Numerical treatment of nonlinear partial differential equations of integrable type
Developing numerical methods to compute the scattering data of the Zakharov-Shabat system associated to initial value problem for the nonlinear differential equations of integrable type.
In the publication L. Fermo, C. van der Mee and S. Seatzu. Scattering data computation for the Zakharov-Shabat system with non smooth potentials, Applied Numerical Mathamatics , 116 195-203, 2017 a numerical method has been developed in the case when we handle with the nonlinear Schrödinger equation and nonsmooth initial potential.
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.
Numerical methods for singular integral equations
Developing new numerical methods to approximate the solution of integral equations which are classified as "singular" in virtue of the pathologies of the kernel function. My last publication on the topic deals with the Cauchy integral equation on the square:
L. Fermo, M.G. Russo and G. Serafini. Numerical Methods for Cauchy Bisingular Integral Equations of the First Kind on the square, Journal of Scientific Computing , First Online: 3 October 2018, in press.
Numerical treatment of integral models applied to geophisics
Developing efficient algorithms to treat from a numerical point of view integral models which are widely used in the electromagnetic induction techniques.
In the publication P. Diaz de Alba, L. Fermo, van der Mee and G. Rodriguez. Recovering the electrical conductivity of the soil via a linear integral model, Journal of Computational and Applied Mathematics 352, 132-145, 2019 different collocation methods combined with regularization techniques are developed to treat a linear model consisting of two integral equations of the first kind.