Mawia Osman

In this paper, we provide the fuzzy time-fractional Navier-Stokes equations (NSEs) by substituting the time derivative with the Katugampola fractional derivative (KFD), which is expressed in Caputo type. The simplified fuzzy generalized Laplace residual power series method (GLRPSM) is used to derive an analytical solution to the fuzzy time-fractional NSEs. The proof of convergence to the proposed solution is established. Additionally, we present a novel method that combines the residual power series approach with the Laplace transform, abbreviated as (LRPSM), to produce a series solution for the fuzzy time-fractional NSEs. The suggested technique for solving fuzzy fractional physical equations was constructed using Laurent series. The new procedure’s most notable features are its speed and precision in deriving an exact or approximate solution. Two fuzzy time-fractional NSEs that describe the motion of a pipe’s flow were analyzed by the suggested approaches. Our findings are compared to the results of previous studies that were conducted to show the technique’s accuracy and effectiveness. In addition, the fractional derivatives are used to present the fuzzy fractional modified Korteweg-de Vries equation (fmKdV) and fKdV. The technique is predicated mostly on the approximate two-dimensional differential transform method (DTM). The technique is suitable for numerous problems and has the capacity to reduce the amount of computing work. In the sense of the Caputo derivative, the fractional derivative is described. Finally, we establish the approximate analytical solutions to fmKdV; the fuzzy homotopy-perturbation method (HPM) is used. To produce the fuzzy fmKdV, replace the first-order time and space derivatives with fractional derivatives of order α and β with α > 0, β ≤ 1. The provided approaches to solving diverse nonlinear fuzzy fractional-order partial differential equations are interesting, easy, and very accurate.

The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the symmetrical properties of functions. These inequalities also generalize some former famous inequalities in the literature. Two examples as applications will be provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problems for nonlinear integro-differential equations and differential equations which can be seen in graphs. This research work will ensure opening new opportunities for studying nonlinear dynamic inequalities on a time-scale structure of a varying nature.

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

This paper is dedicated to exploring the existence, uniqueness and Ulam stability analysis applied to a specific class of mathematical equations known as nonlinear impulsive Volterra Fredholm integro-dynamic adjoint equations within finite time scale intervals. The primary aim is to establish sufficient conditions that demonstrate Ulam stability for this particular class of equations on the considered time scales. The research methodology relies on the Banach contraction principle, Picard operator and extended integral inequality applicable to piecewise continuous functions on time scales. To illustrate the applicability of the findings, an example is provided.

Activation energy can be elaborated as the minimal energy required to start a certain chemical reaction. The concept of this energy was first presented by Arrhenius in the year 1889 and was later used in the oil reservoir industry, emulsion of water, geothermal as well as chemical engineering and food processing. This study relates to the impacts of mass transfer caused by temperature differences (Soret) and heat transport due to concentration gradient (Dufour) in a Carreau model with nanofluids (NFs), mixed convection and a magnetic field past a stretched sheet. Moreover, thermal radiation and activation energy with new mass flux constraints are presumed. All chemical science specifications of nanofluid are measured as constant. As a result of the motion of nanofluid particles, the fluid temperature and concentration are inspected, with some physical description. A system of coupled partial differential frameworks is used mathematically to formulate the physical model. A numerical scheme named the Runge–Kutta (R-K) approach along with the shooting technique are used to solve the obtained equations to a high degree of accuracy. The MATLAB R2022b software is used for the graphical presentation of the solution. The temperature of the nanofluid encompasses a quicker rate within the efficiency of a Dufour number. An intensifying thermal trend is observed for thermophoresis and the Brownian motion parameter. The Soret effect causes a decline in the fluid concentration, and the opposite trend is observed for rising activation energy. In addition, the local Nusselt number increases with the Prandtl number. Further, the comparative outcomes for drag force are established, with satisfying agreement with the existing literature. The results acquired here are anticipated to be applied to improving heat exchanger thermal efficiency to maintain thermal balancing control in compact heat density equipment and devices.

In this work, we have adopted an advanced Julia-based function that helps not only in sorting out hidden wings in the fractals of chaotic systems, but can also generate an extra wing in chaotic systems based on a single wing. For verification, two examples Lorenz and modified stretch-twist-fold (STF) systems based on more than one wing, whereas chemical reaction-based chaotic system with a unique wing is considered. The existence of another wing in chaotic systems based on a single wing was a big question mark on the creation of multi-wings in the theory of fractals, but with the aid of advanced Julia functions, we have elaborated in detail that the existence of a second wing in such systems is also possible. The stretching and squeezing in the trajectories of fractals are also integral parts of our findings. Moreover, our study has solved another problem related to fractals. In the past, authors have shown multi-wings in chaotic systems with empty space inside all the time. In this study, we have shown for the first time that these inner spaces have special meaning due to the existence of hidden wings. Furthermore, we have shown that fractals can be divided into outer and inner wings, where the inner wings reflect the answer to the question about the empty space in between the outer wings of chaotic systems. For convenience, an extra file named as MiddleSpace.pdf is attached as a supplementary file to better understand the concept of covering empty space inside fractals.