Classical integer-order differential equations are not capable of capturing the property of memory. Researchers have been drawn to fractional differential equations because they can be used to model a wide range of phenomena, including those relating to epidemic diseases, biomedical applications, viscoelasticity, biology, hydrology, electricity, chemical physics, electrochemistry, probability theory, dynamical systems, heat conduction, and others. The theory of fractional-order calculus has been used for practical applications, being applied to signal processing, chaos theory, optics, noisy environments, and other areas. With the swift growth in digital computer knowledge, many researchers began to work on the theory and applications of fractional calculus. The foundation for fractional calculus was laid by numerous innovative investigations that provided various definitions of fractional calculus. Fractional calculus has recently developed into a potent tool as a result of its advantageous characteristics. The origins of fractional calculus can be found in a letter from l’Hopital to Leibniz in 1695, in which he speculated on the meaning of the symbol d 1 2 x ( t ) d t 1 2, which denotes the semiderivative of x ( t ) with respect to t. The general extension of the integer-order calculus to arbitrary order is known as fractional calculus, which includes integration and differentiation with noninteger order.

The numerical results obtained using the suggested techniques show that the methods are both simple and have excellent computational merit. The present study also illustrates error analysis. The results of applying the current methodologies revealed strong connections to the precise resolution of the problem under investigation. We then produced graphical representations that allowed us to visually check all of the outcomes in the proposed case for various fractional order values.

These techniques create recurrence relations representing the proposed problem’s solution. We looked at a single case and contrasted it with the actual result to validate the methodologies. We used the Yang transform with the Adomian decomposition process and homotopy perturbation procedure on the time-fractional Sawada–Kotera–Ito problem to obtain the solution. The fractional derivative is taken into account in the Caputo sense. brasiliensis stems can be used in food supplements or as natural antioxidants in the food industry.This article uses the Yang transform decomposition method and the homotopy perturbation transform method to study the seventh-order time-fractional Sawada–Kotera–Ito equation. Thus, extract and fractions obtained from the S. brasiliensis stems demonstrated to be a rich source of polyphenols compounds and exhibited high antioxidant potential without toxicity. brasiliensis since these compounds were identified for the first time in the stems of this species. This study contributes to the phytochemical study of S. The EE, DCMF, ACF, and HEF exhibited great antioxidant potential by DPPH (IC 50 1.71 – 32.83 µg/mL) and FRAP (IC 50 0.63 – 6,71 µg/mL) assays. The EE, DCMF, and ACF showed high total contents of phenolic compounds (112.99, 175.71, and 524.02 µg of GAE/mg, respectively), and in the ACF and DCMF a great content of flavonoids was also quantified (50.08 and 31.49 µg of QE/mg, respectively). The EE and DCMF, ACF, and HEF were analyzed by liquid chromatography coupled to a diode array detector and mass spectrometer (LC–DAD–MS), and the identified constituents included glycosylated (rutin, 3- O-β-galactopyranosyl quercetin, 3- O-β-glucopyranosyl quercetin, O-deoxyhexosyl-hexosyl quercetin, O-deoxyhexosyl-hexosyl kaempferol, O-deoxyhexosyl-hexosyl O-methyl quercetin, and others), and non-glycosylated (quercetin) flavonoids, phenylpropanoids (3- O- E-caffeoyl quinic acid, 5- O- E-caffeoyl quinic acid, O-caffeoyl shikimic acid, and others), neolignan, steroidal saponin (dioscin), and N-feruloyltyramine. Fatty acid esters, hydrocarbons, and phytosterols were identified in the HEXF analyzed by gas chromatography – mass spectrometry (GC–MS). The chemical composition was determined, the contents of phenolic compounds and flavonoids were quantified, and the antioxidant potential and the cytotoxic effect on Artemia salina were evaluated. In this study, the ethanol extract (EE) and the hexane (HEXF), dichloromethane (DCMF), ethyl acetate (ACF), and hydroethanol (HEF) fractions of the stems were obtained. Smilax brasiliensis Sprengel is a monocotyledon of the Smilacaceae family, native to the Brazilian Cerrado, popularly known as “salsaparrilha” or “japecanga”.

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