A new fractal model of the convective-radiative fins with temperature-dependent thermal conductivity

Abstract

In this paper, the convective-radiative fins of rectangular profile with temperature-dependent thermal conductivity are considered. By studying the conventional heat transfer equation, its modified fractal form, which can describe the problem in the porous medium, is presented based on He?s fractal derivative for the first time. The fractal two-scale transform method together with the Taylor series are applied to deal with fractal model, and an analytical approximate solution is obtained. The impact of the different fractal orders on the thermal behavior of the fins is also elaborated in detail. In addition, a comparison between our solution and the existing one is given to prove the correctness of the proposed method, which shows that the proposed method is easy but effective, and are expected to shed a bright light on practical applications of fractal calculus.

The fractal active low-pass filter within the local fractional derivative on the Cantor set

“…Thus, the fractional calculus has been applied in many fields, and attracted more and more attention. Recently, the local fractional derivative (LFD) (Yang, 2011; Yang et al , 2015a, 2015b) is a new definition of the fractional derivative and has been successfully applied to describe many complex phenomena involving in the wave (Yang et al , 2019), diffusion (Yang et al , 2017a, 2017b), physics (Zhang and Yang, 2016; Wang and Si, 2023; Yang, 2017; Wang and Shi, 2023a, 2023b; Wang, 2023a, 2023b) and so on (Yang et al , 2014; Yang et al , 2016). As an important field, the fractional calculus has been used widely to model the fractal electrical systems such as the fractal LC electric circuit (Yang et al , 2017a, 2017b), fractal RC circuit (Zhao et al , 2017), fractal RL high-pass filter (Wang and Li, 2020), fractal high-pass filter (Wang, 2020) and so on (Banchuin, 2023; Banchuin, 2022; Wang et al , 2020).…”

“…The fractional differential equations are more general than integral differential equations, and can more accurately describe objective laws and the essence of things under the extreme conditions such as the unsmooth boundary (He et al , 2021a, 2021b; Wang, 2022a, 2022b, 2022c; Wang et al ., 2022; Wang, 2022a, 2022b, 2022c), microgravity space (He, 2020; He et al , 2021a, 2021b), porous media (Xiao et al , 2019; Wang, 2023a, 2023b; Xiao et al , 2021; Wang and Shi, 2023a, 2023b) and so on (Atangana, 2016; Wang and Zhu, 2022; Ghanbari and Abdon, 2020; Wang, 2022a, 2022b, 2022c) than integral differential equations. In addition, the fractional differential equations have greatly enriched the content of mathematical theory and penetrated into many fields of natural science.…”

The pulse narrowing nonlinear transmission lines model within the local fractional calculus on the Cantor sets

“…Recently, the fractal and fractional calculus have received considerable attention in a number of different fields because they can model many complex phenomena occurring in extreme conditions such as the unsmooth boundary (He et al , 2021a; Wang, 2022; He et al , 2021b; Wang, 2023a), microgravity space (He, 2020; He et al , 2021c), porous media (Xiao et al , 2019; Wang, 2023b; Xiao, 2021; Wang, 2023c) and so on (Atangana, 2016; Wang et al , 2023a). Among the different fractional derivatives, the local fractional derivatives (LFDs) play an increasingly important role in the electrical and electronic engineering involving in the fractal LC-electric circuit (Yang et al , 2017a), fractal RC circuit (Zhao et al , 2017), fractal filter (Wang and Shi, 2023) and so on (Banchuin, 2023; Banchuin, 2022; Sikora and Pawłowski, 2018). Inspired and encouraged by the research results in electrical and electronic engineering, here in this paper, we aim to develop a new fractional pulse narrowing nonlinear transmission lines model based on the LFD for the first time and present a novel method to seek for the nondifferentiable (ND) exact solutions.…”

On the zero state-response of the ℑ-order R-C circuit within the local fractional calculus

“…In recent decades, researchers pay more and more attention to the fractional calculus since it can accurately describe some strange phenomena in many scientific research fields, such as the porous media (Xiao et al , 2019; Wang, 2023a; Xiao et al , 2021; Wang and Shi, 2023a), non-smooth boundary (He et al , 2021; Wang et al , 2023a; Wang, 2022a), image analysis (Ghamisi et al , 2012), control (Ladaci and Charef, 2006), diffusion (Ammi et al , 2019; Atangana, 2016) and so on (Wang et al , 2023a; Ghanbari and Abdon, 2020; Wang, 2022b). In recent years, as a new theory of fractional calculus, the local fractional calculus has been successfully used to explain many non-differentiable (ND) scientific problems, for instance, the shallow water surfaces (Yang et al , 2016), rheological (Yang et al , 2017a), physics (Wang et al , 2023b, 2023c; Yang, 2017; Wang and Shi, 2023b; Wang, 2023c), circuits (Yang et al , 2017b; Zhao et al , 2017; Wang, 2023b; Banchuin, 2022; Banchuin, 2023; Wang et al , 2020), vibration (Yang and Srivastava, 2015) and others. Inspired by the recent research results on fractal circuits, the purpose of this article is to derive a new ℑ-order R-C zero state-response circuit (ZSRC) within the local fractional derivative (LFD).…”