Grant: Discovery Grants Program - Individual

This research program aims to bridge classical analysis and physics via Special Functions tools and the Asymptotic Iteration Method (AIM). Many phenomena, in nature, are modelled using linear difference and differential equations with polynomial coefficients. The polynomial solutions to these equations are the seeds of Special Functions. These functions have intensive uses in physics, chemistry, engineering, and the life and social sciences. The AIM is an iterative and constructive approach introduced to solve linear differential equations with coefficients that are not necessarily polynomials. It was introduced to overcome the complexity of dealing with eigenvalue problems of singular Hamiltonians in quantum mechanics (QM). This program is expected to develop two parallel theories: one based on a continuous version of AIM and the other based on discrete and q-difference versions of AIM. The continuous version aims to develop a framework of accessible and practical techniques to solve spectral problems in mathematical physics. In particular, the spectral problems associated with quasi-exactly solvable problems and Fuchsian differential equations. Another purpose of the continuous version is to gain a deeper understanding of the mathematical theory that underlies the exceptional orthogonal polynomials and emphasize their impact on physics applications. The importance of this research program comes from the applications of its findings in different areas of mathematical physics. The applications, for example, of Heun's differential equations (Fuchsian equation with four regular singular points) in general relativity, astrophysics, quantum, plasma, atomic, molecular, and nanophysics indicate the need for fresh and comprehensive approaches. Difference equations are essential for understanding many physical phenomena and have found applications in all fields of science. The q-Calculus generalizes many subjects, like hypergeometric series, complex analysis, and particle physics. The theory of q-difference equations and related q-functions has shown remarkable success in quantum group theory. The findings of the present program are expected to contribute substantially in these directions. The research program covers five research topics: 1) Analytical and computational study of the eigenvalue problems in QM using AIM. 2) Classify the solutions of the Fuchsian equations based on the equation singularity, developing a computational treatment for the spectral problem of the Heun equation. 3) To integrate my recent findings into the existing supersymmetry QM and Darboux transformations framework and improve on the current trends of these topics. 4) Deliver a complete theory of the discrete AIM to treat the q-difference and discrete difference equations and offer practical solutions in physics and related fields. 5) To study the Appell hypergeometric series Fj, j = 1, 2, 3, 4 and their uses in theoretical physics, especially in evaluating Feymann integrals