Unit DIFFERENTIAL EQUATIONS
- Course
- Mathematics
- Study-unit Code
- 55A00088
- Curriculum
- Didattico-generale
- Teacher
- Tiziana Cardinali
- Teachers
-
- Tiziana Cardinali
- Hours
- 42 ore - Tiziana Cardinali
- 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
- Fixed point theory. Existence theorems for differential equations and inclusion equations.
- Reference texts
- I will use pieces of :
- L. C. PICCININI, G. STAMPACCHIA,G. VIDOSSICH, Equazioni differenziali ordinarie in R^n, Ed. Liguori, 1978.
- J.M. A. TOLEDANO, T. D. BENAVIDES, G.L. ACEDO, Measures of Noncompactness in Metric Fixed Point Theory, Birkhauser, 1997.
- L. GASINSKI, N. S. PAPAGEORGIOU. Exercises in analysis. Part 1. Problem Books in Mathematics. Springer, Cham, 2014.
- L. GASINSKI, N. S. PAPAGEORGIOU. Exercises in analysis. Part 2. Nonlinear analysis. Problem Books in Mathematics. Springer, Cham, 2016.
The teacher has prepared handouts in Italian that collect the topics presented in class. These handouts will be made available on the Unistudium platform.
Working students, non-attending students, disabled and / or SLD students are invited to report this to the teacher in charge of the course in order to better interact with the student. - Educational objectives
- The aim of the course is a critical learning both of the methodologies used in literature to study the existence of solutions for problems in which differential equations or differential inclusions are present, and of the classical demonstrative lines on the theory of fixed points.
The course aims to develop a critical spirit useful both for teaching preparation and for developing an interest in research.
On successful completion of the course, students should be able to:
- have knowledge of theorems on the existence of local or global solutions for problems involving differential equations or differential inclusions and of classical theorems on the theory of fixed points
- have the ability to autonomously organize the exhibition on the topics studied;
- to think critically and express mathematical concepts precisely in writing;
- to apply the knowledge gained in this course to other situations and disciplines
- autonomously develop solutions to problems inherent to the course program
- to know how to recognize correct demonstrations and identify incorrect reasoning;
- to communicate the mathematical knowledge acquired in the course;
- to read and understand texts of Differential Equations and of Fixed Point Theory,
- to be able to arrive at a mathematical proof of simple propositions with the help of the arguments acquired in the course.
- apply the knowledge and skills acquired to analyze and manage new situations related to economics and control theory.
The competencies are set out in my view essential to a mathematician who wants to dedicate to teaching, as well as for a mathematician who is rather interested in research. - Prerequisites
- This course assumes that the student has a good working knowledge of Mathematical Analysis topics of a Bachelor Degree in Mathematics.
- Teaching methods
- Lectures (frontal hours) - exercise sessions - office hours.
- The course consists into 42 hours of theory, together with different examples and counterexamples.
The aim of course is:
- to invite students to a critical approach to the study of existence of solutions of differential equations (by using examples and counterexamples in order to compare definitions and theorems)
- to show methods in order to obtain a solution for problems involving differential equations.
- to show methods in order to obtain the existence of a fixed point for a map.
- Tutorial service is given in office hours. or online on Teams): two hours per week will be dedicated to a more personalized reception in which students are invited to participate in order to continue the discussion and understanding of the topics presented during the lessons.
- It is the intention of the teacher to prepare two exemption tests that invite students to study the topics in a calibrated way and to also have the possibility with a study distributed throughout the course to be able to take the exam more easily and in the exam sessions exam set in the academic year in progress.
- In the lessons the teacher invites to interact as he considers it very useful both for a preparation for research and for facing a future didactic activity. For this reason it is highly recommended to attend classes.
- If there are working students, non-attending students or students with disabilities, the teacher has prepared slides and handouts in Italian, available on the Unistudium platform. In any case, I advise non-attending students to report this to the teacher at the beginning of the lessons to decide together the most suitable strategy to arrive at a good preparation for the exam. - Other information
- The course takes place over 42 hours and the calendar of educational activities is available on the page web: http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-magistrale/orario-lezioni)
- Classroom: (see: http: //www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-magistrale/orario-lezioni) of the Department of Mathematics and Computer Science of the University of Perugia , Via Vanvitelli.
- Attendance of the lectures is warmly recommended.
- Tutorial service is given in office hours or online on Teams. Tutorial service for the Differential Equations course is organized according to the timetable indicated on the web page
http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-magistrale/ricevimento-e-tutorato
- During the reception hours the students will be followed in a personalized way.
- The teacher will distribute lectur notes and slides (in Italian) useful for a better understanding of the course, in order to help and to let the students pass easily the exam. See: https://www.unistudium.unipg.it/unistudium/login/index.php
8 exams, see: web http://www.dmi.unipg.it/files/matematica/doc-magistrale/mate_calendarioesami_lm_iisem2019-2020
Examining board: T. Cardinali, I. Benedetti (A. Boccuto, R. Filippucci, P. Pucci, P. Rubbioni, A. Sambucini, E. Vitillaro).
- The aim of course is to invite students to a critical approach to the study of existence of solutions of differential equations. Some exercises show methods in order to prove the existence of solutions for problems involving differential equations.
- Some lecture notes will be provided by the teacher in Italian.
There is a Web page which contains this course description as well as other information related to this course, see:
https://www.unistudium.unipg.it/unistudium/login/index.php
Advice:
Don't miss class. Ask questions. Go to office hours as often as necessary.
You need to know the terminology used throughout this course. - Learning verification modality
- The exam includes only an oral test with joint performance of some critical exercises. If the student is interested, the oral exam can be divided into two partial tests.
The oral exam consists of a discussion of about 40/50 minutes on three topics proposed to the student by the Commission, aimed at ascertaining the level of knowledge and understanding reached by the student on the theoretical contents indicated in the program.
The oral exam will also allow to verify the communication skills of the student with language properties and the ability to organize autonomous exposure on the same topics with theoretical content. It provides for the request by the members of the Commission for detailed clarifications with the aim of ascertaining, in addition to the ability to know and understand and that of exposure, also the ability to apply the skills acquired and to develop solutions independently. it is necessary that the student will have to know all the definitions, theorems, proofs, examples and counterexamples introduced in the course. Furthermore, the student will have to demonstrate that he has understood the methods presented in the demonstrations and, above all, that he knows how to comment and motivate them.
The critical exercises required during the oral exam are inspired by the comments, reflections and comparisons between the theorems and the methods of proof presented during the lectures.
The oral examination allows the teacher to verify the performance of the student and his/her ability to organize the presentation in autonomy.
It is possible to divide the oral examination in two parts (each lasting 30/40 minutes): his subdivision is personalized, to be agreed with the teacher.
In this case the final grade is given by an arithmetic average of the two grades obtained on the two partial tests if the student prefers to divide the exam into two parts.
In the event that it is necessary according to the University regulations, the exams and partial exams could take place on the online platform.
For the preparation of the exam it is important to actively attend, that is by asking the teacher questions about the program carried out and interacting with the teacher even during class hours.
The right place for clarification is the lesson hours which are organized in such a way as to encourage participation and obviously the hours dedicated by the teacher to receive students.
For Erasmus students, if they wish, it is possible to take the oral exam also in written form in English.
For the needs of didactic programming, being a teaching
without the obligation to attend, based on the requirements set out in the Syllabus (Rev. 3 of 11 March 2022) the following is specified:
"In the event that the student intends to anticipate the exam in a year prior to that
programmed in the study plan, it is recommended to attend the cycle of lessons e
to take the exam in the first useful session after the lessons themselves are
completed, thus respecting the semester of teaching planning ".
Students with DSA certification must submit the same at least two weeks prior to the test.
Information on support services for students with disabilities and / or DSA see: http://www.unipg.it/disabilita-e-dsa - Extended program
- Fixed point theory. Existence theorems for problems involving differential equations or differential inclusions. Selections theorems for multimaps. Hints of applications to the existence of equilibrium points for deterministic or random abstract economies Hints of problems that can be studied with differential inclusions.