This contribution aims to optimize nonlinear thermal flow for Darcy-Forchheimer Maxwell fuzzy $$\left( {{\text{Al}}_{{2}} {\text{O}}_{{3}} + {\text{Cu }} + {\text{TiO}}_{{2}} {\text{/EO}}} \right)$$Al 2 O 3 + Cu + TiO 2 /EO tri-hybrid nanofluid flow across a Riga wedge in the context of boundary slip. Three types of nanomaterials, $${\text{Al}}_{{2}} {\text{O}}_{{3}} ,$$Al 2 O 3 , Cu and $${\text{TiO}}_{2}$$TiO 2 have been mixed into the basic fluid known as engine oil. Thermal properties with the effects of porous surface and nonlinear convection have been established for the particular combination $$\left( {{\text{Al}}_{{2}} {\text{O}}_{{3}} + {\text{Cu}} + {\text{TiO}}_{{2}} {\text{/EO}}} \right){.}$$Al 2 O 3 + Cu + TiO 2 /EO . Applying a set of appropriate variables, the set of equations that evaluated the energy and flow equations was transferred to the dimensionless form. For numerical computing, the MATLAB software's bvp4c function is used. The graphical display is used to demonstrate the influence of several influential parameters. It has been observed that flow rate decay with expansion in porosity parameter and nanoparticles volumetric fractions. In contrast, it rises with wedge angle, Grashof numbers, Darcy-Forchheimer, nonlinear Grashof numbers, and Maxwell fluid parameter. Thermal profiles increase with progress in the heat source, nanoparticles volumetric fractions, viscous dissipation, and nonlinear thermal radiation. The percentage increases in drag force for ternary hybrid nanofluid are 13.2 and 8.44 when the Modified Hartmann number takes input in the range $$0.1 \le {\text{Mh}} \le 0.3$$0.1 ≤ Mh ≤ 0.3 and wedge angle parameters $$0.1 \le m \le 0.3$$0.1 ≤ m ≤ 0.3 . For fuzzy analysis, dimensionless ODEs transformed into fuzzy differential equations and employed symmetrical triangular fuzzy numbers (TFNs). The TFN makes a triangular membership function (M.F.) that describes the fuzziness and comparison. This study compared nanofluids, hybrid nanofluids, and ternary nanofluids through triangular M.F. The boundary layer flow caused by a wedge surface plays a crucial role in heat exchanger systems and geothermal.

The study of hybrid nanoliquids can aid in developing numerous advanced features that facilitate heat transmission, such as pharmaceutical processes, hybrid-powered engines, microelectronics, engine cooling, and domestic refrigerators. In the current study, a mathematical model is designed to elaborate the physical inception of an unsteady second-grade hybrid nanofluid with Al2O3−Cu/SA, a combination concentrated over the permeable exponentially heated stretching/shrinking sheet under hydromagnetic, heat source/sink, and viscous dissipation implications. The set of similarity transforms is used to convert underlying partial differential equations into the system of ordinary differential equations. The well-known homotopy analysis method is applied to tackle the formulated differential system in the MATHEMATICA program, which can obtain non-uniqueness outcomes. The imprecision of nanofluid and hybrid nanofluid volume fractions was modeled as a triangular fuzzy number [0%, 5%, 10%] for comparison. The double parametric approach was applied to deal with the fuzziness of the associated fuzzy parameters. The nonlinear ordinary differential equations are converted into fuzzy differential equations, and the homotopy analysis method is used for the fuzzy solution. In terms of code validity, our results are matched to previous findings. The features of several parameters against the velocity, surface-friction coefficient, heat transfer, and Nusselt number are described via graphs. Furthermore, the nanoparticle volume fraction magnifies the fluid temperature and retards the flow profile throughout the domain, according to our findings. Thermal profiles increase with progress in the heat source, nanoparticles volumetric fractions, viscous dissipation, and nonlinear thermal radiation. The percentage increase in the drag force and heat transfer rate are 15.18 and 5.54 when the magnetic parameter takes input in the range 0.1 ≤ M ≤ 0.3 and nanoparticle volume fraction inputs 0.01 ≤ ϕ1 ≤ 0.15. From our observation, the hybrid nanofluid displays the maximum heat transfer compared to nanofluids. This important contribution will support industrial growth, particularly in the processing and manufacturing sectors.

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 work uses the collocation approximation method to solve a specific type of backward stochastic Volterra integral equations (BSVIEs). Using Newton’s method, BSVIEs can be solved using block pulse functions and the corresponding stochastic operational matrix of integration. We present examples to illustrate the estimate analysis and to demonstrate the convergence of the two approximating sequences separately. To measure their accuracy, we compare the solutions with values of exact and approximative solutions at a few selected locations using a specified absolute error. We also propose an efficient method for solving a triangular linear algebraic problem using a single integral equation. To confirm the effectiveness of our method, we conduct numerical experiments with issues from real-world applications.

This article is devoted to the structured and unstructured condition numbers for the total least squares with linear equality constraint (TLSE) problem. By making use of the dual techniques, we investigate three distinct kinds of unstructured condition numbers for a linear function of the TLSE solution and three structured condition numbers for this problem, i.e., normwise, mixed, and componentwise ones, and present their explicit expressions under both unstructured and structured componentwise perturbations. In addition, the relations between structured and unstructured normwise, componentwise, and mixed condition numbers for the TLSE problem are investigated. Furthermore, using the small-sample statistical condition estimation method, we also consider the statistical estimation of both unstructured and structured condition numbers and propose three algorithms. Theoretical and experimental results show that structured condition numbers are always smaller than the corresponding unstructured condition numbers.