Problem solving for ordinary and partial differential equations. Manipulation of basic methods for solution (separation of variables, Fourier series, Fourier transform) and knowledge of qualitative properties of solutions (maximum principles and regularity properties). These topics will be presented in the perspective of their relevant applications to differential models in Chemical Engineering.

The teaching activity of this course includes lectures hours and training sessions. Expectation of achievements and abilities resulting from a successsful attendance of this course  are those rassociated  with the Dublin descriptors DD1, DD2 an DD3.

1. Partial differential equations - Knowledge of main examples of linear partial differential equations of Mathematical Physics, together with the related notions of boundary value problems and well posed problem.

2. Conservation Laws - Solving the initial value problem for the linear transport equation with source term, with/or without decay, exploiting the method of charactestics and explicit formulas of the solution. 

3. Diffusion - Knowledge of diffusion and heat conduction modeling and maximum principle. Solving the Cauchy problem for the heat equation in R^n by closed formulas. Solving the Cauchy problem for the heat equation in bounded domains by separation of variables. 

4. Transforms - Evaluation of Fourier transforms of functions, evaluation of Fourier series of periodic signals. Ability to apply transform-based methods for solving differential equations with initial and/or energy conditions. Ability to apply Fourier series tecniques to find periodic solutions of ordinary differential equations.

5. The Laplace Equation - Definition and main properties of harmonic functions. Solution of the Laplace equation in bounded domains by separation of variables. Solution of elliptic equations in R^n by using Fourier transform. The fundamental solution in the physical space: Newtonian potential. 

6. Waves and Vibrations - Solution of the one dimensional wave equation with D’Alembert formula. Knowledge of: types of waves, domain of dependence and range of influence of data for the wave equation in the physical space.

Lecture hours will allow students to know and understand the topics listed above (DD1), and to apply their knowledge focussing on the acquisition and critical use of basic tools (DD2, DD3).

Training sessions will allow studentsto become familiar with the theoretical concepts presented during lecture hours (DD1, DD2, DD3),

and help developing ability to make judgements (DD3).

1. Introduction to partial differential equations - Mathematical modeling. Examples of partial differential equations. Well posed problems. Boundary value problems: Dirichlet and Neuman boundary conditions. Initial value problems. Classification of second order linear PDE. 

2. Conservation Laws - Pollution in a channel. Linear transport equation. Distributed source. Decay and localized source. Characteristics. Inflow and outflow. First order quasi-linear equations. 

3. Diffusion - The diffusion equation. Heat conduction. Well posed problems. Separation of variables. Maximum principle and uniqueness. The fundamental solution. The Dirac distribution. The Cauchy problem for the heat equation. Energy estimates. An example of nonlinear diffusion: the porous medium equation.

4. Transforms - Fourier transform. Fourier series of periodic signals. Function spaces: C^k, L^1, L^2. Distributions. Convolution. Transform-based methods for solving differential equations with initial and/or energy conditions.

5. The Laplace Equation - Well posed problems and uniqueness. Harmonic functions and their properties. Separation of variables. Eigenvalue problems. The fundamental solution in the physical space: Newtonian potential. Solution of Laplace equation in R^n.

6. Waves and Vibrations - Types of waves. Transverse waves in a string. The one dimensional wave equation. D’Alembert formula. Domain of dependence and range of influence. Huygens principle. Energy conservation.

Students are supposed to know basic notions concerning multiple integrals, integrals on surfaces, Divergence Theorem, Fourier series of periodic functions, functions of complex variable, Cauchy-Lipschitz Theorem, linear second order ordinary differential equations, Euler's differential equation. 

The final (written) exam consists in solving exercises and answering questions related with the topics of the course,

in order to assess the knowledge of theorems and ability to exploit the mathematical tools presented in the frontal lessons:

all exam tasks will have prescribed scores: basic exercises are aimed at evaluating DD1, while advanced exercises and the theoretical question

are aimed at evaluating DD3. 

Additional score can be achieved by a (facultative) theory question and a (facultative) final drill in MATLAB laboratory.

Precisely the exam will check the acquired skills concerning:

1- Knowledge of main examples of linear partial differential equations of Mathematical Physics, together with the related notions of boundary value problems and well posed problem.

2 - Solving the initial value problem for the linear transport equation with source term, with/or without decay, exploiting the method of charactestics and explicit formulas of the solution. 

3 - Knowledge of diffusion and heat conduction modeling and maximum principle. Solving the Cauchy problem for the heat equation in R^n by closed formulas. Solving the Cauchy problem for the heat equation in bounded domains by separation of variables. 

4 - Evaluation of Fourier transforms of functions, evaluation of Fourier series of periodic signals. Ability to apply transform-based methods for solving differential equations with initial and/or energy conditions. Ability to apply Fourier series tecniques to find periodic solutions of ordinary differential equations.

5 -Definition and main properties of harmonic functions. Solution of the Laplace equation in bounded domains by separation of variables. Solution of elliptic equations in R^n by using Fourier transform. The fundamental solution in the physical space: Newtonian potential. 

6 - Solution of the one dimensional wave equation with D’Alembert formula. Knowledge of: types of waves, domain of dependence and range of influence of data for the wave equation in the physical space. 

English text is also available as an E-book. A printed version in Italian language is available too.