Fractional derivatives, dimensions, and geometric interpretation: An answer to your worries
In this study, I look into the study of fractional calculus in the mathematical modeling of nonlinear complex systems. I began by analyzing the dimensional aspects of fractional derivatives, in particular, the Caputo-Fabrizio and Atangana-Baleanu derivatives, and demonstrated that the fractional ord...
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| Format: | Article |
| Language: | English |
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AIMS Press
2025-02-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2025119 |
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| Summary: | In this study, I look into the study of fractional calculus in the mathematical modeling of nonlinear complex systems. I began by analyzing the dimensional aspects of fractional derivatives, in particular, the Caputo-Fabrizio and Atangana-Baleanu derivatives, and demonstrated that the fractional order can be interpreted as a distinct temporal dimension. Provided a thorough study of the mathematical kernels describing such derivatives, emphasizing their contrasting memory effects and the manner in which they impact the dynamics of the system. In particular, the Atangana-Baleanu derivative has the Mittag-Leffler kernel, which has smoother decay and is long-range compared to any power-law kernel or exponential kernels, with short-memory effects. Starting only from the energy and entropy responses related to each kernel, we showed how the fractional derivatives provide a more comprehensive description of the energy lost and the entropy gained. The kernels had stunning convolution properties that were analyzed to understand how the history and external heat effect the system dynamics via the kernel dependencies. As a new extension of artificial intelligence against DML, a novel method utilizing fractional calculus was proposed in LED lifespan modeling with varying ambient conditions. The model has been developed using several fractional kernels so that thermal cycling, humidity, mechanical stress, and electrical stress can be used to analyze and capture the degradation of the LED. The icing on the cake is that these fractional kernels inherently include memory effects and can be used to realistically and tailorably predict LED lifetime in a variety of environments. This study illustrates the possibility of using fractional derivatives framework to go beyond delivering physical understanding regarding time dependency of degradation processes. |
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| ISSN: | 2473-6988 |