Fractional order theory of Cattaneo-type thermoelasticity using new fractional derivatives

A Two-Level Crank-Nicolson Difference Scheme and Its Richardson Extrapolation Methods for a Magneto-Thermo-Elasticity Model

“…[5][6][7]). To satisfy various applications, there have been growing many magneto-thermo-elasticity models with more general forms, such as phase-lag Green-Naghdi models [15][16][17][18], fractional GN II thermoelasticity models [19] and Cattaneo-type thermoelasticity [20]. Some detailed reviews about the classical and extended theories, please refer to the references [8,9,21] and the related references therein.…”

Dual‐phase‐lag thermoelastic diffusion analysis of a size‐dependent microplate based on modified fractional‐order heat conduction model

“…Introducing the thermal phase lag of temperature gradient $${\tau _\theta }$$ into Equation (), the time fractional DPL model is derived as $$\begin{equation}\left( {1 + {\tau _q}{D_q}} \right)\;{q_i} = - \kappa {I^{\alpha - 1}}\left( {1 + {\tau _\theta }{D_\theta }} \right){\theta _{,i}}\end{equation}$$where $${I^{\alpha - 1}}$$ denotes an integral operator, α denotes the fractional‐order parameter. For $$0 < \alpha < 1$$, the definition of TC fractional derivative is written as [37] $$\begin{equation}{I^{\alpha - 1}}f\left( t \right) = \frac{{{{\rm{e}}^{ - \chi t}}}}{{\Gamma \left( {\alpha - 1} \right)}}{\int_0^t {\left( {t - p} \right)} ^{\alpha - 2}}\frac{{{\rm{d}}\left[ {{{\rm{e}}^{\chi p}}f\left( p \right)} \right]}}{{{\rm{d}}p}}{\rm{d}}p\end{equation}$$in which $$\Gamma (\alpha - 1)$$ is the Gamma function.…”

“…It can be found that the fractional-order heat conduction models were defined by the Captuo-type fractional derivative [36]. The Caputo-type fractional derivative exists singularity [37]. To overcome such shortcoming, Yu et al [25] established a unified fractional order thermoelastic model by incorporating different fractional derivative models, that is, Caputo-Fabrizio [38], Atangana-Baleanu [39], and Tempered-Caputo (TC) [40] type fractional derivatives.…”

“…It can be found that the fractional‐order heat conduction models were defined by the Captuo‐type fractional derivative [36]. The Caputo‐type fractional derivative exists singularity [37]. To overcome such shortcoming, Yu et al.…”

Iterative parameter and order identification for fractional‐order nonlinear finite impulse response systems using the key term separation

“…However, some complex systems are difficult to describe accurately by the traditional integer‐order systems. Under the framework of the fractional‐order derivative theory, the traditional integer‐order systems are extended to fractional‐order systems 25,26 . Djamah et al studied an output‐error identification approach and analyzed statistical performance by using Monte‐Carlo simulation at various signal‐to‐noise ratios 27 .…”