Semestre 2021-2
Ecuaciones Diferenciales Parciales
Posgrado en Ciencias Matemáticas (UNAM)
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Online classes
Introduction and General Information
Section I: First order PDEs
Transport Equation Method of Characteristics I Method of Characteristics II: Quasi-linear first order PDEs
Method of Characteristic III: loss of regularity Method of characteristics in dimension n>2. Fully nonlinear equations
Scalar conservation laws I Scalar conservation laws II: Rankine-Hugoniot condition Scalar conservation laws III: entropy condition
Section II: Laplace's equation
Laplace and Poisson equations Mean-value formulas Properties of harmonic functions I Properties of harmonic functions II
Green's function Exterior domains. Perron method. Dirichlet's principle Weak solutions for elliptic equations
Section III: Heat (or diffusion) equation
Derivation of heat equation and fundamental solution Cauchy problem and Fourier Transform
Nonhomogeneous problem and Duhamel's principle Mean-value formula and maximum principle Uniqueness for IBVP and IVP
Energy methods. Separation of variables I Separation of variables II Reaction-Diffusion equations Viscous Burgers equation
Section IV: wave equation
Wave equation and d'Alembert's formula A reflection method. Spherical means Kirchhoff’s and Poisson's formulas
Solutions for general dimensions Nonhomogeneous problem. Energy methods Separation of variables Second Order Linear PDEs
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Tareas
Tarea 1: First order PDEs Solutions
Tarea 2: Laplace's equation Solutions