Dual binary-order fractional operators and transforms

Abstract The paper introduces fractional dual binary-order operators as well as associated weighted operators alongside novel fractional transforms to address complex mathematical challenges in mathematics, physics and engineering. The framework establishes dual binary-order transforms with corresponding inverse operations, enabling bidirectional analysis of systems governed by fractional calculus. A weighted operator is formulated as a hybrid concept, blending functions with their dual binary-order fractional derivatives and integrals to unify analytical approaches. The methodology demonstrates versatility through applications in solving fractional differential equations (particularly inhomogeneous types), modelling fractional densities for statistical distributions, and enhancing anomaly detection in data-driven systems. Additionally, the operators enable information encryption/decryption by leveraging chaotic waveforms derived from the principles presented. This unified approach bridges, amongst other things, theoretical constructs with practical implementations in cryptography, signal processing, and statistical modelling.