Mathematical Analysis IV B
Code
5006
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
António Patrício Alexandre
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
- The program deals with the study of ordinary differential equations. We hope that the students be able to determine the general solution and particular solutions of various types of ordinary differential equations.
- We intend the students use the Laplace Transforms for solving differential equations and corresponding initial value problems.
- In the last part of the program we intend to familiarize the students with Fourier Analysis in such a way they could be able to use it as a tool in solving the most important partial differential equations occuring in engineering.
Prerequisites
- Linear Algebra and Mathematical Analysis I, II, III.
Subject matter
1. First order differential equations. Exact differentials. Integrating factors. Separation of variables. Homogeneous equations. Linear equations. Qualitative theory.
2. Second order differential equatons. Linear equations and Euler equation. Newton''s second law. The free, damped and forced harmonic oscilator. Variation of constants.
3. Solution in series. Bessel, Lagrange and Hermite functions.
4. Linear equations of higher order.
5. Systems of linear equation with constant coefficients. Differential equations in polar coordinates. Linearization of non-linear systens near equilibria.
6. Partial differential equations. Heat, wae and Laplace equations. Laplacian in spherical coordinates.
7. Fourier series and Fourier transform and its use in differential equations.
8. Laplace transform and its use in differential equations. Dirac delta.
9. Introduction to variational calculus. The law of sinus (Snell-Descartes). Catenary. The principle of minimum action. Euler-Lagrange equation and the lagrangean. Brachistochone curve.
10. Introduction to inverse problems. Radon transform.
Bibliography
- M. BRAUN, Differential Equations and Their Applications. Springer-Verlag.
- R. DIPRIMA & W.E. BOYCE, Elementary Differential Equations and Boundary Value Problems. John Wiley & Sons.
- D.G. ZILL, Equações Diferenciais com Aplicações em Modelagem. Pioneira Thomson Learning.
- E. KREYSZIG, Advanced Engineering Mathematics. John Wiley & Sons.
- F. BRAUER & J.A. NOHEL, Introduction to Differential Equations with Applications. Harper & Row, Publishers.
- N.H. ASMAR, Partial Differential Equations with Fourier Series and Boundary Value Problems. Pearson Prentice Hall.
- G. BIRKHOFF & GIAN-CARLO ROTA, Ordinary Differential Equations. John Wiley & Sons.
- T.G. FEEMAN, The Mathematics of Medical Imaging: A Beginner''s Guide. Springer-Verlag
Teaching method
General (3hs/week) and exercises (2hs/week) class. Homework and examples will be solved in exercises class.
Evaluation method
- Students must attend, at least, all problem-solving sessions, with the possible exception of one third.
- The evaluations consists in three tests. If the grade of the third test is equal or superior to seven, then the final classification is the unweighted average of the three tests, rounded to the near integer (n.5 is rounded to n+1).
- For students that fail in the evaluation, there is a final exam.
(More detailed rules are available in the Portuguese version.)