| Code |
14787
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| Year |
3
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
NA
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Learning outcomes |
(i) To understand concepts and fundamental results from the theory of ordinary and partial differential equations;
(ii) To use results from ordinary differential equations theory to analyse equations or systems of ordinary differential equations;
(iii) To understand and to use some results from partial differential equations theory, with incidence in the wave, heat and Laplace equations;
(iv) To analyse and understand mathematical proofs;
(v) To communicate using mathematical language, written and orally.
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Syllabus |
Introduction - Examples and classification of DE.
ODE - Solution of a DE; IVP and BVP; Existence and uniqueness of the solution.
Scalar ODEs - Separable and homogeneous equations; Linear and Bernoulli equations; Exact and reducible to exact equations; Linear equations of order higher than 1; Method of undetermined coefficients; Method of variation of parameters.
Systems of linear ODEs - The solution space of linear systems; Matrix exponentials and Jordan canonical form; Solving systems of linear equations.
Notions of stability for autonomous equations - Orbit, phase portrait, and tangent field; Stability of equilibrium points; Flow and local behavior.
PDE - Definitions and classification; Boundary value problems; Second-order equations.
Separation of variables and Fourier method - Fourier series; Separation of variables; Fourier transform; Fourier method.
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Main Bibliography |
- Braun M (1993). Differential Equations and Their Applications. Springer
- Ross S (1984). Differential Equations. John Wiley and Sons
- Chicone C (2006). Ordinary Differential Equations with Applications, 2nd edition (Texts in Applied Mathematics, 34). Springer
- Doering CI e Lopes AO (2016). Equações Diferenciais Ordinárias, 6.a edição (Coleção Matemática Universitária). IMPA
- Hirsch MW, Smale S and Devaney RL (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd edition. Elsevier Inc.
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Teaching Methodologies and Assessment Criteria |
The classes will be theoretical-practical. The teacher presents the concepts and enunciates the results, demonstrating many of them. It also illustrates the theory with examples and applications. The student is encouraged to participate in classes, interacting with the teacher and sometimes solving exercises and problems. Autonomous work will be encouraged, mainly consisting of exercises, problem-solving and mathematical demonstrations.
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Language |
Portuguese. Tutorial support is available in English.
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