# Physical Modeling

Professor | Danani Andrea Pezzuto Simone Favino Marco |

Course program | MSc |

Year | 1 |

Semester | Fall |

Category | Fundamental |

ECTS | 6 |

Academic year | 2016/2017 |

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**Objectives**

The goal of the course is to provide the student a working knowledge in basic mathematical concepts and show how these tools are used in applied problems. It is divided in two main parts.

Part I is devoted to the study of ordinary differential equations (ODEs). Initially, the Cauchy problem and the study of its stability is introduced. Then, the main focus is on classical and modern discretisation methods for the simulation of ODEs and system of ODEs: one-step methods, multistep methods, Runge-Kutta, and spectral deferred correction methods.

In part II, after a brief recap about basic linear algebra concepts (vector spaces, scalar product, norm, orthogonal projection), the focus will be on the theory of transforms: Fourier series and integrals, Discrete and Fast Fourier Transform, Laplace, Zeta and Wavelet transforms.

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**Contents**

### Part I (Dr. Pezzuto and Dr. Favino)

Cauchy problem

Basic single step methods

Stability, consistency and convergence

Absolute stability

Runge-Kutta

Multistep methods

Spectral deferred correction methods

### Part II (Prof. Danani)

Linear Algebra

Signal theory

Fourier series and transforms

Interpolation

DFT and FFT

Laplace Transform

Zeta Transform

Wavelets

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**References**

- Class handouts (slides)
- Arieh Iserles, A first course in The numerical Analysis of Differential Equations
- Ernst Hairer, Syvert Paul Nørsett, Gerhard Wanner, Solving Ordinary Differential Equations
- P. Lancaster and K. Salkauskas, Transform Methods in Applied Mathematics: An Introduction, Wiley-Interscience, 1996.