T. Telksnys “Construction of solitary solutions to differential equations via operator techniques” doctoral dissertation defence

Thesis Defense

Author, Institution: Tadas Telksnys, Kaunas University of Technology

Science area, field of science: Natural Sciences, Informatics N 009

Scientific Supervisor: Prof. Dr. Habil. Minvydas Ragulskis (Kaunas University of Technology, Natural Sciences, Informatics N 009).

Scientific Advisor: Prof. Dr. Zenonas Navickas (Kaunas University of Technology, Natural Sciences, Informatics N 009).

Dissertation Defence Board of Informatics Science Field:

Prof. Dr. Habil. Rimantas Barauskas (Kaunas University of Technology, Natural Sciences, Informatics N 009) – chairman;
Prof. Dr. Habil. Raimondas Čiegis (Vilnius Gediminas Technical University, Natural Sciences, Informatics N 009);
Prof. Dr. Habil. Remigijus Leipus (Vilnius University, Natural Sciences, Informatics N 009);
Prof. Dr. Gintaras Palubeckis (Kaunas University of Technology, Natural Sciences, Informatics N 009);
Prof. Dr. Miguel A.F. Sanjuan (University of Rey Juan Carlos, Natural Sciences, Informatics N 009).

The doctoral dissertation is available on the internet and at the libraries of Kaunas University of Technology (K. Donelaičio St. 20, Kaunas), Vytautas Magnus University (K. Donelaičio St. 58, Kaunas), and Vilnius Gediminas Technical University (Saulėtekio al. 11, Vilnius).

Annotation:

A novel mathematical and computational framework for the construction of solitary solutions to various types of nonlinear differential equations, including ordinary differential equations (ODE), partial differential equations (PDE), and fractional differential equations (FDE) is presented in this thesis. Solitary solutions have a plethora of unique physical and mathematical properties that influence the behavior of the considered system. The developed techniques are based on the application of symbolic computations to generalized differential operators. Such operators can be used to generate the coefficients of the series solution to differential equations, after which symbolic computations and linear recurring sequences are used to transform the series solution into the closed form solution. Necessary and sufficient conditions that differential equations must satisfy in order for the developed techniques to be applicable are discussed, as well the value of such techniques in applications. A real-world system of hepatitis C virus evolution is considered – it is shown that this system admits solitary solutions when the system parameters take biologically significant values. Fractional-order differential equations are also considered: it is demonstrated under what conditions these equations can be transformed into more complex ordinary differential equations and that the developed techniques can be used to construct solitary solutions.

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