Ongoing Research

      My principal research interests lie in the field of Applied Analysis and applied Mathematics: Ordinary and Functional Differential Equations, Numerical Methods, Inverse Problems and Boundary Value Problems.


Also, I am interested in Mathematical Physics, in particular in the theory of integrable systems. My research activity concerns symmetries, coherent structures (solitons), algebraic structures of integrable equations and their applications to physics and selected biophysical systems.


Most of my research accomplishments regard the analysis of nonlinear PDEs (both from a theoretical view point and for applications). My contribution in this area has been to classify special algebraic and rational solutions of integrable equations, to produce new examples of this class of nonlinear differential equations and to study the asymptotic behavior of their solutions.


      Variable-coefficient nonlinear evolution equations have attracted considerable attention to reflect the inhomogeneities of media, nonuniformities of boundaries, and external forces. My research is concerned with variable-coefficient PDEs which can be used to model shallow water waves, nonlinear optical pulses, currents in electrical networks, nerve pulses, waves in the atmosphere, etc. I have developed a simplified bilinear method to obtain the N-soliton solutions of such equations. Current work deals with the explicit functions which describe the evolution of the amplitude, phase and velocity of the waves, the dynamical behaviors for nonautonomous waves in a periodic distributed and dispersion decreasing systems, propagation characteristics and interactions among the waves.


 Whilst direct formulations consist of determining the effect of a given cause, in inverse formulations the situation is completely or partially reversed. The interest is into the research of inverse problems for partial differential equations governing phenomena in fluid flow, elasticity, acoustics, heat transfer, mechanics of aerosols, etc. Typical practical applications relate to flows in porous media, heat conduction in materials, thermal barrier coatings, heat exchangers, corrosion, etc. My future research plans are to investigate the existence, uniqueness and stability of the solution to the problem that mathematically models a physical phenomenon under investigation, and to develop new convergent, stable and robust algorithms for obtaining the desired solution. The analyses concern inverse boundary value problems, inverse initial value problems, parameter identification, inverse geometry and source determination problems.