A Novel Approach to Solve Fractal--Fractional Differential Equations with Exponential Kernel and Time-Varying Fractal Dimension

A Novel Approach to Solve Fractal--Fractional Differential Equations with Exponential Kernel and Time-Varying Fractal Dimension | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 7 August 2025 V1 Latest version Share on A Novel Approach to Solve Fractal--Fractional Differential Equations with Exponential Kernel and Time-Varying Fractal Dimension Authors : Mahesh B. Nagpurkar 0009-0000-6869-6946 and Krunal Kachhia 0000-0001-8270-7162 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175455730.04434763/v1 481 views 199 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 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. Supplementary Material File (mahesh_4_numerical for variable ff.pdf) Download 1.79 MB Information & Authors Information Version history V1 Version 1 07 August 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords chaotic systems exponential kernel fractal–fractional derivatives numerical simulation predictor–corrector method variable-order differential equations Authors Affiliations Mahesh B. Nagpurkar 0009-0000-6869-6946 P D Patel Institute of Applied Sciences View all articles by this author Krunal Kachhia 0000-0001-8270-7162 [email protected] P D Patel Institute of Applied Sciences View all articles by this author Metrics & Citations Metrics Article Usage 481 views 199 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Mahesh B. Nagpurkar, Krunal Kachhia. A Novel Approach to Solve Fractal--Fractional Differential Equations with Exponential Kernel and Time-Varying Fractal Dimension. Authorea . 07 August 2025. 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