2023
Theory of Ordinary Differential Equations
Name: Theory of Ordinary Differential Equations
Code: MAT14361M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese, English
Regime de Frequência: Presencial
Presentation
Differential equations play an important role in the link and interaction of mathematics with other sciences. It is intended that students learn the fundamental results of the theory of ordinary differential equations, linear and nonlinear, including existence and stability of solutions.
Sustainable Development Goals
Learning Goals
It is intended that students acquire skills on a theoretical level of different methods for linear and non-linear ordinary differential equations, including: initial value problems and boundary value problems; solution existence and location results; stability of solutions.
Contents
- Initial Value Problems. Inequalities and convergences. Uniqueness Theorems. Differential inequalities and extremal solutions.
- Boundary Value Problems. Linear Boundary Value Problems. Maximum Principles. Development in series of eigen-functions. Nonlinear Boundary Value Problems.
- Lower- and upper-solutions method. Application of the lower and upper-solutions method to periodic problems. Nagumo's condition and a priori estimations. Problems with separate boundary conditions.
- Stability of solutions. Preliminary concepts. Stability of quasi-linear systems. Planar autonomous systems. Limit cycles and periodic solutions. Lyapunov method for autonomous systems. Lyapunov method in non-autonomous systems.
- Applications. Lidstone Problems. Functional Problems. Impulsive problems.
- Boundary Value Problems. Linear Boundary Value Problems. Maximum Principles. Development in series of eigen-functions. Nonlinear Boundary Value Problems.
- Lower- and upper-solutions method. Application of the lower and upper-solutions method to periodic problems. Nagumo's condition and a priori estimations. Problems with separate boundary conditions.
- Stability of solutions. Preliminary concepts. Stability of quasi-linear systems. Planar autonomous systems. Limit cycles and periodic solutions. Lyapunov method for autonomous systems. Lyapunov method in non-autonomous systems.
- Applications. Lidstone Problems. Functional Problems. Impulsive problems.
Teaching Methods
Classes are theoretical and practical, using a methodology based on the structured presentation of the syllabus, supported by materials available to students, and examples of applications to real life phenomena, with possible use of appropriate mathematical software.
The students 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. The final grade is calculated as the average of the works' grades.
2) Evaluation by exam, with the possibility of consulting material produced by the student.
The students 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. The final grade is calculated as the average of the works' grades.
2) Evaluation by exam, with the possibility of consulting material produced by the student.
Teaching Staff
- Feliz Manuel Barrão Minhós [responsible]
Certificações
