Speciality:
01.01.01; 01.01.02 (Calculus; Differential equations)
Birth date:
20.07.1948
Keywords: fractional calculus; singular integral equations; nonlinear integral equations; integral transforms with special function kernels; H-function; Mittag-Leffelr type functions; integral operators on special function spaces; partial differential equations of mixed type; differential equations of fractional order.
Subject:
Firstly in world monographic literature classical and modern results in the theory of fractional calulus of function of one and several variables are presented, and applications are given to integral and differential equations and to function theory. Necessary and suficient conditions for the solvability of some classes of singular integral equations of the first kind with weak singularities and differential equations of fractional order are established, and their solutions in closed form are given. The theory of integral transforms with H-function kernels, including the boundedness, the representation, the range and the inversion, is constructed in weighted spaces of p-summable functions, and applicatuions are given to obtain the corresponding results for integral transforms with functions of hypergeometric and bessel type kernels. Compositional and asymptotic properties are investigated for the H-function, for Mittag-Leffler type functions and for functions generalizing the modified Bessel function of third kind.
Main publications:
Samko S. G., Kilbas A. A., Marichev O. I. Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publishers, Yveron, 1993. 1012 p.
Kilbas A. A., Saigo M. On solution of a nonlinear Abel–Volterra integral equation // Journal of Mathematical Analysis and Applications. 1999. V. 229, no. 1. P. 41–60.
Glaeske H.-J., Kilbas A. A., Saigo M. Modified Bessel type integral transform and its compositions with fractional calculus operators on spaces $F_{p,\mu }$ and $F_{p,\mu}'$ // Journal of Computational and Applied Mathematics. 2000. V. 118, no. 1–2. P. 151–168.
Kilbas A. A., Trujillo J. J. Differential equations of fractional order: methods, results and problems - I // Applicable Analysis. 2001. V. 78, no. 1–2. P. 153–192.
