Unit GEOMATHEMATICS
- Course
- Geosciences for risk and environment management
- Study-unit Code
- GP004871
- Location
- PERUGIA
- Curriculum
- Geologia applicata alla salvaguardia e alla pianificazione del territorio
- Teacher
- Luca Zampogni
- Teachers
-
- Luca Zampogni
- Hours
- 42 ore - Luca Zampogni
- CFU
- 6
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Affine/integrativa
- Area
- Attività formative affini o integrative
- Academic discipline
- MAT/05
- Type of study-unit
- Opzionale (Optional)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- First and second order ordinary differential equations. Fouries series and applications. Partial differential equations: traffic equation, and the method of characteristics. The wave equation: Cauchy and Cauchy-Dirichlet problems. Application to geology: the equation of sismology, the Korteveg-de Vries equation and the Tsunami equation
- Reference texts
- James Stewart: Calcolo, funzioni di più variabili, ed. Apogeo.
Fabio Scarabotti: Equazioni alle derivate parziali: teoria elementare e applicazioni, ed. Esculapio. - Educational objectives
- The student should acquire a basic knowledge of qualitative analysis of a Cauchy problem and of the theory necessary to deal with simple partial differential equations. The course matter is part of the contents of a standard reformed second level course for Italian master degrees in Geological Sciences. Even the setting is reformed, and the textbooks used are rich of examples and counterexamples, and therefore seem to be optimal to achieve a good understanding of the topics starting from exercises, that is from applications.Main knowledge acquired will be:
- To know the main topics of mathematical analysis and how to apply them to the earth sciences,
- To know the basilar theory of parabolic partial differential equations,
To know several tools to solve elementary and basilar exercises,
- To read and understand basilar texts of Mathematical Analysis,
- To work in teams, but also in autonomy.
The skills listed above are set out in the framework of the professions related to both a traditional Geologist, and a Geologist oriented to technical and/or industrial activities.Main competence (i.e. ability to apply the main knowledge acquired) will be:- To apply the theory to the resolution of exercises or problems based on models developed during lessons
- To analyze and treat simple qualitative problems in order to understand the appropriate second level analysis to perform,
- To solve some easy mathematical problems in the field of applied mathematics, indipendently. - Prerequisites
- To understand the topics covered in this course, the student should be familiar with all the contents of the courses Matematica (mod. 1 e mod. 2) included in the first level degree. In particular, a robust knowledge of the main definitions and elementary functions (polynomial functions, exponential function, logarithm and circular functions), a good knowledge in calculating limits and derivatives, and finally good knowledge of direct integrations, of integration of rational functions, of integration by parts and by substitution.
- Teaching methods
- Face-to-face and practical training
- Other information
- Optional but recommended attendance
- Learning verification modality
- The exam is written with open answers. In particular the written test consists of solving (generally) 4 exercises on the main topics of the programme.
- Extended program
- A. Some notions about differential calculus on more variables.
B. Ordinary differential equations:
1b. Examples
2b Cauchy problems
3b Existence and uniqueness
4b separation of variables for ordinary equations
5b first order diff. equations
6b second order differential equations
7b varition of constants method
8b the Bernoulli equation
9b the energymethod for some diff. equations
10 examples and applications
C. Fourier series
1c pointwise convergence and uniform convergence. Limit theorems
2c The trigonometric system
3c The Fourier series. Pointwise and uniform convergence, Bessel inequality
4c Total convergence and Parseval relation
5c Asymptotic properties of the Fourier coefficients
D. Partial differential equation (PDE)
1d introduction and general clssification of PDEs
2d the unidirectional wave equation: examples and the genaral formula of the solution with the method of characteristics
4d. First order PDEs in normal form. Sketch of the general method of characteristics.
5d Traffic equation: the continuity equation, velocity field. Burgers equation
6d Wave equation: transversal vibrations of a wire
7d D'Alambert formula for the Cachy problem in the upper half-plane
7e wave equation: separation of variables for the Cauchy-Dirichlet homogeneous problem
E. Examples and applitions
1e wave equation: examples
2e the fundamental equation of sismology: derivation and solutions
3e shallow water waves: the K-dV equation and the Tsunami equation