In this article, we consider the natural (free) convection flows of viscous carbon nanotubes nanofluids in the presence of transverse magnetic field in a vertical cylinder. The fractional derivative is introduced by means of generalized Fourier's law (Caputo–Fabrizio and Atangana–Baleanu fractional derivatives) in the thermal equation. The solutions of the equations are found by means of the Laplace and finite Hankel transforms, which satisfied all imposed initial and boundary conditions. The effects of fractional and physical parameters are graphically underlined.
The fractional integral inequalities are crucial to deal applied problems. The present paper deals with the generalize midpoint type inequalities for a certain class of convex functions, namely, MT-convex functions in the setting of weighted fractional integral. It can be observed from the remarks given in the paper that our results or more generalized than existing results of literature and many of them can be obtained immediately from our results.
The study of convex functions is an interesting area of research due to its huge applications in pure and applied mathematics special in optimization theory. The aim of this paper is to introduce and study a more generalized class of convex functions. We established Schur (S), Hermite-Hadamard (HH), and Fejér (F) type inequalities for introduced class of convex functions. The results presented in this paper extend and generalize many existing results of the literature.
Fractional integral inequalities help to solve many difference equations. In this paper, we present some fractional integral inequalities for generalized harmonic nonconvex functions. Moreover, we also present applications of developed inequalities.
To overcome the limitation of unfolding-based methods
and handle
the multiple data set and limited data problems in the complex processes,
such as multigrade batch processes, a novel tensor-based common and
special feature extraction method and a comprehensive monitoring framework
are proposed. In the proposed method, the uneven-length three-dimensional
data are directly analyzed by the comprehensive tensor-based method
without unfolding. To handle the multiple data set modeling problem,
the tensor-based common feature extraction methods are first proposed
to obtain the common features shared among different grades. The special
features are sequentially determined by conducting tensor principal
component analysis (PCA) on the residuals of each grade. The data
are thus divided into common, special, and residual subspaces. Three
monitoring statistics are established respectively in each subspace
for online fault detection. The merits and effectiveness of the proposed
method are demonstrated by an injection molding process with both
even-length and uneven-length data in comparison with traditional
methods.
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