Superdiffusive Wave Behavior, Analysis, and Control of Fractional in Time Schrödinger Equations

Authors

Luis Caicedo Torres, Florida International UniversityFollow

Document Type

Dissertation

Degree

Doctor of Philosophy (PhD)

Major/Program

Mathematical Sciences

First Advisor's Name

Ciprian Gal

First Advisor's Committee Title

Committee Chair

Second Advisor's Name

Svetlana Roudenko

Second Advisor's Committee Title

Committee Member

Third Advisor's Name

Gueo Grantcharov

Third Advisor's Committee Title

Committee Member

Fourth Advisor's Name

Wim Cosyn

Fourth Advisor's Committee Title

Committee Member

Keywords

partial differential equations, caputo derivative, nonlinear PDEs, schrodinger, wave equation, nonlocal PDEs, integro-differential equations, dirichlet forms, energy forms, mittag-leffler function

Date of Defense

5-22-2024

Abstract

The scientific literature has produced many examples of the phenomenon known as anomalous diffusion, where particles don’t behave/move normally. We seek to model anomalous quantum wave behavior in the Fractional-in-Time Superdiffusive Schrödinger Equation (FNLS). This equation will serve as a proper interpolation between the classical Schrödinger and Klein-Gordon/Wave equation by making use of nonlocal fractional time derivative operators of order between one and two. While interpolating between the Schrödinger Equation and the Wave Equation, the (FNLS) will exhibit wave-like behavior but enjoy none of the usual behaviors expected such as conserved quantities or an evolution group property. In this work, we introduce the fractional time derivative, the F-SNLS, expose the behavior and well-posedness theory for the inhomogenous and nonlinear problems. We then draw inspiration from classical optimal control and our well-posedness results to develop comprehensive strategies for the additive control of these superdiffusive problems.

Identifier

FIDC010987

ORCID

0000-0003-1289-2539

Recommended Citation

Caicedo Torres, Luis, "Superdiffusive Wave Behavior, Analysis, and Control of Fractional in Time Schrödinger Equations" (2024). FIU Electronic Theses and Dissertations. 5198.
https://digitalcommons.fiu.edu/etd/5198