Capturing complexities with composite operator and differential operators with non-singular kernel

Area of Science:

• Mathematics
• Applied Mathematics
• Dynamical Systems

Background:

• The composite operator, a concept from functional analysis, has real-world applications.
• Modeling chaotic phenomena and natural processes often involves advanced mathematical techniques like fractional calculus.
• Composing equations, analogous to composing functions, can potentially lead to novel dynamics and problem-solving approaches.

Purpose of the Study:

• To explore the creation of new chaotic attractors by composing existing ones using the composite operator.
• To investigate the resulting linear integro-differential equations, including classical and fractional forms in the Caputo-Fabrizio sense.
• To develop and validate a new numerical scheme for solving these novel equations.

Main Methods:

• Application of the composite operator to combine two distinct chaotic attractors.
• Derivation of linear integro-differential equations (classical and fractional) in the Caputo-Fabrizio sense.
• Development of a numerical scheme employing finite difference, Simpson, and Lagrange polynomial approximations.
• Validation of the numerical scheme through comparison with exact solutions for specific examples.

Main Results:

• Successful generation of new chaotic attractors through the composition of existing ones.
• Formulation of novel fractional integro-differential equations.
• Demonstration of a highly accurate numerical scheme with errors on the order of 10^-4.
• Validation of the proposed numerical method against exact solutions.

Conclusions:

• The composite operator provides a novel method for generating new chaotic attractors.
• The proposed numerical scheme accurately solves the newly derived fractional integro-differential equations.
• This approach offers a promising tool for modeling complex phenomena and solving real-world problems with enhanced dynamics.