B002376 - DIFFERENTIAL EQUATIONS

Linear ODE of I and II order, with constant and non-constant coefficients. First order quasi-linear EDP and II order linear EDP with constant coefficients.
Separation variables method.
Fourier Series and Transform.

Lecture Notes written by the lecturer. They can be found at:
http://www2.de.unifi.it/anum/zecca/ED/index.html

Modelling physical problems (potential, wave and heat transport) and their solution in the simplest cases of simple domains.

Courses of Analysis I & II and Geometry.

Teorical lessons and calssroom exercises

Written exam

I order ODE
• Separation of variables
• Classification of ODE's and vector fields;
• First order linear equations;
• Exact equations;
• Numerical solutions: Euler method.
II order ODE
• Linear Equations;
• Order reduction method;
• Homogeneous equations with constant coefficients;
• Non-homogeneous equations with constant coefficients;
• Euler'equations;
• Harmonic motion. Forced and damped equations.
Series solutions for ODE. Special functions
• Singular points for linear II order ODE';
• Frobenius' method;
• Special cases;
• Bessel Functions and Legendre polynomials
• .
Boundary value problems for ODE's
• Spinning rope and bar;
• Curvature of a column under coaxial load;
• Orthogonality of characteristic functions;
• Series of functions w.r.t. orthogonal families of functions;
• Boundary value problems for non-homogeneous equations.
PDE's
• Quasi-linear first order equations;
• Characteristic curves for I order equations;
• Linear and quasi-linear equations of II order;
• Linear equation of II order with constant coefficients, their characterization;
• Separation variables method;
• One dimensional heat equation;
• One dimensional wave equation;
• Potential equation;
• two and three dimensional heat and wave equations;
• Duhamel integral;
• Non homogeneous boundary conditions. Variation parameter method.