The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented.

Lactuca sativa L. is an economically important vegetable that contains numerous phytochemicals. This study aimed to determine the phytochemicals in three lettuce cultivars (red oak, red coral, and butterhead) and assess their total phenolics, total flavonoids, and antioxidant activities. The dried leaves of each lettuce cultivar were macerated with hexane, ethyl acetate (EtOAc), and 95% ethanol (EtOH). Total phenolics, total flavonoids and antioxidant activities from the three solvent extracts were measured. The phytochemical screening showed that the leaves from the three lettuce cultivars contained flavonoids, hydrolyzable tannins, coumarins, steroids, and phenolic compounds. While the EtOAc fraction of the red coral lettuce showed the highest total phenolic content (9.747 ± 0.021 mg gallic acid equivalent/g) and the hexane fraction of the butterhead lettuce contained the highest flavonoids (7.065 ± 0.005 mg quercetin equivalent/g). In the DPPH (2,2-diphenyl-1-picryl-hydrazyl-hydrate) assay, the EtOAc fraction of the red coral lettuce had the highest antioxidant capacity with an IC50 of 0.277 ± 0.006 mg/mL, whereas, in the ABTS (2,2’-azino-bis(3- ethylbenzothiazoline-6-sulfonic acid)) assay, the 95% EtOH of the red coral lettuce had the highest antioxidant capacity with an IC50 of 0.300 ± 0.002 mg/mL. All three lettuce cultivars contained high levels of phenolic content and flavonoids, which are the source of antioxidant activities. These lettuce cultivars, especially the red coral lettuce, are a potential source of natural antioxidants. Further research on the application of natural antioxidants is required to investigate the therapeutic or the neutraceutical implication of the lettuce cultivars.

Let α, b and T be positive numbers, D = (0, ∞),D = [0, ∞), and Ω = D × (0, T ]. This article studies the first initial-boundary value problem, u t − u xx = αδ(x − b)f (u(x, t)) in Ω, u(x, 0) = ψ(x) onD, u(0, t) = 0 = lim x→∞ u(x, t) for 0 < t ≤ T, where δ (x) is the Dirac delta function, and f and ψ are given functions. We assume that f (0) ≥ 0, f (u) and its derivatives f (u) and f (u) are positive for u > 0, and ψ(x) is nontrivial, nonnegative and continuous such that ψ (0) = 0 = lim x→∞ ψ (x), and ψ + αδ(x − b)f (ψ) ≥ 0 in D. It is shown that if u blows up, then it blows up in a finite time at the single point b only. A criterion for u to blow up in a finite time and a criterion for u to exist globally are given. It is also shown that there exists a critical position b * for the nonlinear source to be placed such that no blowup occurs for b ≤ b * , and u blows up in a finite time for b > b *. This also implies that u does not blow up in infinite time. The formula for computing b * is also derived. For illustrations, two examples are given.

Let v and T be positive numbers, D = (0, ∞), Ω = D × (0, T ], and D be the closure of D. This article studies the first initial-boundary value problem,where δ (x) is the Dirac delta function, and f and ψ are given functions. It is shown that if the solution u blows up in a finite time t b , then it blows up only at the point x = vt b . A criterion for u to exist globally and a criterion for u to blow up in a finite time are given. Furthermore, the problem is shown to have a critical speed v * of the moving nonlinear source such that no blowup occurs for v ≥ v * and blowup occurs in a finite time for v < v * . The formula for computing v * is also derived.

This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for (x,t)∈(0,1)×(0,T), where f and g are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given.