2026
Functional Differential Equations
Name: Functional Differential Equations
Code: MAT11698D
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Sustainable Development Goals
Learning Goals
The main goal is to place students in contact with some methods and techniques of Functional Differential Equations, as well as some of its applications. Thus the student will make a first approach to this area of nonlinear analysis and obtain technical skills to make research in this field and related topics.
The functional dependence either in the differential equation or in the possible boundary conditions allows to generalize a wide range of equations (integro-differential, with delay or advances, ...) and boundary conditions (multi-point, impulsive, with delay or advances, with arguments of maximum or minimum, ...).
On the other hand there are a variety of applications that can only be studied by models that include a functional and global dependence..
The functional dependence either in the differential equation or in the possible boundary conditions allows to generalize a wide range of equations (integro-differential, with delay or advances, ...) and boundary conditions (multi-point, impulsive, with delay or advances, with arguments of maximum or minimum, ...).
On the other hand there are a variety of applications that can only be studied by models that include a functional and global dependence..
Contents
1. Linear Functional Differential Equations
Delay-differential equations and Neutral differential equations . Generalized delay-differential equations
2. Equations in finite dimension spaces and applications
Green´s functions and Green´s operator.
Higher order problems (scalar case)
Multi-point problems.
Higher order impulsive problems
Equations with generalized Volterra's operator
3. Functional Differential Equations Oscillation
Comparison theorems and oscillation.
Nonlinear neutral differential equations with variable coefficients
Existence of non-oscillatory solutions
4. Functional Impulsive Problems and Stability
Stability of solutions in Lyapunov sense.
Global stability
Stability on a parameter
Applications: Population models, Neural networks, Economic models
Delay-differential equations and Neutral differential equations . Generalized delay-differential equations
2. Equations in finite dimension spaces and applications
Green´s functions and Green´s operator.
Higher order problems (scalar case)
Multi-point problems.
Higher order impulsive problems
Equations with generalized Volterra's operator
3. Functional Differential Equations Oscillation
Comparison theorems and oscillation.
Nonlinear neutral differential equations with variable coefficients
Existence of non-oscillatory solutions
4. Functional Impulsive Problems and Stability
Stability of solutions in Lyapunov sense.
Global stability
Stability on a parameter
Applications: Population models, Neural networks, Economic models
Teaching Methods
Theoretical and practical lectures, tutorial sessions, with each student being responsible for reading the bibliography advised. For some topics there may be seminars focused in specific points of the course program.
The evaluation process will consider partial tests (Chapters 1,2,3 and Chapters 4,5,6) and a final exam.
The evaluation process will consider partial tests (Chapters 1,2,3 and Chapters 4,5,6) and a final exam.
Accreditations and Certifications
