This study intends to significantly enhance the capacity of decision experts (DEs) to capture their judgment in a larger area. In order to accomplish this, we propound the r, s, t-spherical fuzzy set (r, s, t-SFS), an expansion of the t-spherical fuzzy set. In r, s, t-SFS the sum of the rth power of membership grade, sth power of neutral grade and the tth power of non-membership grade is less than or equal to 1, where r, s and t are natural numbers. Due to the inclusion of the extra parameters r and s, the r, s, t-SFS is able to describe assessment information in a more flexible and comprehensive manner. This work begins by defining r, s, t-SFS and demonstrating that it is an extension of various existing fuzzy sets. The fundamental operations, score, and accuracy functions of r, s, t-SFS are then introduced, and their mathematical features are examined. Also, we study some distance measures between r, s, t-SFSs and their required properties. Next, to aggregate r, s, t-spherical fuzzy data, r, s, t-spherical fuzzy weighted averaging (r, s, t-SFWA) and r, s, t-spherical fuzzy weighted geometric (r, s, t-SFWG) operators are bring forward along with some of their essential properties. Based on the proposed distance measure, maximizing deviation method is combined with r, s, t-spherical fuzzy information to establish the criteria weight determination method. Following this, we present r, s, t-spherical fuzzy VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method using the grounds of classical VIKOR method depending upon two focal properties, namely, group utility and individual regret of opponent. To demonstrate the use of the framed approach and exhibit its validity, we present a case study of arc welding robot selection. Besides, the effectiveness and accuracy of the proposed VIKOR are proved by parameter analysis and comparison analysis findings.

In this paper, we used the Natural decomposition approach with nonsingular kernel derivatives to explore the modified Boussinesq and approximate long wave equations. These equations are crucial in defining the features of shallow water waves using a specific dispersion relationship. In this research, the convergence analysis and error analysis have been provided. The fractional derivatives Atangana-Baleanu and Caputo-Fabrizio are utilised throughout the paper. To obtain the equations results, we used Natural transform on fractional-order modified Boussinesq and approximate long wave equations, followed by inverse Natural transform. To verify the approach, we focused on two systems and compared them to the exact solutions. We compare exact and analytical results with the use of graphs and tables, which are in strong agreement with each other, to demonstrate the effectiveness of the suggested approaches. Also compared are the results achieved by implementing the suggested approaches at various fractional orders, confirming that the result comes closer to the exact solution as the value moves from fractional to integer order. The numerical and graphical results show that the suggested scheme is computationally very accurate and simple to investigate and solve fractional coupled nonlinear complicated phenomena that exist in science and technology.

In this paper, the steady electrically conducting hybrid nanofluid (CuO-Cu/blood) laminar-mixed convection incompressible flow at the stagnation-point with viscous and gyrotactic microorganisms is considered. Additionally, hybrid nanofluid flow over a horizontal porous stretching sheet along with an induced magnetic field and external magnetic field effects that can be used in biomedical fields, such as in drug delivery and the flow dynamics of the microcirculatory system. This investigation can also deliver a perfect view about the mass and heat transfer behavior of blood flow in a circulatory system and various hyperthermia treatments such as the treatment of cancer. The simple partial differential equations (PDEs) are converted into a series of dimensional ordinary differential equations (ODEs), which are determined using appropriate similarities variables (HAM). The influence of the suction or injection parameter, mixed convection, Prandtl number, buoyancy ratio parameter, permeability parameter, magnetic parameter, reciprocal magnetic prandtl number, bioconvection Rayleigh number, coupled stress parameter, thermophoretic parameter, Schmidt number, inertial parameter, heat source parameter, and Brownian motion parameter on the concentration, motile microorganisms, velocity, and temperature is outlined, and we study the physical importance of the present problem graphically.

The p,q -quasirung orthopair fuzzy set ( p,q -ROFS) is a potent tool for representing fuzziness and uncertainty compared to the q -rung orthopair fuzzy set (q-ROFS). This research aims to introduce the aggregation technique of p,q -ROFSs with the help of Aczel–Alsina (AA) operations. We first propound several novel AA operations of p,q -ROFSs such as AA sum, AA product, AA scalar multiplication, and AA exponentiation. Following these operations, we develop a range of p,q -quasirung orthopair fuzzy averaging and geometric aggregation operators to efficiently aggregate p,q -quasirung fuzzy data. Additionally, we construct different features of these operators, discuss certain special cases, and study their fundamental results. Afterward, we use these operators to design a method for dealing with multi-criteria decision-making with p,q -quasirung orthopair fuzzy information. Finally, we provide a case study to demonstrate the suggested method’s practicality followed by parameter analysis and comparison study.

In this paper, the new iterative transform method and the homotopy perturbation transform method was used to solve fractional-order Equal-Width equations with the help of Caputo-Fabrizio. This method combines the Laplace transform with the new iterative transform method and the homotopy perturbation method. The approximate results are calculated in the series form with easily computable components. The fractional Equal-Width equations play an essential role in describe hydromagnetic waves in cold plasma. Our object is to study the nonlinear behaviour of the plasma system and highlight the critical points. The techniques are very reliable, effective, and efficient, which can solve a wide range of problems arising in engineering and sciences.