I and II ODE. nth ODE. Existence and uniqueness Theorems to the Initial Value Problem (IVP). Qualitative analysis and stability. Functions spaces and Fourier series. Probability spaces. Discrete and continuous random variables of one and many dimensions.
1) G.Borgioli: Lecture Notes, on the webpage http://www.unifi.it/detmod/mdswitch.html
2) G. Borgioli - Modelli Matematici di Evoluzione ed Equazioni Differenziali - CELID, Torino (1996).
3) M.Bramanti, C.D.Pagani, S.Salsa - Matematica – Zanichelli
4) P. Baldi, Introduzione alla probabilità con elementi di statistica, McGraw-Hill.
5) G. Modica, L. Poggiolini - Note di Calcolo delle Probabilità - Pitagora
6) S.M. Ross, Probabilità e statistica per l'ingegneria e le scienze - Apogeo
7) R. Giuliano, L. Ladelli, P. Baldi, Laboratorio di statistica e probabilità, McGraw-Hill.
8) G.C.Barozzi - Matematica per l'Ingegneria dell'Informazione - Zanichelli
9) M.Marini - Metodi Matematici per lo studio delle Reti Elettriche - CEDAM
10) W. E. Boyce, R. C. DiPrima - Elementary Differential Equations and Boundary Value Problems,John Wiley & Sons, Inc.
11) W. Navidi, Probabilità e statistica per l'ingegneria e le scienze, McGraw-Hill.
On the webpage: http://www.unifi.it/detmod students can find:
- Notes and exercises.
- The lectures diary .
- Written tests of past exams.
Learning Objectives
The aim of the course is providing basic knowledge in dealing Ordinary Differential Equations, in Fourier Analysis and in Probability Theory. Students learn methods to solve EDO of I and II (linear) order. They are able to face simple qualitative analysis of ODE and Fourier series. Moreover students are able to solve problems in Probability, know main Probability distributions of discrete and continuous random variables of one and many dimensions.
Prerequisites
The subjects from the course of Analisi Matematica.
Teaching Methods
Lectures and Exercises
Further information
EXAMS SCHEDULE:
08/01/2013, 9:30 a.m., Hall 001 Centro Didattico Morgagni
29/01/2013, 9:30 a.m., Hall 001 Centro Didattico Morgagni
19/02/2013, 9:30 a.m., Hall 001 Centro Didattico Morgagni
11/06/2013, 9:30 a.m., Hall 005 Centro Didattico Morgagni
25/06/2013, 9:30 a.m., Hall 005 Centro Didattico Morgagni
09/07/2013, 9:30 a.m., Hall 005 Centro Didattico Morgagni
11/09/2012, 3:30 p.m., Hall 005 Centro Didattico Morgagni
REMARK:
Students must register for the written test on the webpage http://sol.unifi.it/prenot/prenot
Registration usually starts three weeks before the date of the test and stops two days before the test
The written test (3 hours of duration) consists in solving some exercises.
Students are admitted at the oral exam if the written test is positevely evaluated (a mark equal or greater than 18/30).
Oral exam is about theoretical subjects of the course.
EXAMINATION BOARD
G. BORGIOLI, L. POGGIOLINI, S. MATUCCI, M. MARINI, M. CECCHI, M. LANDUCCI, G. MODICA, G. FROSALI, M. MODUGNO, F. MUGELLI, M. SPADINI.
Type of Assessment
A written test consisting in solving some exercises and an oral exam.
Course program
1- ORDINARY DIFFERENTIAL EQUATIONS (ODE)
1.1-Definitions and terminology; the normal form; vector equation y'(x)=f(x,y(x)) as a general representation of n-th order ODE and systems of ODE; the Cauchy problem or Initial Value Problem (IVP); The existence and uniqueness Theorem.
1.2-First order ODE: solving methods for separable equations, homogeneous equations, linear equations, Bernoulli equations, exact equations and integrating factors.
1.3 Second order ODE: equations convertible to first order; linear equations with constant coefficients, homogeneous and nonhomogeneous: the undetermined coefficients method and the variation of constants method.
1.4-Linear equations in R^n: general solutions. Linear spaces of functions: the space generated by the solutions of linear homogeneous ODE.
1.5-Geometric interpretation and phase space analysis for second order ODE and systems of ODE.
1.6-Stability of solutions with respect to the initial conditions: Liapunov stability: Stability of equilibrium and of evolution equations;Asymptotic stability; stability properties of linear ODE and linear systems of ODE; the phase space; phase portrait for two-dimensions equations and systems: center, saddle point, spiral point, node; stability of almost linear systems; nonlinear ODE and systems: Liapunov’s second method and the Liapunov function; the phase portrait and the “energy-balance” plane.
1.7- Mathematical models in Mechanics and in Theory of Electric Circuits: the harmonic oscillator, the linearly damped harmonic oscillator (LDHO), the forced LDHO and the linear resonance;
Mathematical models in Theory of Populations: the Malthus and the logistic model, the prey-predator (Lotka-Volterra) model.
2-FOURIER SERIES (FS)
2.1- Functions Spaces equipped with inner product (Unitary Spaces); norm of a function. Schwartz, Minkowski and Bessel inequalities. Hilbert Space L^2. Space of the functions piecewise continuous on an interval. Trigonometrical polynomials and Fourier polynomials.
Real and complex approximating basis.
2.2 - Sequences of functions: pointwise convergence and uniform convergence. Series of functions, power series, radius of convergence, Abel's criterium.
2.3-Real and complex Fourier series; computation of the coefficients; FS of periodic functions and of functions defined on an interval; quadratic convergence; the Parseval equality; Dirichlet conditions for the pointwise convergence; pointwise convergence of the derivative and the integral series; even and odd functions and thei FS; the Gibbs phenomenon and the uniform convergence.
2.4-Applications: one-dimension Diffusion (Heat) equation and Wave equation; solution of an initial value and boundary (in a slab) problem.
3-PROBABILITY SPACES
Deterministic and random problems. Probability Spaces: definitions and elementary properties. Conditional probability, the Bayes formula, the total probabilities formula, independent events. The Bernoulli scheme (success-failure scheme). Combinatorial calculus: dispositions, permutations, combinations. Hypergeometric Law.
3.1-Discrete Models (success-failure scheme).
Random variables (RV) and partition functions. Discrete RV and their density. Examples: binomial density (success-failure scheme with replacement), success-failure scheme without replacement, geometric density, modified geometric density, Poisson distribution. Many-dimension discrete RV. Combined density and marginal densities. Example: two-dimension RV with the same marginal density but different combined probability. Multinomial density. Independent RV. Conditional density. Functions of RV and independence. Computations of densities: density of the sum of discrete independent and not independent RV, density of the sum of Bernoulli and Poisson independent RV. Partition function of the maximum and the minimum of two independent discrete RV. Mathematical hope (or average or expected value) of a discrete RV; compositions, linearity, mathematical hope of the product of independent RV, monotonicity. Examples: computation of the expected value of Bernoulli, binomial, hypergeometric, Poisson, modified geometric RV. K-order moments and k-order central moments, variance, covariance and their properties. Chebyshev inequality. Examples variance of Bernoulli, binomial, hypergeometric, Poisson, modified geometric RV. Correlation coefficient.
3.2-Continuous Models
Partition function and its properties. RV with continuous partition functions. Density of a continuous RV. Examples: uniform density on an interval, exponential density. Quantiles. Functions of continuous RV: partition function and density for f(X)=X^2 and f(X)=aX+b. Independent RV. Density of Z=max(X,Y), W=min(X,Y), when X,Y are RV of known density. Normal Laws, Gamma Laws and their main properties. Mathematical hope of RV with continuous density. Linearity and monotonicity properties. Mathematical hope of composition and product of independent RV. Moments and central moments, variance, covariance, Chebyshev inequality. Examples: mathematical hope and variance of the uniform distribution on an interval, normal RV, exponential RV.
3.3. - Sequences of Random variables
Weak and strong law of large numbers. The Central Limit Theorem