- home
- about
- founder & publisher
- editorial boards
- advisory board
- for authors
- call for papers
- archive
- online first
- editorial policy
- participation fee
- contacts
THERMAL SCIENCE
International Scientific Journal
Find this paper on
ANALYTICAL SOLUTIONS FOR A CLASS OF FRACTAL KORTEWEG-DE VRIES TYPE EQUATION
ABSTRACT
The present study focuses on a class of Korteweg-de Vries (KdV)-type equations involving the time-space fractal scaling law derivative. The objective of this investigation is to explore their exact analytical solutions. The employment of fractal scaling law derivatives, calculus theory, and Jacobi elliptic functions, in conjunction with variable substitutions and equation transformations, facilitates the attainment of precise analytical solutions for this equation type under various conditions. The findings of the research endeavor have yielded two notable outcomes. Firstly, they have augmented the solution system for KdV-type equations. Secondly, they have furnished an effective method reference for solving other non-linear fractal partial differential equations. These contributions are instrumental in fostering the advancement of the application of fractal calculus in the domain of mathematical physics.
KEYWORDS
PAPER SUBMITTED: 2024-10-08
PAPER REVISED: 2025-05-02
PAPER ACCEPTED: 2025-05-05
PUBLISHED ONLINE: 2026-04-12
DOI REFERENCE: https://doi.org/10.2298/TSCI2602879D
CITATION EXPORT: view in browser or download as text file
REFERENCES
[1] Kato, T., On the Korteweg-de Vries Equation, Manuscripta Math., 28 (1979), Jan., pp. 89-99
[2] Miles, J. W., The Korteweg-de Vries Equation: A Historical Essay, Journal of Fluid Mechanics, 106 (1981), May, pp. 131-147
[3] Wazwaz, A.-M., The KdV Equation, Handbook of Differential Equations: Evolutionary Equations, 4 (2008), 1, pp. 485-568
[4] Seadawy, A. R., New Exact Solutions for the KdV Equation with Higher Order Non-Linearity by Using the Variational Method, Computers and Mathematics with Applications, 10 (2011), 62, pp. 3741-3755
[5] Guan, Y. Z., et al., Variational Formulations for a Coupled Fractal - Fractional KdV System, Fractals, 32 (2024), 24500543
[6] Shang, C. H., Yi, H. A., Solutions of the KdV - mKdV Equations Arising in Non - Linear Elastic Rods under Fractal Dimension Thermal Science, 28 (2024), 3A, pp. 2125-2133
[7] Sun, J. S., Fractal Solitary Waves of the (3+1)-Dimensional Fractal Modified KdV - Zakharov - Kuz-netsov, Thermal Science, 28 (2024), 3A, 1967-1974
[8] He, J.-H., Wu, X. H., Exp - Function Method for Non-Linear Wave Equations, Chaos Solitons & Frac-tals, 30 (2006), 3, pp. 700-708
[9] Sajid, N., et al., Implementation of the Exp - Function Approach for the Solution of KdV Equation with Dual Power Law Non-Linearity, Computational & Applied Mathematics, 41 (2022), 338
[10] Abdalla, M. Y. Y., et al., Multiple Soliton Solutions of the Model mKdV with Time - Dependent Variable Coefficients, Thermal Science, 28 (2024), 6B, pp. 5029-5036
[11] He, J.-H, A Short Review on Review on Analytical Methods for a Fully Fourth-Order Nonlinear Integral Boundary Value Problem with Fractal Derivatives, International Journal of Numerical Methods for Heat and Fluid Flow, 11 (2020), 30, pp. 4933-4943
[12] Manddelbrot, B., How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, Science, 156 (1967), 3775, pp. 636-638
[13] Yang, X.-J., New Insight into the Fourier-Like and Darcy-Like Models in Porous Medium, Thermal Science, 24 (2020), 6A, pp. 3847-3858
[14] Yang, X.-J., The Scaling-Law Flows: An Attempt at Scaling-Law Vector Calculus, Fractals, 32 (2024), 2340126
[15] Grunau, D. W., et al., Domain Growth, Wetting, and Scaling in Porous Media, Physical Review Letters, 71 (1993), 4198
[16] Hariti, Y. et al., Scaling Law of Permeability and Porosity for Fluid Transport Phenomena in Porous PCM Media, Journal of Applied and Computational Mechanics, 7 (2021), 1, pp. 84-92
[17] Werner J. P., et al., Crisis and Stochastic Resonance in Shinriki's Circuit, Physica D: Nonlinear Phenomena, 237 (2008), 6, pp. 859-865
[18] Paradisi, P., Allegrini, P., Scaling Law of Diffusivity Generated by a Noisy Telegraph Signal with Fractal Intermittency, Chaos, Solitons and Fractals, 81 (2015), Dec., pp. 451-462
[19] Rafe, A., Razavi, S. M. A., Scaling Law, Fractal Analysis and Rheological Characteristics of Physical Gels Cross-Linked with Sodium Trimetaphosphate, Food Hydrocolloids, 62 (2017), Jan., pp. 58-65
[20] Yuan, C., et al., The Emergence of Scaling Law, Fractal Patterns and Small-World in Wireless Networks, IEEE Access, 5 (2017), Jan., pp. 3121-3130
[21] Meyer, K. R., Jacobi Elliptic Functions from a Dynamical Systems Point of View, The American Math-ematical Monthly, 108 (2001), 8, pp. 729-737
[22] Chouikha, A. R., On Properties of Elliptic Jacobi Functions and Applications, Journal of Nonlinear Mathematical Physics, 12 (2005), 2, pp. 162-169
[23] Ain, Q. T., et al., ABC Fractional Derivative for the Alcohol Drinking Model Using Two-Scale Fractal Dimension, Complexity, 2022 (2022), 8531858
[24] Zhang, Y. R., et al., Fast and Accurate Population Forecasting with Two-Scale Fractal Population Dynamics and its Application to Population Economics, Fractals, 32 (2024), 2450082
[25] He, J.-H., et al., A Fractal-Based Approach to the Mechanical Properties of Recycled Aggregate Concretes, Facta Universitatis, Series: Mechanical Engineering, 22 (2024), 2, pp. 329-342
[26] He, J.-H., Liu, C., Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete's Strength, Facta Universitatis Series: Mechanical Engineering, 21 (2023), 1, pp. 137-150
[27] Liu, Y. P., et al., Clover-Inspired Fractal Architectures: Innovations in Flexible Folding Skins for Sustainable Buildings, Fractals, 33 (2025), 25500410
PDF VERSION [DOWNLOAD]
© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence