Conformable fractional Dirac system on time scales

Area of Science:

• Spectral theory
• Fractional calculus
• Time scale calculus

Background:

• Classical Dirac systems have well-established spectral properties.
• Extending these properties to fractional and time scale domains is an active research area.
• Conformable fractional calculus offers a novel approach to generalizing differential equations.

Purpose of the Study:

• To investigate the conformable fractional (CF) Dirac system with separated boundary conditions on arbitrary time scales.
• To extend fundamental spectral properties of the classical Dirac system to the CF context.
• To derive asymptotic estimates for eigenfunctions within the CF Dirac eigenvalue problem.

Main Methods:

• Analysis of the conformable fractional Dirac system on time scales.
• Extension of spectral properties from classical to CF Dirac systems.
• Derivation of asymptotic estimates for eigenfunctions using established mathematical techniques.

Main Results:

• Established spectral properties for the CF Dirac system on time scales.
• Obtained asymptotic estimates for the eigenfunctions of the CF Dirac eigenvalue problem.
• Developed a constructive procedure for solving the CF Dirac eigenvalue problem.

Conclusions:

• The study successfully extends spectral theory to conformable fractional Dirac systems on time scales.
• The derived asymptotic estimates provide valuable insights into eigenfunction behavior.
• This work consolidates the connection between fractional calculus, time scale calculus, and spectral theory.