23346 - Mathematical Physics II

Learning outcomes

At the end of the course the tudent is expected to get the basic knowledge of the classical linear and non linear PDE's  (with particular regard to diffusion and wave propagation), integral equations of Fredholm and  Volterra type, asymptotic and perturbation methods for special functions and differential equations,per funzioni speciali, green functions and Sturm-Liouville problems,  fractional calculus.. If time is allowed the student can be informed about stochastic (non-Gauusaian and non-Msarkovian) processes. The studrnt will be able -to solve in a analytical and/otr asymptotic way some integral and differential equations, - to develop simple mathematical models of diffusion and wave propagation, - use the technalities of fractional calculus to simulate  random walk models related to non-Gaussian and non-Markovian stochastic processes.

Course contents

Integral Equations  of  Fredholm, Volterra and  Abel type. Boundary value problems and Green functions . Integrsal Transforms:  Laplace, Fourier e Mellin.. Special Functions:Gamma, Beta, Bessel, Error, Exponential Integrals, Mittag-Leffler, Wright); Characteristic functiosn in probability   (stable and infinitely divisible distributions); Introduction to Fractional Calculus  (Integrals and Derivatives of non-integer order in R^+ e in R). Asymptotic and perturbation methods  (for integrals and differential equations) . PDE's of Mathematical Physics and their classification. Fundamental solutions of diffusion and wabve equations.  Linear dispesive waves (phase, group, signal velocities); Waves in viscoelastic media and surface waves in liquids. Non linear waves (KdV,  Burgers, Schroedinger).

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Readings/Bibliography

Lecture notes in English: Methods and Problem in Mathematical Physics

Teaching methods

Oral lectures with exercises. Compouter slides

Links to further information

http://www.fracalmo.org

Office hours

See the website of Francesco Mainardi