In this paper, we propose a novel numerical method for solving fractal--fractional differential equations with variable-order and exponential kernel. The method integrates the variable-order fractal derivative with the memory-preserving properties of the exponential kernel to better model complex and chaotic dynamical systems. A new predictor--corrector algorithm is developed to accommodate the nonlocal and nonlinear structure of fractal--fractional operators. The proposed scheme is shown to be accurate, stable, and computationally efficient through a detailed stability analysis. To validate the method, we apply it to several nonlinear systems, including classical and generalized chaotic models such as the Rucklidge, Chua--Hartley, Arnedo, and Wang--Sun systems. Numerical simulations demonstrate that the scheme captures intricate dynamic behavior and chaotic attractors under both constant and variable-order settings. The graphical results confirm that the approach is robust and adaptable for a wide range of applications involving memory-dependent and scale-invariant phenomena. This work contributes a reliable and flexible numerical tool for advancing the study of complex systems governed by variable-order fractional dynamics.

This study introduces a novel financial model based on fractal–fractional calculus with exponential decay kernels and Riemann-type derivatives, designed to capture memory-dependent processes and erratic time dynamics inherent in financial systems. The model effectively describes the dynamic interplay among three critical economic variables: price indices, investment demand and interest rates. Using a newly developed predictor–corrector numerical scheme tailored for fractal–fractional operators, we analyze the system’s local and global behavior, including equilibrium structure and Lyapunov spectra. The results reveal the presence of chaotic dynamics, driven by the fractional order µ and fractality index ν. Parametric analysis confirms that varying these parameters transitions the system from stability to chaos. This modeling approach offers a versatile and powerful framework for capturing nonlinear feedback, memory and multiscale characteristics of complex financial phenomena.

In this paper, the generalized iterative method has been successfully implemented for nding exact solutions of the non linear time fractional Klein-Gordon equations using com- posite fractional derivatives. Graphical representations also been studied. Illustrations related to Klein-Gordon equations using composite fractional derivative viz. Riemaan- Liouville fractional derivatives, Hilfer fractional derivatives, Caputo fractional derivatives has been discussed.