A spectral collocation method for fractional chemical clock reactions
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A spectral collocation method for fractional chemical clock reactions Mohamed M. Khader1,2 · Khaled M. Saad3,4 Sunil Kumar7
· Dumitru Baleanu5,6,8
·
Received: 22 August 2020 / Revised: 13 October 2020 / Accepted: 1 November 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract We implement an efficient computational scheme to study the effect of precursor consumption on chemical clock reactions. The proposed model is formulated as a system of FDEs with power kernel. This paper considers the fractional derivatives of Liouville–Caputo (LC). We use the spectral collocation method (SCM) with the help of the third-kind Chebyshev polynomials. This scheme generates the fast convergent series solutions with conveniently determinable coefficients. We compute the residual error function (REF) to satisfy the accuracy of the introduced technique. This approach is an easy and efficient tool for implementing the study of such these models. We introduce a comparison between the obtained approximate solutions and those which occurred using a previously published method and excellent agreement is reported. Keywords Chemical clock reactions · LC-fractional derivative · Third-kind Chebyshev polynomials · Convergence analysis Mathematics Subject Classification 26A33 · 34A08 · 65N12 · 65N35
1 Introduction In the last 3 decades, many authors are interested in fractional calculus (Kilbas et al. 1993). Some of them modified the singular kernels to non-singular kernels and found a new definition of the fractional derivatives (Ghanbari et al. 2020; Alshabanat et al. 2020; Veeresha et al. 2020; Goufo et al. 2020; Morales-Delgado et al. 2018). Among these definitions is LCfractional derivatives which used to express the models in a general real world. In different sciences, such as physics, fluid mechanics, biology, and engineering, we need to use these definitions of the fractional-order derivatives (Kilbas et al. 2006). A clock reaction is a chemical reaction which gives rise to a significant induction period during which one of the chemical species, the clock chemical, has a very low concentration (Stephen 2002). The end of the induction period is marked by a rapid increase in concentration
Communicated by José Tenreiro Machado. Extended author information available on the last page of the article 0123456789().: V,-vol
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of the clock chemical. This rapid growth can produce remarkable effects experimentally such as sudden dramatic color changes in solution phase reactions. One particular class of reaction even produces a flash of light. Examples of clock reactions include the arsenic(III) sulfide clock reaction, the formaldehyde clock reaction, the iodine bisulphate clock, and the hydration of carbon dioxide. Also, chemical clock behaviour has obvious parallels with the induction period observed during the hydration and setting of cements: a slurry remains liquid for a characteristic time before rapidly hardening into a porous solid (Jones et al. 1964, 198
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