2025
Compact Operators and Integral Equations
Name: Compact Operators and Integral Equations
Code: MAT14358M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Presentation
This UC deepens knowledge of Operator Theory in general and compact operators in particular, and applies these methods to various areas of Analysis such as Ordinary, Partial and Functional differential equations. These operators are suitable also for problems with Integral Equations.
Sustainable Development Goals
Learning Goals
Objectives:
- Deepen knowledge of Operator Theory in general, and compact operators in particular
- Apply linear and non-linear Functional Analysis results to Operators' theory.
- Acquire skills to apply these methods to various areas of Analysis, such as Ordinary, Partial, and/or Functional Differential Equations, as well as Integral Equations.
Skills:
- Develop abstract thinking to solve, in a simpler and more general way, concrete problems in Analysis.
- Ability to abstract, creative intuition, model building, and critical thinking.
- Take advantage of the lesser regularity requirement of the Integral Equations to guarantee the solvency of problems with an initial value or with values at the boundary.
- Deepen knowledge of Operator Theory in general, and compact operators in particular
- Apply linear and non-linear Functional Analysis results to Operators' theory.
- Acquire skills to apply these methods to various areas of Analysis, such as Ordinary, Partial, and/or Functional Differential Equations, as well as Integral Equations.
Skills:
- Develop abstract thinking to solve, in a simpler and more general way, concrete problems in Analysis.
- Ability to abstract, creative intuition, model building, and critical thinking.
- Take advantage of the lesser regularity requirement of the Integral Equations to guarantee the solvency of problems with an initial value or with values at the boundary.
Contents
- Compact Operators in Banach Spaces. Compact Linear Operators. Nonlinear Compact Operators and Boundary Value Problems. Compact operators defined in compact and non-compact intervals. Fixed point theory on non-compact sets.
- Compact operators in cones.
- Volterra and Hammerstein integral equations.
- Boundary Value Problems and Integral Equations.
- Integral Equation Systems.
- Lotka-Volterra systems and predator-prey models.
- Compact operators in cones.
- Volterra and Hammerstein integral equations.
- Boundary Value Problems and Integral Equations.
- Integral Equation Systems.
- Lotka-Volterra systems and predator-prey models.
Teaching Methods
Classes are theoretical and practical, using a structured methodology for the presentation of the syllabus, supported by materials made available to students, and in examples of applications of the main results.
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.
Certificações
