Fractional differential inequalities have emerged as powerful tools for modeling and analyzing dynamic systems with fractional-order derivatives, offering a sophisticated framework to capture the complexities of real-world processes. Among the various analytical techniques, the comparison principle stands out as a fundamental approach in understanding the behavior of solutions to fractional differential inequalities. This study focuses on the development and analysis of comparison principles for some of the fractional differential inequalities involving the variable-order Caputo fractional derivative which is a generalization of the classical Caputo derivative that allows the order of differentiation to vary with respect to time or space. Such flexibility is important for modeling systems whose memory characteristics change over time or space. We formulate both weak and strong versions of the comparison principle with variable order Caputo fractional derivative. Our approach combines analytical techniques from fractional calculus and the theory of differential inequalities to establish some results. To have the applicability and relevance of our theoretical work, we provide an example demonstrating the effectiveness of the proposed comparison theorems. The findings of this paper not only contribute to the theoretical advancement of fractional differential inequalities with variable order but also applicable to systems where dynamic memory effects are prominent.
| Published in | Applied and Computational Mathematics (Volume 14, Issue 4) |
| DOI | 10.11648/j.acm.20251404.14 |
| Page(s) | 216-222 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Comparison Principle, Fractional Differential Inequalities, Variable Order Caputo Derivative, Volterra Fractional Integral Equation, Fractional Calculus
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APA Style
Patil, J. V., Janjal, M. M. (2025). Comparison Principle for Fractional Differential Inequalities with Variable Order Caputo Derivative. Applied and Computational Mathematics, 14(4), 216-222. https://doi.org/10.11648/j.acm.20251404.14
ACS Style
Patil, J. V.; Janjal, M. M. Comparison Principle for Fractional Differential Inequalities with Variable Order Caputo Derivative. Appl. Comput. Math. 2025, 14(4), 216-222. doi: 10.11648/j.acm.20251404.14
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Download@article{10.11648/j.acm.20251404.14,
author = {Jayashree Vinayakrao Patil and Monali Manikrao Janjal},
title = {Comparison Principle for Fractional Differential Inequalities with Variable Order Caputo Derivative
},
journal = {Applied and Computational Mathematics},
volume = {14},
number = {4},
pages = {216-222},
doi = {10.11648/j.acm.20251404.14},
url = {https://doi.org/10.11648/j.acm.20251404.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251404.14},
abstract = {Fractional differential inequalities have emerged as powerful tools for modeling and analyzing dynamic systems with fractional-order derivatives, offering a sophisticated framework to capture the complexities of real-world processes. Among the various analytical techniques, the comparison principle stands out as a fundamental approach in understanding the behavior of solutions to fractional differential inequalities. This study focuses on the development and analysis of comparison principles for some of the fractional differential inequalities involving the variable-order Caputo fractional derivative which is a generalization of the classical Caputo derivative that allows the order of differentiation to vary with respect to time or space. Such flexibility is important for modeling systems whose memory characteristics change over time or space. We formulate both weak and strong versions of the comparison principle with variable order Caputo fractional derivative. Our approach combines analytical techniques from fractional calculus and the theory of differential inequalities to establish some results. To have the applicability and relevance of our theoretical work, we provide an example demonstrating the effectiveness of the proposed comparison theorems. The findings of this paper not only contribute to the theoretical advancement of fractional differential inequalities with variable order but also applicable to systems where dynamic memory effects are prominent.
},
year = {2025}
}
TY - JOUR T1 - Comparison Principle for Fractional Differential Inequalities with Variable Order Caputo Derivative AU - Jayashree Vinayakrao Patil AU - Monali Manikrao Janjal Y1 - 2025/08/05 PY - 2025 N1 - https://doi.org/10.11648/j.acm.20251404.14 DO - 10.11648/j.acm.20251404.14 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 216 EP - 222 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20251404.14 AB - Fractional differential inequalities have emerged as powerful tools for modeling and analyzing dynamic systems with fractional-order derivatives, offering a sophisticated framework to capture the complexities of real-world processes. Among the various analytical techniques, the comparison principle stands out as a fundamental approach in understanding the behavior of solutions to fractional differential inequalities. This study focuses on the development and analysis of comparison principles for some of the fractional differential inequalities involving the variable-order Caputo fractional derivative which is a generalization of the classical Caputo derivative that allows the order of differentiation to vary with respect to time or space. Such flexibility is important for modeling systems whose memory characteristics change over time or space. We formulate both weak and strong versions of the comparison principle with variable order Caputo fractional derivative. Our approach combines analytical techniques from fractional calculus and the theory of differential inequalities to establish some results. To have the applicability and relevance of our theoretical work, we provide an example demonstrating the effectiveness of the proposed comparison theorems. The findings of this paper not only contribute to the theoretical advancement of fractional differential inequalities with variable order but also applicable to systems where dynamic memory effects are prominent. VL - 14 IS - 4 ER -
Department of Mathematics, Vasantrao Naik Mahavidyalaya, Chhatrapati Sambhajinagar, India
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Chhatrapati Sambhajinagar, India
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