 ## NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS

Teachers

##### Rossana Vermiglio
6
Second Period
###### Course Type
Supplementary
Prerequisites. The contents of the course “Approximation theory and practice”
Teaching Methods. Frontal lectures and laboratory activities in Matlab to solve simple case studies. “English-taught course” when there are international students.
Verification of Learning. Oral examination on the fundamental thoerems with proofs and computational propseriies of the algorithms presented in the course.. The students can agree with the teacher to substitute a part of the program with an in-depth study on a specific topic also related to the numerical modeling of partial differential equations or integral equations. The final mark will take into account the clarity and the accuracy of the presentation.
More Information. The course could also be offered in English and / or with notes in English
##### Objectives
The course provides the students with the basic numerical methods to solve initial value probelms and boundary value problems for ordinary differential equations. Having in mind in particular the application to finantial mathematics and to structured epidemiology models, the course introduces some numerical techniques for partial differential equations. Case studies in MATLAB, a mathematical software used in scientific research and working enviroment, will be used to experimentally analyze the theoretical properties and the performance of numerical methods. Such skills aim to develop the maturity of judgment and the critical sense.

The acquired skills allow to continue studying numerical analysis at advanced level, and providemathematical tools useful in various applications fields. Infact differential models real phenomena which arise in physics, engineering, medicine, biology, economics.

##### Contents
The first part of the course is devoted to the general techiques for solving Initial Value Problems of Ordinary Differential Equations (ODEs), like Runge-Kutta (RK) and linear multistep (LM) methods.

The RK methods: consistency and convergence. Stiff Problems. Stability analysis: A-stability, AN-stability, algebraic stability and BN-stability. Implementation aspects of RK-methods.

Linear Multistep (LM) methods: consistency, zero-stability and convergence. Predictor-Corrector methods. Stability analysis. Implementation aspects of LM-methods.

In the second part the course treats Boundary Value Problems for Ordinary Differential Equations. The shooting method. Elementary finite difference methods. Collocation methods. Weak formulation. Lax- Milgram Theorem. Sobolev space. Galerkin finite-element method. Strang’s lemma.

In the last part an overview of the numerical methods for Partial Differential Equations (PDEs). The course includes laboratory activities with MATLAB and the numerical solution of a case study, for instance the Black-Scholes model, which has many applications in mathematical finance, structured models whcih arise in ecology and epidemiology.

##### Texts
Teacher’s note available on-line on e-learning and the support of (part of the) books

P. Deufhard, F. Bornemann Scientific Computing with Ordinary Differential Equations. Springer , New York 2002.

E. Hairer, S.P. Norsett, G. Wanner: Solving Ordinary Differential Equations I Sringer, Berlin 1987.

E. Hairer, G. Wanner: Solving Ordinary Differential Equations II Springer, Berlin 1996

E. Hairer, C. Lubich, G. Wanner: Geometric Numerical Integration Springer, Berlin 2001.

J.D. Lambert: Numerical methods for ordinary differential equations. Wiley 1991.

A. Quarteroni Modellistica Numerica per problemi differenziali. Springer, New York 2000.