Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OB | 3 | 2 |
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Mathematical analysis in one and several variables, Linear Algebra and a first course on Differential Equations and modeling.
This assignment is the second part of a course on fundamental differential equations. Similar to the assignment Differential Equations and Modeling I, this second part has both a theoretical (which will be explored in theory and problem-solving classes) and an applied (which will be introduced in theory classes and applied both in problem-solving and practice classes) components. It is important that the students understand how to apply the concepts of the quantitative theory of differential ordinary equations to problems and have a knowledge of the basic partial differential equations. In addition to applying a number of the well-established and well explored findings from Equations Differential and Modeling I, new techniques for investigating the named differential equations will also be introduced.
There are three sections to this topic. The first one focuses specifically on planar autonomous systems and discusses the qualitative theory of ordinary differential equations. It serves as an introduction to a topic that will later be covered in greater detail in the Dynamical systems course. The second and third one, have continuity with the course Partial differential equations and are a first study of the most well-known partial differential equations.
1 Autonomous systems in the plane.
1.1. Autonomous systems in R^n. Geometric interpretation. Orbits' structure. First integrals. Invariant surfaces. Phase portraits and conjugation.
1.2. Integrable systems. Phase portrait of planar integrable systems: potential systems, Hamiltonian systems, the model of Lotka-Volterra.
1.3. Non-integrable systems: flow box theorem, qualitative analysis of equilibrium points, limit behavior of the orbits, Bendixson-Poincaré theorem, Lyapunov functions. Limit cycles. Criterion of Bendixon-Dulac. Models of ecology. Van der Pol system.
2 First order partial differential equations.
2.1. Introduction to partial differential equations(PDE).
2.2. Linear and quasi-linear PDE of first order.
3 Second order partial differential equations.
3.1. The wave equation on an infinite string. D'Alembert's formula. Boundary value problems.
3.2. The heat equation. The case of a finite bar.
3.3. Variable separation and Fourier series.
3.4. The Laplace's equation.
There will be three different kinds of interactive activities: theoretical classes, problem-solving classes, and practical classes.
The teacher will use the theoretical classes to motivate the class with the study subjects, explain the material, and incorporate motivating examples.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Classes of problems | 15 | 0.6 | |
Classes of theory | 30 | 1.2 | |
Type: Supervised | |||
Practical classes | 6 | 0.24 | |
Type: Autonomous | |||
Personal studies | 88 | 3.52 |
Continuous assessment /Ongoing evaluation:
•A mandatory delivery of practices. We referred to the note on 10 received with the deliveries as PR. This is a non-recoverable activity.
• A partial first exam halfway through the theory and problems course. We refer to the note on 10 as E1.
• A second partial exam on theory and problems at the end of the course. We refer to the note on 10 as E2.
• In the theoretical classes, there may be two optional assignments that are added to E1 and/or E2, respectively, and have a maximum point value of 0.5 before the first and second partial exams. It is necessary for the student to receive a grade higher than or equal to 3.5 in the respective partial in order to add this qualification.
Unique assessment:
Recovering exam for both cases above:
Course qualification:
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Final test | 80% | 4 | 0.16 | 4, 1, 3, 2 |
First test | 40% | 3 | 0.12 | 4, 1, 3, 2 |
Practical exercices | 20% | 0 | 0 | 4, 1, 3, 2 |
Second test | 40% | 4 | 0.16 | 4, 1, 3, 2 |
Single examination (SE) - for the students who has requested a single evaluation method (4h) | 80% | 0 | 0 | 4, 1, 3, 2 |
For the course's first section, the required reading list will be as follows:
Regarding the second and third topics:
As an auxiliary bibliography for the three topics, the following is suggested:
For the practical classes, we will use SAGE.