Grant: Discovery Grants Program - Individual

The theory of pseudoanalytic functions, developed independently by L. Bers and I. N. Vekua, became one of the important impulses for developing the general theory of elliptic systems. Roughly speaking, these functions are solutions of generalized Cauchy-Riemann equations Among other results, Bers obtained analogues of the Taylor series for pseudoanalytic functions. The long term goal of this research program is to find and evaluate solutions of some differential equations in terms of the Bers' theory of Taylor series for pseudoanalytic functions. This means to construct an infinite system of exact solutions explicitly. The first class of differential equation considered will be two-dimensional, but we also intent to consider the multidimensional cases. The other long term goal, tied to the first one, is to develop Bers theory for the bicomplex pseudoanalytic functions theory. This will allow us to find solutions of differential equations containing complex-valued functions. Finally, since the procedure is algorithmically simple, we intent to implement the procedure with the aid of a computer system of symbolic or numerical calculation. This research program has several applications in such areas as quantum mechanics, elastic scattering between composite systems (e.g. ions, atoms and molecules), electromagnetic wave propagation in disordered media, acoustic, helioseismology, fluid mechanics, geophysical prospecting as well as in medical imaging for the electrical impedance tomography.