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Syllabus

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Autores: Manuel Carretero, Luis L. Bonilla, Filippo Terragni, Sergei Iakunin, Rocío Vega
Syllabus de la asignatura: Temas que forman parte de la asignatura.

 

Lecture 1: First-order ordinary differential equations.

  • Basics. First-order linear ordinary differential equations (ODEs). First-order non linear ODEs. Direction fields. Existence and uniqueness of solutions of the initial value problem (IVP). Numerical methods: Euler, Heun and Runge-Kutta.

Lecture 2: Second-order ordinary differential equations.

  • Linear second-order ODEs: Homogenous. Variation of paremeters and inhomogeneous ODEs. Method of undetermined coefficients. Linear ODEs with constant coefficients. Equidimensional ODEs (Euler-Cauchy). Reduction of order.
  • SUPPLEMENTARY MATERIAL: Linear oscillator and resonance.

Lecture 3: Systems of differential equations.

  • Systems of first-order ODEs: Linear systems. Autonomous linear systems. Two-dimensional homogeneous linear systems. Inhomogeneous linear systems. Variation of parameters. Undetermined  coefficients.
  • SUPPLEMENTARY MATERIAL: Reduction to normal modes.

Lecture 4: Boundary value problems.

  • Existence of solutions of a boundary value problem (BVP). BVP for nonlinear ODEs.
  • SUPPLEMENTARY MATERIAL: Shooting method for linear BVPs.

Lecture 5: Fourier series and separation of variables: Heat equation.

  • Classification of second-order Partial Differential Equations (PDEs). Heat equation. Fourier series method for the heat equation: Separation of variables for the homogeneous heat equation. Eigenvalue problem. Homogeneous heat equation with Neumann boundary conditions and with periodic boundary conditions. Inhomogeneous problems. Properties of Fourier series.
  • SUPPLEMENTARY MATERIAL: Finite difference solution of the heat equation.

Lecture 6: Fourier series and separation of variables: Wave equation.

  • Derivation of the 1D wave equation. Boundary conditions. Fourier series method for the wave equation: Separation of variables for the homogeneous wave equation. Inhomogeneous wave equation. Resonance.
  • SUPPLEMENTARY MATERIAL: Finite difference solution of the wave equation.

Lecture 7: Fourier series and separation of variables: Laplace equation.

  • 2D Laplace equation. Laplace equation for a rectangular region and for a disk. Cualitative properties of the Laplace equation.
  • SUPPLEMENTARY MATERIAL: Poisson equation.
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