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Abstract The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work.

In this analysis, Sutterby fluid model over a curved surface is considered. The main mechanisms that contribute to the improvement of the convection characteristics of the nanofluid are categorized as Brownian motion and thermophoresis. The radiation and slip mechanism have been studied at curved stretchable surface. The suction/injection impacts also studied. The partial differential equations are converted into ordinary differential equations through transformations.The numerical solution of the specified mathematical model is obtained using the built-in bvp4c tool in MATLAB. The effects of various parameters related to the system of ordinary differential equations are illustrated in the graphs. The influence of some intended parameters through the physical quantities are presented through tabular form.

Based on the notion of class of CG simulation functions, we launch the concept of (ρ,ϑ)-ZG-contraction on the setting of w-distance mappings. We employ our new concept to formulate and prove several fixed point results over the complete metric spaces. Also, we provide some concrete examples to show the usability of the obtained results. Furthermore, we study the existence of a unique solution of nonlinear fractional differential equations, nonlinear integral equations and spring-mass system problems. The results presented in this manuscript generalize many of the results known in the literature.

Abstract A steady flow of Maxwell Sutterby fluid is considered over a stretchable cylinder. The magnetic Reynolds number is considered very high and induced magnetic and electric fields are applied on the fluid flow. Joule heating and radiation impacts are studied under the temperature-dependent properties of the liquid. Having the above assumptions, the mathematical model has been evolving via differential equations. The differential equations are renovated in the dimensionless form of ordinary differential equations using the appropriate transformations. The numerical results have been developed employing numerical techniques on the ordinary differential equations. The impact of involving physical factors on velocity, induced magneto hydrodynamic, and temperature function is debated in graphical and tabular form. The velocity profile is boosted by thicker momentum boundary layers, which are caused by higher values of the magnetic field factor. So, the fluid flow becomes higher velocity due to enlarging values of the magnetic field factor. Heat transfer factor and friction at surface factor boosted up for increment of $${\gamma }_{0}$$ γ 0 (Magnetic field factor). The $${\gamma }_{0}$$ γ 0 (Magnetic field factor) is larger which better-quality of heat transfer at surface and also offered the results of friction factor boosting up in both cases of stretching sheet/cylinder. The $${\lambda }_{0}$$ λ 0 (Magnetic Prandtl number) increased which provided better-quality of heat transfer at surface.

In this study, we considered the flow of a micropolar fluid over a vertical Riga sheet. The non linear stretching sheet is considered. The effects of variable thermal conductivity and radiation on the Riga sheet are taken into account. Additionally, we also debated the Brownian motion and thermophoretic. To simplify the partial differential equations, we converted them into dimensionless ordinary differential equations using suitable similarity variables and solved dimensionless system numerically using the bvp4c function. The impact of some intended parameters on the dimensionless velocity, microrotation, temperature, and concentration distributions graphically are presented and the numerical outcomes of physical quantities like skin friction, Nusselt number, Sherwood number, and couple stress have been presented in tabular form. The micropolar parameter increased which increased the couple stress and friction at surface. Because, the fluid rotation increased which increased friction at surface and also increased the couple stress. The transfer of mass decayed and transfer of heat heightened by larger values of variable thermal conductivity. Thermal conductivity improved which improved the heat transfer phenomena, so transfer of heat at surface becomes larger while also reducing the transfer of mass.

The two-dimensional flow of Sutterby-Casson fluid flow over an exponentially curved sheet is considered. Incompressible and steady flow is taken into account. The magnetic and electric fields are ignored due to the small Reynolds number in the flow assumptions. Magnetic and chemical dipoles are considered at stretchable curved sheet. Heat generation and slip phenomena have been considered in the present analysis. The governing models of the above assumptions are reduced by implementing the boundary layer approximations and gained nonlinear coupled differential equations. The couple differential equation system is reduced in a dimensionless system (ordinary differential equations) using appropriate transformations. The dimensionless system is solved through a numerical scheme due to finding the results involving physical factors. The results are highlighted in graphs and as well as tables. Concentration achieved higher values by enlargement in the Brownian factor. The movement of particles becomes faster due to the enlargement of the Brownian factor, ultimately increment in concentration. Energy activation factor values enlarged which gained lesser values of Nusselt factor and mass transfer factor while friction factor remains constant. The reaction of chemical factor impact on mass transfer, friction factor, and heat transfer have been debated.

In this analysis, Sutterby nanofluid flow with an induced magnetic field at a nonlinear stretching cylinder is deliberated. The effects of variable thermal conductivity, Darcy resistance, and viscous dissipation are discussed. Thermal radiation and chemical reaction are considered to analyze the impact on the nonlinear stretching cylinder. The governing model of the flow problem is developed under the boundary layer approximation in terms of partial differential equations. Partial differential equations are transformed into ordinary differential equations by performing the suitable transformations. A numerical structure is applied to explain ordinary differential equations. The impact of each governing physical parameters on the temperature, concentration, skin friction, Sherwood, and Nusselt number is presented in graphs and tabular form. Increment in Prandtl number, which declined the curves of the temperature function. Temperature declined because the Prandtl number declined the thermal thickness as well as reduce the temperature of the fluid. Temperature curves showed improvement as Eckert number values increased because the Eckert number is a ratio of kinetic energy to the specific enthalpy difference between the wall and the fluid. As a result, increasing the Eckert number causes the transformation of kinetic energy into internal energy via work done against viscous fluid stresses.

In the current analysis, steady incompressible Sutterby fluid flows over a stretching cylinder are studied. The influence of variable thermal conductivity is considered in the presence of thermal slip, Darcy resistance, and sponginess. The impact of the induced magnetic field is considered to analyze the results at the cylindrical surface. The governing equations are established as partial differential equations using the boundary layer approximation. Appropriate transformations are used to convert partial differential equations into ordinary differential equations. The numerical technique, namely (bvp4c), is applied to ordinary differential equations to develop the results. The numerical results, such as heat transfer rate and skin friction, are revealed by tabular form to demonstrate the physical impact of governing factors. The physical impact of governing factors on induced magnetic hydrodynamic, velocity, and temperature profiles is presented through various graphs. The velocity function deteriorated due to the augmentation of the Sutterby fluid parameter.

In this analysis, the generalized Fourier and Fick's law for Second-grade fluid flow at a slendering vertical Riga sheet is examined along with thermophoresis and Brownian motion effects. Boundary layer approximations in terms of PDE's (Partial Differential Equations) are used to build the mathematical model. An appropriate transformation has been developed by using the Lie symmetry method. PDE's (Partial Differential Equations) are transformed into ODE's (Ordinary Differential Equations) by implementing the suitable transformation. A numerical method called bvp4c is used to explain the dimensionless system (ODE's). Graphs and tables are used to interpret the impact of the significant physical parameters. The curves of temperature function declined due to enchanting the values of the thermophoresis Parameter. The temperature is produced at a low level due to enchanting the values of thermophoresis because this force transports burn at a low 10 μm diameter so the temperature becomes lessor. Increments of thermophoresis parameter which enhanced the values of concentration Function. As the concentration boundary layer increased which declined the mass transfer due increment in thermophoresis. The curves of temperature function are increasing due to enhancing the values of the Brownian parameter because addition in the Brownian motion, improved the movement of particles ultimately increasing the kinematic energy of fluid which improved the heat transfer phenomena. Increments of Brownian parameter which declined the values of concentration function. Physically, the kinematic energy improved which declined the mass transfer rate near the surface.

The purpose of this paper is to define the notion of extended convex ℱ contraction by imposing less conditions on the function ℱ satisfying certain contractive conditions. We prove the existence of fixed points for these types of mappings in the setting of b-metric spaces. In addition, some illustrative examples are provided to show the usability of the obtained results. Lastly, we use the obtained fixed-point results to find the fractals with respect to the iterated function systems in the framework of b-metric spaces. Furthermore, the variables involved in the b-metric space are symmetric, and symmetry plays an important role in solving the nonlinear problems defined in operator theory.