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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. [ABSTRACT FROM AUTHOR]

The conditioning theory of the generalized inverse C A ‡ is considered in this article. First, we introduce three kinds of condition numbers for the generalized inverse C A ‡ , i.e., normwise, mixed and componentwise ones, and present their explicit expressions. Then, using the intermediate result, which is the derivative of C A ‡ , we can recover the explicit condition number expressions for the solution of the equality constrained indefinite least squares problem. Furthermore, using the augment system, we investigate the componentwise perturbation analysis of the solution and residual of the equality constrained indefinite least squares problem. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator to devise the first algorithm and the small-sample statistical condition estimation method for the other two algorithms. In the end, the numerical examples illuminate the obtained results. [ABSTRACT FROM AUTHOR]

In this article, first we will present the new rigorous perturbation bounds with normwise perturbation for the generalized Cholesky block downdating problem by combining the unified matrix–vector equation approach with the method of Lyapunov majorant functions and the Banach fixed point theorem. Then, we will derive the explicit expressions for the six distinct kinds of condition numbers, i.e. four normwise ones, mixed and componentwise ones. Furthermore, with the help of probabilistic spectral norm estimator and the small-sample statistical condition estimation method, these condition numbers can be estimated with high accuracy. At the end, the obtained results are illuminated by the numerical examples. [ABSTRACT FROM AUTHOR]

The explicit expressions of normwise, mixed and componentwise condition numbers for the K-weighted pseudoinverse L K † are first presented. With the intermediate result, i.e. the derivative of L K † , we can recover the explicit expressions of condition numbers for solution of least squares problem with equality constraint. Then, we investigate the statistical estimation of condition numbers of L K † using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, numerical examples are provided to illustrate these results. [ABSTRACT FROM AUTHOR]

In this paper, we give the explicit expressions for the strong and the weak rigorous multiplicative perturbation bounds for the block Cholesky downdating problem. First, we derived the strong rigorous multiplicative perturbation bounds by combining the modified matrix-vector equation approach with the strategy for Lyapunov majorant function and the Banach fixed point theorem. Then, the weak rigorous multiplicative bounds are obtained with the help of the matrix-equation approach. The obtained results are illustrated by numerical tests. [ABSTRACT FROM AUTHOR]

Some improved rigorous perturbation bounds with normwise perturbation for the block Cholesky downdating problem are first derived by combining the modified matrix-vector equation approach with the strategy for Lyapunov majorant function and the Banach fixed point theorem. Then, we investigate four distinct kinds of condition numbers, i.e. two normwise ones, and mixed and componentwise ones, for this problem, and present their explicit expressions. Furthermore, using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, we also consider the statistical estimation of these condition numbers and design two algorithms. The obtained results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Condition number plays an important role in perturbation analysis, the latter is a tool to judge whether a numerical solution makes sense, especially for ill-posed problems. In this paper, perturbation analysis of the Tikhonov regularization of total least squares problem (TRTLS) is considered. The explicit expressions of normwise, mixed and componentwise condition numbers for the TRTLS problem are first presented. With the intermediate result, i.e. normwise condition number, we can recover the upper bound of TRTLS problem. To improve the computational efficiency in calculating the normwise condition number, a new compact and tight upper bound of the TRTLS problem is introduced. In addition, we also derive the normwise, mixed and componentwise condition numbers for TRTLS problem when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed. We choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method to estimate these condition numbers with high reliability. Numerical experiments are provided to verify the obtained results. [ABSTRACT FROM AUTHOR]

The explicit expressions of the normwise, mixed, and componentwise condition numbers and their upper bounds for the generalized Cholesky factorization are first obtained. Then, some improved rigorous perturbation bounds with normwise or componentwise perturbation in the given matrix are derived by bringing together the modified matrix-vector equation approach with the method of Lyapunov majorant function and the Banach fixed point theorem. Theoretical and experimental results show that these new bounds are always tighter than the corresponding ones in the literature. [ABSTRACT FROM AUTHOR]

The present investigation aims to use entropy analysis to analyze the unsteady Magnetohydrodynamic (MHD) flow in a second-grade fuzzy hybrid Al 2 O 3 - Cu/SA nanofluid over an exponentially shrinking/stretching surface. The model for hybridization of the mixture of alumina Al 2 O 3 and copper (Cu) nanoparticles in the sodium alginate (SA) base fluid under heat source/sink, nonlinear thermal radiation, and viscous dissipation. The fundamental partial differential equations (PDEs) are simplified using an appropriate similarity conversion to generate the ordinary differential equations (ODEs). The analytical computation occurs in the MATHEMATICA program implementing the homotopy analysis method (HAM). In terms of code validity, our results are preferable to previous findings. The features of several parameters against the velocity, surface friction coefficient, entropy, temperature, and Nusselt number are described through graphs. According to our findings, the rise in the Brinkman and Reynolds numbers enhanced the total entropy of the system. Furthermore, the nanoparticle volume fraction and viscus dissipation magnifies the fluid temperature while retards the flow profile throughout the domain. Fluid velocity declined due to the Lorentz force using magnetic impact applications. The imprecision of nanofluid and hybrid nanofluid volume fractions was modelled as a triangular fuzzy number (TFN) [0%, 1%, 2%] for comparison. The double parametric approach was applied to deal with the fuzziness of the associated fuzzy parameters. The nonlinear ODEs convert into fuzzy differential equations (FDEs) and use HAM for the fuzzy solution. 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 percentage increase in skin friction factor is 18.3 and 15.0 when α and β take input in the ranges of 0 ≤ α ≤ 0.8 and 0 ≤ β ≤ 1, respectively. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This contribution aims to optimize nonlinear thermal flow for Darcy-Forchheimer Maxwell fuzzy 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, Al 2 O 3 , Cu and 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 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 ≤ Mh ≤ 0.3 and wedge angle parameters 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. [ABSTRACT FROM AUTHOR]

This contribution aims to optimize nonlinear thermal flow for Darcy-Forchheimer Maxwell fuzzy 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, Al 2 O 3 , Cu and 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 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 ≤ Mh ≤ 0.3 and wedge angle parameters 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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the accelerated expansion of the Universe in the context of f (Q) modified theory of gravity, where Q is a non-metricity scalar which characterizes the gravitational interaction by using parametrization of the deceleration parameter q = α − β H with f (Q) = ξ + λ Q , where ξ and λ are free parameters constrained by the 57 points of H (z) datasets, 1048 points of Pantheon, 10 points from Baryon Acoustic Oscillations (BAO) datasets and the shift parameters from Planck 2018 of Cosmic Microwave Background (CMB). In the purpose of validating our model, we proceed by the Om diagnostic and the energy conditions. Later we discussed how our model statistically supports Λ CDM using AIC c criterion analysis. [ABSTRACT FROM AUTHOR]