Chemical Engineering

Activity type

Attività formative affini o integrative

SSD

MAT/06

Year

N/D

Semester

N/D

CFU

3

Hours distribution

30 classroom hours

Lecturers

ANNA VIDOTTO

Activity type

Attività formative affini o integrative

SSD

MAT/05

Year

N/D

Semester

N/D

CFU

6

Hours distribution

60 classroom hours

Lecturers

ROBERTO CONTI

Upon completion of the course, students will have acquired the basic concepts (definitions, statements, and proofs) related to the theory of partial differential equations, including all preliminary calculus tools. They will be able to understand related topics by applying the acquired concepts, even in interdisciplinary contexts.

They will be able to independently apply the mathematical models learned to solve problems that may arise during the design or development of processes (Riemann problem, heat equation, Laplace equation, wave equation).

They will have consolidated their command of rigorous mathematical language and will have acquired the tools to self-assess and plan their personal workload.

Differential calculus for function of one real variable and for functions of several real variables

Module: MATHEMATICAL METHODS FOR CHEMICAL ENGINEERING I
Introduction to probability theory: events and probability measures, conditional probabilities and the Bayes theorem; random variables, probability distributions, joint distributions, functions of random variables. Expected value and second moments, covariance matrices.
Limit theorems and applications: central limit theorem, convergence in probability, law of large numbers.
Markov chains: Chapman-Kolmogorov equations and classification of states, limit theorems, transitions among classes, branching processes, applications of Markov chains; time-reversible Markov chains, semi-Markov processes; continuous time Markov chains, the Kolmogorov differential equations, limiting probabilities, time reversibility.
The Poisson process: interarrival and waiting time distribution, non-homogeneous Poisson process.



Module: MATHEMATICAL METHODS FOR CHEMICAL ENGINEERING II
Chapter 1 Review and basic tools.

Review of basic concepts in vector-space algebra: Algebra of vectors and its representation, scalar product, vector product, projections, exponential of a matrix.
Review of Vector Calculus: functions of several variables, integral theorems on curves, integral on surfaces, Gauss-Green Formula, Divergence and Stokes Theorems, Helmholtz decomposition.
Review of Ordinary differential equations (ODE): Existence and uniqueness of solutions for general ODE's of first and second order, phase space and qualitative behavior of solutions,
explicit solutions of first ODE's and of second order ODE's with constant coefficients.
Banach and Hilbert Spaces: Dual Space, Lebesgue measure and Lebesgue spaces, weak derivatives and Sobolev spaces ($H^1$), Motivation: Dirichlet Principle (Sketch).
Review of Fourier Transform: Plancherel Theorem, Inverse Fourier Transform and applications to ODE's (Sketch).
Review of Fourier series: Parseval Identity and convergence theorems.

Chapter 2: Partial Differential Equations.

First order PDE's: method of characteristics, shocks, entropy solutions.
Second order PDE's: Hyperbolic Equations: wave equations in dimension 1 and $N\geq 2$;
Parabolic Equations: Fundamental solution, Maximum principle, Cauchy problem in dimension 1;
Elliptic Equations: Harmonic functions, Green functions (space, disk, semispace...), minimization and weak formulation of elliptic problems, Lax--Milgram Theorem.

Module: MATHEMATICAL METHODS FOR CHEMICAL ENGINEERING I
Author: Sheldon M. Ross
Title: Stochastic Processes, 2nd Edition
Editor: Wiley



Module: MATHEMATICAL METHODS FOR CHEMICAL ENGINEERING II
Salsa, Sandro: Partial differential equations in action. From modelling to theory. Springer

Module: MATHEMATICAL METHODS FOR CHEMICAL ENGINEERING I
N/D
Module: MATHEMATICAL METHODS FOR CHEMICAL ENGINEERING II
N/D

Talk and chalk

Optional, but suggested

Traditional written exam at the end of the teaching period, which requires the solution of three exercises on topics discussed in the course, and oral exam (which includes a discussion of the written exam)

Solve a homogeneous wave equation on the real line with given initial conditions

Module: MATHEMATICAL METHODS FOR CHEMICAL ENGINEERING I

• Introduction to probability theory: probability spaces, events, random variables, distributions of random variables, first- and second-order moments.
  • Books: Sheldon M. Ross, "Stochastic Processes", 2nd edition, Wiley


• Limit theorems and applications: central limit theorem, convergence in probability, law of large numbers. 
  • Books: Sheldon M. Ross, "Stochastic Processes", 2nd edition, Wiley


• Markov chains: Chapman-Kolmogorov equations and classification of states, limit theorems, transitions among classes, branching processes, applications of Markov chains; time-reversible Markov chains, semi-Markov processes; continuous time Markov chains, the Kolmogorov differential equations, limiting probabilities, time reversibility.
  • Books: Sheldon M. Ross, "Stochastic Processes", 2nd edition, Wiley


• The Poisson process: interarrival and waiting time distribution, non-homogeneous Poisson process. 
  • Books: Sheldon M. Ross, "Stochastic Processes", 2nd edition, Wiley

• metric spaces, normed spaces, inner product spaces.
  • Books: Salsa, "PDE in action", Springer, 2012


• review of ordinary differential equations.
  • Books: Salsa, "PDE in action", Springer, 2012


• Fourier theory: series and transform.
  • Books: Salsa, "PDE in action", Springer, 2012


• Heat equation.
  • Books: Salsa, "PDE in action", Springer, 2012


• Laplace and Poisson equations.
  • Books: Salsa, "PDE in action", Springer, 2012


• Partial differential equations of the first order.
  • Books: Salsa, "PDE in action", Springer, 2012


• wave equation.
  • Books: Salsa, "PDE in action", Springer, 2012