


Politecnico di Torino  
Academic Year 2009/10  
02BQZDR, 02BQZAX, 02BQZAY, 02BQZDS Mathematical methods for engineering II 

1st degree and Bachelorlevel of the Bologna process in Mechanical Engineering  Vercelli 1st degree and Bachelorlevel of the Bologna process in Civil Engineering  Vercelli Master of sciencelevel of the Bologna process in Civil Engineering  Vercelli Espandi... 





Objectives of the course
The course provides basic knowledge about ordinary and partial differential equations and their applications.

Expected skills
Being able to integrate ordinary differential equations (ODE) with forcing terms. Computing eigenfunctions and eigenvalues of simple linear differential operators. Reducing a partial differential equation (PDE) to its canonic form. Solving the wave equation using D'Alembert's method. Deriving the equation for small vibrations of a string or membrane, the heat equation, the equations of momentum and energy balance for a perfect fluid. Solving simple linear PDE (wave equation, heat equation) with a variety of forcing terms, boundary conditions, and initial conditions, both in the transitory and stationary regime.

Prerequisites
Calculus I, II, Linear algebra, Mathematical methods I

Syllabus
Ordinary differential equations (ODE) with costant coefficients with a forcing term. Damped springmass system. Resonance. Principle of superposition. Conservative ODE's. Eigenvalues and eigenfunctions of linear differential operators. Linear partial differential equations (PDE). Variable change in a PDE: geometric meaning, transformation of derivative terms. Equations of characteristic curves. PDE classification: elliptic, parabolic, hyperbolic PDE's. Main examples: wave equation, heat equation, Laplace equation. Small, transverse vibrations of a string. Small, longitudinal vibrations of a bar. Divergence theorem. Small, transverse vibrations of a membrane. Hydrodynamic equations for perfect fluids: Euler equations, continuity equation, equation of state. Linearization of hydrodynamic equations close to equilibrium, wave equation. Boundary conditions for PDE's. Theorem of uniqueness for the equation of small vibrations. Small vibrations of an infinite string, D'Alembert's method. Small vibrations of a finite string, method of images. SturmLiouville problems: eigenvalues, eigenfunctions, distribution of energy on normal modes.

Laboratories and/or exercises
Exercises about the arguments of the course are discussed and solved in the class.

Bibliography
Haberman, 'Elementary applied partial differential equations with Fourier series and boundary value problems ', Prentice Hall , 1987
Tikhonov, Samarskii, 'Equations of Mathematical Physics', Dover, 1990 Kreyszig, 'Advanced Engineering Mathematics', Wiley, 1998 
Revisions / Exam
The exam consists of written exercises and questions about the arguments discussed in the course.

