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.
First element of calculus of variations
Lecture Notes written by the lecturer. They can be found at:
http://www. math.unifi.it/users/zecca/ED/index.html
Learning Objectives
Modelling physical problems (potential, wave and heat transport) and their solution in the simplest cases of simple domains.
Elements of calculus of variations.
Prerequisites
Courses of Analysis I & II and Geometry.
Teaching Methods
Teorical lessons and calssroom exercises
Type of Assessment
Written exam
Course program
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.
Elements of caluculus of variations: Eulero equation, variations with parameters.