In this study, the suggested residual power series transform method is used to compute the numerical solution of the fractional-order nonlinear Gardner and Cahn-Hilliard equations and the result is discovered in a fast convergent series. The leverage and efficacy of the suggested technique are demonstrated by the test examples provided. The achieved results are proved graphically. The current method handles the series solution in a sizable admissible domain in a powerful way. It provides a simple means of modifying the solution's convergence zone. Results with graphs expressly demonstrate the effectiveness and abilities of the suggested method.

The selection of parameters plays a vital role in the multi-attribute decision-making process. In some situations, it is observed that the nature of parameters is ambiguous and a multi-decisive opinion is necessary for managing such parametric uncertainty. In the literature, there is no suitable model that can cope with such situations. This study was purposed to develop a novel context called the fuzzy parameterized fuzzy hypersoft expert set (FPFHSE-set), which is capable of managing the uncertain nature of parameters and the multi-decisive opinion of experts collectively in one model. In this way, the proposed model may be described as the generalization of the existing model fuzzy parameterized fuzzy soft expert set (FPFSE-set). Theoretic, axiomatic and algorithmic approaches have been employed for the characterization of the basic notions of the FPFHSE-set. In order to handle multi-attribute decision-making, two algorithms are proposed and then validated by applying them to some real-world scenarios in the FPFHSE-set environment. The merits and superiority of the new algorithms are presented by comparing them with some existing fuzzy decision-making models. According to the proposed FPFHSE-set-based decision-making approaches, the experts have more freedom in specifying their preferences and thoughts according to their expertise, and they can process new types of data. Therefore, this paper presents a state-of-the-art improvement that provides a holistic view to understand and handle the multi-attribute decision-making issues focused on the objective of classifying alternatives according to multiple attributes by multiple experts.

Incorporating self-diffusion and super-cross diffusion factors into the modeling approach enhances efficiency and realism by having a substantial impact on the scenario of pattern formation. Accordingly, this work analyzes self and super-cross diffusion for a predator-prey model. First, the stability of equilibrium points is explored. Utilizing stability analysis of local equilibrium points, we stabilize the properties that guarantee the emergence of the Turing instability. Weakly nonlinear analysis is used to get the amplitude equations at the Turing bifurcation point (WNA). The stability analysis of the amplitude equations establishes the conditions for the formation of small spots, hexagons, huge spots, squares, labyrinthine, and stripe patterns. Analytical findings have been validated using numerical simulations. Extensive data that may be used analytically and numerically to assess the effect of self-super-cross diffusion on a variety of predator-prey systems.

This paper is presented to investigate the exact solutions to the modified Zakharov-Kuznetsov equation that have a critical role to play in mathematical physics. The $\tan \left(\varphi \left(\zeta \right)/2\right)$-expansion, $(m+G^{\prime }(\zeta )/G(\zeta ))$-expansion and He exponential function methods are used to reveal various analytical solutions of the model. The equation regulates the treatment of weakly nonlinear ion-acoustic waves in a plasma consisting of cold ions and hot isothermal electrons throughout the existence of a uniform magnetic field. Solutions in forms of W-shaped, singular, periodic-bright and bright are constructed.

This article focuses on the controllability of a Hilfer fractional impulsive differential equation with indefinite delay. We analyze our major outcomes using fractional calculus, the measure of non-compactness and a fixed-point approach. Finally, an example is provided to show the theory.

Interval-valued fuzzy hypersoft set ( $\mathbb{I}\mathbb{V}\mathbb{F}\mathbb{H}\mathbb{S}\mathbb{S}$) is considered a pertinent fuzzy set-like model that is the combination of an interval-valued fuzzy set and a hypersoft set. It is more flexible and trustworthy for dealing with information-based uncertainties due to the consideration of interval-based hypersoft settings. This kind of setting enables the decision makers to approximate the alternatives in terms of interval-type opinions by considering multiple arguments concurrently. These features make it a fitting model for dealing with uncertain decision-making scenarios like the recruitment process. The vagueness arises in the recruitment process when the data obtained is hesitant. The analogous educational norms among the candidates may increase its complexity. Evaluation techniques focus on leveling hypersoft sets for grading several alternatives based on multi-arguments. When several alternatives have an identical status, such grading systems frequently encounter problems, making it challenging for decision-makers to select the preeminent alternative. This settlement of such an issue is the basis of this article. Thus, in this study, first the axiomatic notions of similarity measures between $\mathbb{I}\mathbb{V}\mathbb{F}\mathbb{H}\mathbb{S}\mathbb{S}s$ are characterized, and then their relevant theorem is proved. In order to provide a consistent decision-support framework for the recruitment process, a robust algorithm is proposed. Finally, the effectiveness, feasibility, and efficiency of the proposed model are demonstrated through the depiction of recruitment-based pattern recognition.

This paper addresses a mixed and free convective Casson nanofluid flowing on an oscillating inclined poured plate with sinusoidal heat transfers and slip boundaries. As base fluid water is supposed and the suspension of nanofluid is formulated with the combination of individual copper $\left(Cu\right)$, titanium dioxide $\left(Ti{O}_2\right)$ and aluminum oxide $\left(A{l}_2{O}_3\right)$ as nanoparticles, the dimensionless governing equations are generalized based on Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional operators for developing a fractional form. Then, for the semi-analytical solution of the momentum and thermal profiles, the Laplace transformation is utilized. To discuss the influences of various pertinent parameters on governing equations, graphical tablecomparison of the Nusselt number and skin friction is also inspected at different times and numerical schemes. As a result, it has been concluded that both the momentum and energy profiles represent the more significant results for the AB-fractional model as related to the CF-fractional model solution. Furthermore, water-based titanium dioxide $\left(Ti{O}_2\right)$ has a more progressive impact on the momentum as well as the thermal fields as compared to copper $\left(Cu\right)$ and aluminum oxide $\left(A{l}_2{O}_3\right)$ nanoparticles. The Casson fluid parameter represents the dual behavior for the momentum profile, initially momentum field decreases due to the Casson parameter but it then reverses its impact and the fluid flow moves more progressively.

Present research deals with the time-fractional Schrödinger equations aiming for the analytical solution via Shehu Transform based Adomian Decomposition Method [STADM]. Three types of time-fractional Schrödinger equations are tackled in the present research. Shehu transform ADM is incorporated to solve the time-fractional PDE along with the fractional derivative in the Caputo sense. The developed technique is easy to implement for fetching an analytical solution. No discretization or numerical program development is demanded. The present scheme will surely help to find the analytical solution to some complex-natured fractional PDEs as well as integro-differential equations. Convergence of the proposed method is also mentioned.

This article shows how to solve the time-fractional Fisher's equation through the use of two well-known analytical methods. The techniques we propose are a modified form of the Adomian decomposition method and homotopy perturbation method with a Yang transform. To show the accuracy of the suggested techniques, illustrative examples are considered. It is confirmed that the solution we get by implementing the suggested techniques has the desired rate of convergence towards the accurate solution. The main benefit of the proposed techniques is the small number of calculations. To show the reliability of the suggested techniques, we present some graphical behaviors of the accurate and analytical results, absolute error graphs and tables that strongly agree with each other. Furthermore, it can be used for solving fractional-order physical problems in various fields of applied sciences.

The major goal of this research is to use a new integral transform approach to obtain the exact solution to the time-fractional convection-reaction-diffusion equations (CRDEs). The proposed method is a combination of the Elzaki transform and the homotopy perturbation method. He's polynomial is used to tackle the nonlinearity which arise in our considered problems.Three test examples are considered to show the accuracy of the proposed scheme. In order to find satisfactory approximations to the offered problems, this work takes into account a sophisticated methodology and fractional operators in this context. In order to achieve better approximations after a limited number of iterations, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for CRDEs. It has been found that the proposed method's solution converges at the desired rate towards the accurate solution. We give some graphical representations of the accurate and analytical results, which are in excellent agreement with one another, to demonstrate the validity of the suggested methodology. For validity of the present technique, the convergence of the fractional solutions towards integer order solution is investigated. The proposed method is found to be very efficient, simple, and suitable to other nonlinear problem raised in science and engineering.