Numerical treatment of nonlinear partial differential equations of integrable type
Developing a new method to compute the scattering data of the Zakharov-Shabat system associated to initial value problem for the nonlinear Schrödinger equation which is considered to be a prototype of the so-called nonlinear differential equations of integrable type.
In the publication L. Fermo, C. van der Mee and S. Seatzu, Emerging problems in approximation theory for the numerical solution of the nonlinear Schrödinger equation, Publication de l'Institut Mathematique, 96(110):125-141, 2014 a first numerical method has been developed in the reflectionless case.
The method has been subsequently extended to the case where the reflection coefficient does not vanish in the paper L. Fermo, C. van der Mee and S. Seatzu, Scattering data computation for the Zakharov-Shabat system, Calcolo 53(3):487-520, 2016.
Image to the rigth from the website http://www.douglasbaldwin.com/nl-waves.html: Nuevo Vallarta, Mexico by Mark Ablowitz.
Numerical treatment of boundary integral equations
Investigating new methods for the numerical solution of boundary problems (ex: Dirichlet problem, Neumann problem, mixed problem etc.) on special domains which produce well-conditioned linear systems. Prove convergence and stability results and assess the effectiveness of the method by means of numerical results. My latest publications on this topic deal with planar domains with corners:
- L. Fermo and C. Laurita, A Nyström method for a boundary integral equation related to the Dirichlet problem on domain with corners, Numerische Mathematik 130 (1), 35-71 2015;
- L. Fermo and C. Laurita, On the numerical solution of a boundary integral equation for the exterior Neumann problem on domains with corners, Applied Numerical Mathematics 94, 179-200 2015.