Thesis supervisor: Arvet Pedas, professor, University of Tartu, Gennadi Vainikko, professor emeritus, University of Tartu
Opoonents: Andi Kivinukk, Lead Research Fellow, University of Tallinn, Igor Podlubny, professor, Košice University of Technology, Slovakia
The concept of fractional derivatives (i.e. derivatives of non-integer order) has interested scientists since at least the seventeenth century. For a long time considerations regarding these type of derivatives were purely theoretical in nature interesting mainly mathematicians. Recently however, fractional derivatives have seen a remarkable growth in popularity because of interesting new applications in physics, chemistry, mechanics, biology, economics, aerodynamics, etc.
Equations containing divergent integrals (i.e. integrals that diverge under normal conditions) arise in applications quite often. One of the main concepts under discussion has been the summability, i.e. finding the finite part, of divergent integrals.
The present thesis is devoted to the analysis of some classes of singular differential equations involving fractional order derivatives of an unknown function. Also finding the Hadamard finite part of a class of divergent integrals with a logarithmic factor is under consideration. The method for the analysis of singular fractional differential equations is based upon the theory of cordial Volterra integral operators.