The Caputo fractional version of the generalized Newell–Whitehead–Segel model is considered. We introduced a numerical scheme to solve analytically the proposed application. We updated the style of the generalized Taylor series for a reliable treatment of the time-fractional derivative. The effect of the fractional derivative is explored on the obtained solutions for different cases of the problem. A sequential-asymptotic phenomenon has been observed upon varying the order of the fractional derivative from no-memory “\(\alpha=0\)” to full-memory “\(\alpha=1\)”.

In this work, we present a new two-waves’ version of the fifth-order Korteweg–de Vries model. This model describes the propagation of moving two-waves under the influence of dispersion, nonlinearity, and phase velocity factors. We seek possible stationary wave solutions to this new model by means of Kudryashov-expansion method and sine–cosine function method. Also, we provide a graphical analysis to show the effect of phase velocity on the motion of the obtained solutions.

In this work, we obtain appropriate sharp bounds for a certain class of maximal operators along surfaces of revolution with kernels in $L^{q}(\mathbf{S}^{n-1})$ , $q > 1$ . By using these bounds and using an extrapolation argument, we establish the $L^{p}$ boundedness of the maximal operators when their kernels are in $L(\log L)^{\alpha}(\mathbf{S}^{n-1})$ or in the block space $B^{0,\alpha-1}_{q} (\mathbf{S}^{n-1})$ . Our main results represent significant improvements as well as natural extensions of what was known previously.

Let A_i ∈ B(H), (i = 1, 2, ..., n), and T = [ 0 ⋯ 0 A 1 ⋮ ⋰ A 2 0 0 ⋰ ⋰ ⋮ A n 0 ⋯ 0 ] $ T = \left[ {\matrix{ 0 & \cdots & 0 & {A_1 } \cr \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {A_2 } & 0 \cr 0 & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots \cr {A_n } & 0 & \cdots & 0 \cr } } \right] $ . In this paper, we present some upper bounds and lower bounds for w(T). At the end of this paper we drive a new bound for the zeros of polynomials.

In this article, the boundedness of the generalized parametric Marcinkiewicz integral operators ${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}$ is considered. Under the condition that $\mathsf{\Omega }$ is a function in $L^q\left({\mathbf{S}}^{n-1}\right)$ with $q\in (1,2]$ , appropriate estimates of the aforementioned operators from Triebel–Lizorkin spaces to $L^p$ spaces are obtained. By these estimates and an extrapolation argument, we establish the boundedness of such operators when the kernel function $\mathsf{\Omega }$ belongs to the block space $B_q^{0,\nu -1}\left({\mathbf{S}}^{n-1}\right)$ or in the space $L{\left(\log L\right)}^{\nu }\left({\mathbf{S}}^{n-1}\right)$ . Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.