This learning unit is a introduction to PDEs designed for a broad range of students with potentially widely different backgrounds. Although it can be classified as a introductory graduate course, it will move quickly and will require a mathematical maturity commensurate with a beginning PhD course in Mathematics.
Partial differential equations, Mathematical Analysis
Knowledge of the following topics is to be acquired by the students upon completion of this learning unit: (1) basic results and techniques on classical linear PDEs of Mathematical-Physics (heat, transport, waves, and Laplace), as well as (2) several representation techniques for solutions, and (3) some Functional Analysis methods for the study of linear PDEs.
After successfully completion of this learning unit the student must be able to mobilize his knowledge to study equations arising in modelling problems. Together with the abilities to be developed in the learning units Nonlinear
Analysis, Mathematical Modelling I, Mathematical Modelling II, Numerical
Methods for Partial Differential Equations, and Inverse Problems and Medical Imaging, this learning unit will contribute to a solid preparation of the students to tackle modelling issues involving partial differential equations.
1. Introduction: Mathematical modelling and partial differential equations
2. Classical linear equations of Mathematical Physics: heat, transport, waves, and Laplace
3. Nonlinear first order equations: method of characteristics, weak solutions,conservation laws
4. Several ways of representing solutions
5. Sobolev spaces
6. Second order linear elliptic equations
7. Linear evolution equations
1. Evans, L.C.: Partial Differential Equations, 2nd Ed., Graduate Studies in Mathematics, vol. 19, Providence: American Mathematical Society, 2010.
2. DiBenedetto, E.: Partial Differential Equations, Boston: Birkhauser, 1995.
3. Salsa, S. et al: A Primer on PDEs: Models, Methods, Simulations, (UNITEXT) vol. 65, Milan: Springer, 2013.
4. John, F.: Partial Differential Equations, 4th Ed., Applied Mathematical Sciences, vol. 1, New York: Springer-Verlag, 1982.
5. Garabedian, P.: Partial Differential Equations, AMS Chelsea vol. 325, Providence: American Mathematical Society, 1964.
6. Rafael Iório Júnior e Valéria de Magalhães Iório, Equações Diferenciais Parciais: uma Introdução, Projeto Euclides, IMPA, 2018.
Evaluation is made on individual basis and it involves the coexistence of two modes: continuous assessment (at least 60%) and final evaluation (at most 40%). Further information is detailed in the Learning Agreement of the course unit.