13 978-0470-05456-7*

##### Module Modelli Matematici

- Lev D. Landau e Evgenij M. Lifšits, ,
**Fisica Teorica 3 - Meccanica quantistica Teoria non relativistica**. Editori Riuniti, University Press. 2010. - P. A. M. Dirac,
**I princ??pi della Meccanica Quantistica**. Bollati Boringhieri. 1990. - K. Konishi e G. Paffuti,
**Meccanica Quantistica: nuova introduzione**. Pisa University Press. 2005. http://people.disim.univaq.it/?serva/teaching/teaching.html

### Online Teaching Resources

##### Homepage:

##### Teaching material:

### Course page updates

This course page is available (with possible updates) also for the following academic years:*Course information last updated on: 08 ottobre 2018, 15:02*