2024
Qualitative Methods in Differential Equations
Name: Qualitative Methods in Differential Equations
Code: MAT14357M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Presentation
Differential equations model many real-life phenomena. However, there are not always methods for their resolution and, when they do exist, they are not necessarily easy to apply. UC provides techniques and methodologies to obtain qualitative information about solutions without having determined them
Sustainable Development Goals
Learning Goals
Nonlinear differential problems and equations model many different phenomena in real life. However, there are not always methods for its resolution and, if any, they are not necessarily easy to apply.
This course aims to provide techniques and methodologies which is possible to obtain, on the one hand sufficient conditions that guarantee the existence of solution to the equation or problem and, on the other hand, qualitative information about this solution, even without having an explicit or implicit expression for it.
That is, even without an analytical expression of the solution, it is possible to know its sign, range, structure, monotony, convexity, nature, its order relation with other possible solutions.
This course aims to provide techniques and methodologies which is possible to obtain, on the one hand sufficient conditions that guarantee the existence of solution to the equation or problem and, on the other hand, qualitative information about this solution, even without having an explicit or implicit expression for it.
That is, even without an analytical expression of the solution, it is possible to know its sign, range, structure, monotony, convexity, nature, its order relation with other possible solutions.
Contents
- Variational Methods: Deformation Theorem and Palais-Smale conditions. Min-max theorems. Mountain Pass Theorem. Saddle Points. Link Theorems.
- Oscillatory theory. Necessary and sufficient conditions for the existence of oscillatory solutions.
- Resonant Problems.
- Homoclinical and heteroclinic solutions.
- Bifurcation theory.
- Oscillatory theory. Necessary and sufficient conditions for the existence of oscillatory solutions.
- Resonant Problems.
- Homoclinical and heteroclinic solutions.
- Bifurcation theory.
Teaching Methods
Students will have at their disposal in Moodle all material used in classes, which are composed of an initial presentation and discussion of themes to be studied, followed by illustrative practical applications.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of the presentation of four works in which themes or problems related to the syllabus are addressed. Each work has a weight of 25% for the final classification;
2) Evaluation by exam, with the possibility of consulting material produced by the student.
The student can choose one of the following forms of assessment:
1) Continuous evaluation consisting of the presentation of four works in which themes or problems related to the syllabus are addressed. Each work has a weight of 25% for the final classification;
2) Evaluation by exam, with the possibility of consulting material produced by the student.
