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In the present paper, we considered a Korteweg-de Vries type system with weakly damping terms and initial data in the analytic Gevery spaces. The presence of tow functions $ c_1(x), c_2(x) $, called damping coefficients, made the system more interesting from an application point of view due to their great importance in physics. To start, by using the fixed point theorem in Banach space, we investigated the local well-posedness. Additionally, by employing an approximate conservation law, we extended this to be global in time, ensuring that the radius of analyticity of solutions remained uniformly bounded below by a fixed positive number for all time.

The theory of neutral dynamic equations on timescales was based to unify the study of differential and difference equations. The article described several oscillating criteria that will be developed for fourth-order-neutral dynamic equations in the presence of various types of arguments on timescales. The goal was to establish all necessary conditions for the solutions of these models to be oscillatory. To construct observation values, ideas from [Y. Sui and Z. Han, Oscillation of second order neutral dynamic equations with deviating arguments on time scales, Adv. Differ. Equ., 10 (2018)] were used. The research seeked to provide sufficient criteria that ensured the oscillation of solutions to these complex dynamic equations using a technique Riccati transformations generalized, emphasizing their importance in the study of oscillatory processes within various scientific and engineering contexts.

By using linear, bilinear, and trilinear estimates in Bourgain-type spaces and analytic spaces, the local well-posedness of the Cauchy problem for the a Kawahara-Korteweg-de-Vries equation$ \partial_{t}u+\omega\partial_{x}^{5}u+\nu \partial_{x}^{3}u+\mu\partial_{x}u^{2}+\lambda\partial_{x}u^{3}+\mathfrak{d}(x)u = 0, $was established for analytic initial data $ u_{0} $. Besides, based on the obtained local result, together with an analytic approximate conservation law, we prove that the global solutions exist. Furthermore, the analytic radius has a fixed positive lower bound uniformly for all time.

This paper discussed the decay of a thermoelastic laminated beam subjected to nonlinear delay and nonlinear structural damping. We provided explicit and general energy decay rates of the solution by imposing suitable conditions on both weight delay and wave speeds. To achieve this, we leveraged the properties of convex functions and employed the multiplier technique as a specific approach to demonstrate our stability results.

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given.

In this manuscript, by using a new identity, we establish some new Maclaurin-type inequalities for functions whose modulus of the first derivatives are GA-convex functions via Hadamard fractional integrals.

This paper’s major goal is to prove some symmetrical Maclaurin-type integral inequalities inside the framework of multiplicative calculus. In order to accomplish this and after giving some basic tools, we have established a new integral identity. Based on this identity, some symmetrical Maclaurin-type inequalities have been constructed for functions whose multiplicative derivatives are bounded as well as convex. At the end, some applications to special means are provided.

A class for systems of nonlinear second-order differential equations with periodic impulse action are considered. An urgent problem for this class of differential equations is the problem of the quantitative study (existence) in the case when the phase space of the equation is, in the general case, some Banach space. In this work, sufficient conditions for the existence of solutions for a system with parameters are obtained. The results are obtained by using fixed point theorems for operators on a cone. Our approach is based on Schaefer’s fixed point theorem more precisely. In addition, the existence of positive solutions is also investigated.

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper).

In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder’s fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under suitable assumptions on Y1, Y2 and ϖ1, ϖ2, which greatly enriched the existence literature on this system, there is, however, no hope to discuss the uniqueness result in a convex case. In the present study, we analyzed the influence of the impulsive and infinite delay on the solutions to our system. In addition, to the best of our acknowledge, there is no result concerning coupled random system in the presence of impulsive and infinite delay.

Stoke's problems significantly impact several domains, including industrial manufacturing, geophysical flows, chemical engineering, and heat conduction. Stoke's second problem deals with the movement of a semi-infinite viscous incompressible fluid caused by an oscillating flat plate. Thus, this work examines Stoke's second problem for an unsteady hydromagnetic surface-driven flow along an infinite flat plate in the presence of thermal radiation and heat dissipation. The governing momentum and energy equations form an emerging nonlinear system of partial differential equations through dimensionless proxies. MAPLE 2022 is employed to solve the resulting system of nonlinear partial differential equations that control the flow both analytically and numerically in specific situations. The analytical solutions are displayed graphically, along with variations of skin friction and Nusselt number at the plate. It has been observed that momentum and thermal slips significantly diminished the flow characteristics to cause damping flow rate and temperature distributions. Rising the values of Magnetic and velocity slip parameters, an oscillatory motion is observed in skin friction. This is due to the periodic and wavy nature of the boundary wall. Furthermore, a rise in the Prandtl number and radiation value correspondingly boosted the wall heat gradient profiles. Finally, this study will assist industry and engineering in understanding the sensitivity of their working fluids to parameter variations and for the exact prediction of their base fluids.

After introducing the advantage of nanofluids for optimizing heat transfer, many research articles have been written considering the different types of domains. In our article, we will observe the turbulent transport of a hybrid mixture through the three-dimensional L-shaped channel to analyze the characteristic of heat generation via forced convection. The hybrid nanofluid is composed of aluminum oxide copper (Cu) and aluminum oxide (Al2O3), which will be present in the base fluid by the volume fraction of 5%, 15%, 25%, and 35% in the base fluid (water). To analyze the forced convection, the Reynolds number (Re) will be kept at 10,000 to 40,000. The simulation will be developed using the three-dimensional κ−ε turbulence model and energy equations via COMSOL Multiphysics 5.6. The numerical outcomes will be verified using the literature's correlations for the average Nusselt number. It was found that the average Nusselt number at the mid of the channel is increased by increasing the volume fractions of both aluminum oxide and copper, but the rate of increase because of copper is greater than the rate of increase because of aluminum oxide. The study also suggests that to get more benefit from the transport of a hybrid mixture, keep the Reynolds number high. The temperature along the channel length is the function of mean thermal diffusivity, which is directly proportional to the volume fractions of the hybrid mixture.

In this paper, we first prove a new parameterized identity. Based on this identity we establish some parametrized Simpson-like type symmetric inequalities, for functions whose first derivatives are s-tgs-convex via Reimann–Liouville frational operators. Some special cases are discussed. Applications to numerical quadrature are provided.

In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method’s efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory.

This work investigates a weighted Banach space second order pantograph fractional differential equation. The considered equation is of second order, expressed in terms of the Caputo–Hadamard fractional operator, and constructed in a general manner to accommodate many specific situations. The asymptotic stability of the main equation’s trivial solution has been given. The primary theorem was demonstrated in a unique manner by employing the Krasnoselskii’s fixed point theorem. We provide a concrete example that supports the theoretical findings.

AIM: To compare the curative effect of retropupillary iris claw intraocular lens(IOL)implantation and posterior chamber ciliary scleral suture fixed IOL in the aphakic eyes without capsular support.METHODS: The present retrospective study included 70 aphakic eyes without capsular support. According to the different surgical methods, the patients were divided into two groups, including group A(35 eyes)with treatment of retropupillary iris claw IOL implantation, group B(35 eyes)with treatment of posterior chamber ciliary scleral suture fixed IOL. The uncorrected visual acuity(UCVA), best corrected visual acuity(BCVA), intraocular pressure(IOP)and corneal endothelial cell density(CECD)of the two groups were documented and compared before operation and 3d, 1, 3, 6mo and 1a after operation. Furthermore, the stability of IOL in both groups, as well as the intraoperative and postoperative complications were recorded.RESULTS: The patients in the present study were followed up for 12 to 14mo. 3d after surgery, UCVA in group A was significantly improved compared with that before surgery(P0.05).There was no significant difference between whole eye astigmatism and corneal astigmatism in A group and B gruop at 1 and 6mo after operation(P>0.05). Additionally, There was 1 case of IOL dislocation in each group after operation, which was further recovered by operation. Significant tilt and deviation of IOL were not observed in the rest of the patients during the follow up period. The other postoperative complications were mild with no difference in complications between the two groups(P>0.05).CONCLUSION: For aphakic eyes without capsular support, both retropupillary iris claw IOL implantation and posterior chamber ciliary scleral suture fixed IOL are safe and effective surgical methods. Retropupillary iris claw IOL implantation has relatively simple operation, lower damage to the eyeball tissue, the shorter operation time, and the quicker recovery of postoperative visual acuity, which is one of the effective clinical treatments.