In this article, an attempt has been made to explore the potential performance of Al 2 O 3 nanoparticle-based cutting fluid in hard milling of hardened 60Si 2 Mn steel (50-52 HRC) under different minimum quantity lubrication conditions. The comparison of hard milling under minimum quantity lubrication conditions is done between pure cutting fluids and nanofluids (in terms of surface roughness, cutting force, tool wear, and tool life). Hard milling under minimum quantity lubrication conditions with nanofluid Al 2 O 3 of 0.5% volume has shown superior results. The improvement in tool life almost 177%-230% (depending on the type of nanofluid) and the reduction in surface roughness and cutting forces almost 35%-60% have been observed under minimum quantity lubrication with Al 2 O 3 nanofluids due to better tribological behavior as well as cooling and lubricating effects. The most outstanding result is that the uncoated cemented carbide insert can be effectively used in machining high-hardness steels (.50 HRC) while maintaining long tool life and good surface integrity (R a = 0.08-0.35 mm; R z = 0.5-2.0 mm, equivalent to finish grinding) rather than using the costlier tools like coated carbide, ceramic, and (P)CBN. Therefore, using hard nanoparticle-reinforced cutting fluid under minimum quantity lubrication conditions in practical manufacturing becomes very promising.

We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first and fractional orders are also derived. The effectiveness of our methods are shown by applying the results to two interesting models: time fractional Ginzburg-Landau equation, and time fractional Burgers equation, where time and spatial regularity estimates are obtained.

This paper investigates an inverse problem for fractional Rayleigh‐Stokes equations with nonlinear source. The fractional derivative in time is taken in the sense of Riemann‐Liouville. The proposed problem has many applications in some non‐Newtonian fluids. We obtain some results on the existence and regularity of mild solutions.

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atangana and Baleanu fractional derivative in time is taken in the Caputo sense. We obtain results on the existence, uniqueness, and regularity of the solution.

This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space.A representation of the solution is obtained by basing on the terminal value data u(T, x) = 𝜑(x) and the spectrum of the fractional Laplacian operator (−Δ) s∕2 (in a bounded domain X ⊂ R d , 0 < s < 2). First, we show the existence and uniqueness of a mild solution in L p (0, T; L 2 (Ω, V)) ∩ C((0, T]; L 2 (Ω, L 2 (X))), for a suitable sub-space V of L 2 (X). A limitation of this result is the lack of time continuity at t = 0. Second, we study the inverse problem (IP) of recovering u(0, x) when the terminal value data 𝜑 and the source 𝑓 are given. We give an explanation why the time continuity of the solution at t = 0 could not derived. The main reason comes from unboundedness of a solution operator, so the problem (IP) is then ill-posed, that is, recovery u(0, x) cannot be obtained in general. Hence, we propose a truncation regularization method with a suitable choice of the regularization parameter. Finally, we present a numerical example to demonstrate our proposed method.