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In a stagnation point flow, the rate at which heat transfers in fluid containing nanoparticles across a sheet that is stretchable on a surface having pores has been investigated in this research. Magnetohydrodynamic viscous nanofluid flow is considered that is subjected to Brownian movements and the thermophoresis effect. By utilizing a numerical technique, the characteristics of heat transmission in nanofluids are investigated. The model is based on momentum, energy, and concentration equations. To explain the flow model's physical significance, zero mass flux condition has been employed at the surface. Nonlinear partial differential equations are transformed into a collection of linked ordinary differential equations via similarity transformations. Convergent implications of nonlinear systems are produced by MATLAB software's built-in bvp4c algorithm. To indicate the physical importance, a thorough examination of relevant characteristics, such as heat sink/source, porosity, and magnetic parameter is conducted. We have observed the behavior of profiles by fixing the numerical values of the involving parameters as 0.1 ≤ λ ≤ 2.0, 0.1 ≤ Nr ≤ 3.0, 0.1 ≤ R ≤ 0.4, 0.1 ≤ M ≤ 0.4, 0.1 ≤ Rb ≤ 1.5, and 0.1 ≤ Nb ≤ 0.7. The temperature rises yet the rate at which heat transfers at the surface declines due to the increased far-field velocity. The greater nanoparticles concentration at the far field relative to the surface is related to the zero mass flux condition. [ABSTRACT FROM AUTHOR]

The power-law nanofluid's heat and mass transmission over a stretched surface with magnetic field, entropy optimization, and chemical reaction effects is the main focus of the present study. The governing partial differential equations are reduced using similarity transformation into ODEs (ordinary differential equations), which are then numerically solved with the help of the Keller box and finite difference techniques. A substantial evaluation is conducted using the controlling flow factors, which include the Eckert number, chemical reaction factor, Brownian motion parameter, thermophoresis number, power-law index (n), and generalized Prandtl number (Pr), on the fluid velocity profile, flowing fluid temperature, and concentration profile. A decrease in the velocity profile is observed when the chemical reaction process is accelerated, leading to heat absorption. However, when the power-law parameter increases, the fluid's temperature decreases, and the velocity profile increases at specific values of viscous dissipation (Ec = 0.1) and chemical reaction (γ = 0.01). The values of the coefficient of skin friction (Cf), Sherwood factor (Shx), and reduced local Nusselt number ( N u x are also computed and presented in the form of a table for the generalized Prandtl number (Pr = 7.0). The coefficient of skin friction and mass transfer and heat transfer rates are quantitatively compared with those in the literature for particular circumstances, and the results demonstrate good agreement with each other. This research offers insightful information for designing and improving systems that use nanofluids exposed to external magnetic fields, which has ramifications in many technical uses, including heat exchange systems based on nanofluids and thermal management systems. [ABSTRACT FROM AUTHOR]

In this work, we consider the degenerate Frobenius-Euler-Genocchi polynomials utilizing the degenerate exponential function and the degenerate Changhee-Frobenius-Euler-Genocchi polynomials utilizing the degenerate logarithm function. Then, we analyze some summation and addition formulas for these polynomials. In addition, we derive some correlations with degenerate Stirling numbers of both kinds and degenerate Frobenius-Euler polynomials. Moreover, we present difference and derivative operator rules for the generalized degenerate Frobenius-Euler-Genocchi polynomials. Lastly, we show certain zeros of both the degenerate Frobenius-Euler-Genocchi polynomials and the degenerate Changhee-Frobenius-Euler-Genocchi polynomials and provide their beautifully graphical representations. [ABSTRACT FROM AUTHOR]

This study presents a piecewise non-linear binary subdivision scheme based on a four-point linear binary subdivision scheme. This scheme is first transformed into a piecewise linear subdivision scheme and then into a piecewise non-linear subdivision scheme. The non-linear subdivision approach removes artifacts and distortions from limit curves, producing better continuous curves. The study demonstrates that this technique is a workable technique for producing smooth curves, especially when avoiding bends, distortion, and oscillation. [ABSTRACT FROM AUTHOR]

Special functions and special polynomials have been used and studied widely in the context of Geometric function theory of Complex Analysis by the many authors. Here, in our present investigations, making use of the convolution, we first introduce a new subclass of Bazilevivč bi-univalent functions involving the Bernoulli Polynomials. We then find the first two Taylor-Maclaurin coefficient | a 2 | and | a 3 | for our defined functions class. We then calculate the bounds for the Fekete Szegö inequality for the defined function class. Also some known consequences of our main results are highlighted in the form of Corollaries. [ABSTRACT FROM AUTHOR]

In this paper, we present two collocation methods utilizing subdivision schemes and Bernstein polynomials as their basis functions to solve Caputo-type fractional differential equations. These methods transform fractional differential equations into systems of linear equations, which can be solved using various suitable techniques. Notably, these methods are versatile, making them applicable to both boundary value problems and initial value problems. To demonstrate their effectiveness, we applied these methods to three test problems of fractional differential equations, including the Bagley-Torvik equation and fractional oscillation equation. Our results reveal that both methods provide a more accurate approximation to the exact solution when compared to various algorithms found in the existing literature, showcasing their efficiency, accuracy, and consistency. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). Besides, the proposed GFPDEs represent a great generalization of a significant type of FPDEs and their applications, which contain many previous reports as a special case. Moreover, the proposed scheme uses shifted Chebyshev sixth-kind (SCSK) polynomials with spectral collocation approach. The fractional differential derivatives are expressed in terms of the Caputo's definition. Furthermore, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. In addition, the classical fourth-order Runge-Kotta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple and effective numerical method.

Various integral representations of hypergeometric functions have been introduced and investigated due to their important applications in divers fields. In this article, we define some new Euler-type integral representations for the Horn's functions of two variables G1, G2, G3 and H1.

The scourge of water pollution due to organic matter is a well-known fact. In order to address this issue, we employed a highly facile method of solid state reaction for the synthesis of bismuth potassium titanate, Bi0.5K0.5TiO3 (BKT) photocatalyst. Furthermore, the bandgap tailoring of the BKT is performed by photo-reduction of Ag onto the BKT, which results in the formation of hetero-structure junctions at the surface of the BKT. The as-prepared BKT and BKT-Ag photocatalysts are systematically characterized by x-ray diffraction, scanning electron microscopy, and high resolution transmission electron spectroscopy. Furthermore, the photocatalyst activity of the BKT and BKT-Ag was analyzed against the target organic pollutants by using UV–Vis spectroscopy. Three organic dyes such as methylene blue, eriochrome Black T, and methylene orange (MO) were used as target organic pollutants. The anchoring of Ag for the formation of hetero-structure junctions at the surface of BKT rendered it to be an ideal photo-catalyst with a bandgap reduction from 3.37 to 2.08 eV. Therefore, lowering the bandgap resulted in enhanced photocatalytic activity of the BKT-Ag. Among the three target dyes, BKT-Ag showed remarkable photo-degradation of MO and Erb T by degrading ≥90% of MO and Erb T in 120 min at neutral pH values. It is believed that BKT based photocatalysts can provide a unique platform for the removal of organic pollutants from wastewater. [ABSTRACT FROM AUTHOR]

The theme of this study is to investigate the theoretical and numerical simulation of heat and mass transfer of magneto hydrodynamic power-law nanofluid flow over a stretching sheet with surface heat flux and Soret/Dufour effects. The mass flux and energy flux increase with temperature and concentration differences using the Soret/Dufour impact. The similarity transformations were used to transform a physical problem into a non-linear differential equation, and the non-linear equations were solved by the Keller box technique. The Soret/Dufour and magnetic field are incorporated into a nanofluid model. The similarity transformation was used to reduce thermal energy, mass, and momentum in algebraic systems. The impact of nanofluid factors, such as generalized Brownian motion parameter Nb, Dufour parameter Df, Soret parameter Sr, Le, and Pr, are generalized Lewis and Prandtl numbers; the thermophoresis parameter Nt and magnetic parameter on dimensionless stretching surface functions are shown numerically and graphically. The quantitative relationship between heat transfer and skin friction is shown by using Keller box and MATLAB. The skin friction coefficient Cf, Sherwood number Shx, and Nusselt number Nux values were also computed and displayed on the graph. The increment in slip temperature, fluid velocity, and fluid concentration is enhanced with a high Dufour parameter. The temperature variation and fluid concentration are decreased with applied-magnetic effects because the magnetic field acts like an insulating material in heat transfer systems. The enhancement in the Nusselt number and Sherwood number is increased with Soret and Dufour effects. [ABSTRACT FROM AUTHOR]

In this paper, we introduce ten new quadruple hypergeometric series. We also obtain their various properties such that integral representations, fractional derivatives, N-fractional connections, operational relations and generating functions. [ABSTRACT FROM AUTHOR]

The theory of mixed fractional operators is still an uncovered area in fractional modelling. These multi-sided operators result by combining two fractional derivatives with different kernels, that is, the right-sided Caputo's and the left-sided Riemann-Liouville's fractional operators in a sequential manner. Although it may capture different memory phenomenons, there is no general approach to approximate the solution of the mixed type fractional differential equations (MFDEs). In this article, the authors introduce a novel numerical technique that relies on obtaining the matrix representation of the mixed operators. This was done by considering a new extension of the shifted Chebyshev polynomials of the fifth-kind (SFCPs) with a variable domain as a basis of L2 space. Then we employ the spectral collocation method together with the operational matrix to transform the MFDE into the corresponding system of algebraic equations. The convergence analysis of the proposed technique was studied in terms of the generalized Taylor's formula and the Gram determinant. Our findings facilitate further development of this theory by providing such a method of approximation. [ABSTRACT FROM AUTHOR]

This paper introduces a modified method for construction of a new design-control 6-point approximating refinement scheme. The construction of the new scheme is based on translation of the points of 4-point approximating refinement scheme to the new position according to the linear combination of certain displacement vectors. The initial and terminal points of these vectors are the refinement points of two 4-point approximating binary schemes. The new scheme contains three design-control parameters. These parameters increase the efficiency and flexibility of the new scheme. The mathematical and graphical analysis of the refinement scheme show that this scheme is good choice for curve modeling. [ABSTRACT FROM AUTHOR]

We study bivariate Mittag-Leffler function which is an extension of several known Mittag-Leffler functions. We derive the Riemann-Liouville fractional derivatives and integrals of these function and solve a singular integral equation with the bivariate Mittag-Leffler function in the kernel. We also introduce and investigate a fractional integral operator involving the bivariate Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

Recently, Casadei [4] provided an explicit formula for statistical marginal model in terms of Poisson-Gamma mixture. This model involving certain polynomials which play the key role in reference analysis of the signal and background model in counting experiments. The principal object of this paper is to present a natural further step toward the mathematical properties concerning this polynomials. We Ąrst obtain explicit representations for these polynomials in form of the Laguerre polynomials and the conĆuent hyper-geometric function and then based on these representations we derive a number of useful properties including generating functions, recurrence relations, differential equation, Rodrigues formula, Ąnite sums and integral transforms. [ABSTRACT FROM AUTHOR]

In this paper, we present several operator identities by using certain inverse pairs of symbolic operators. Using the obtained operator identities, we establish many decomposition formulas for Gaussian hypergeometric functions in three variables. Furthermore, some transformation formulas are introduced. [ABSTRACT FROM AUTHOR]

In this paper, the higher-order type 2 Daehee polynomials are introduced and some of their relations and properties are derived. Then, some p-adic integral representations of not only higher-order type 2 Daehee polynomials and numbers but also type 2 Daehee polynomials and numbers are acquired. Several identities and relations related to both central factorial numbers of the second kind and Stirling numbers of the first and second kinds are investigated. Moreover, the conjugate higher-order type 2 Daehee polynomials are considered and some correlations covering the type 2 Daehee polynomials of order β and the conjugate higher-order type 2 Daehee polynomials are attained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce partially degenerate Laguerre–Bernoulli polynomials of the first kind and deduce some relevant properties by using a preliminary study of these polynomials. We derive some theorems on implicit summation formulae for partially degenerate Laguerre–Bernoulli polynomials of the first kind L B j (ξ , η | λ). Finally, we derive some symmetry identities for partially degenerate Laguerre–Bernoulli polynomials of the first kind. [ABSTRACT FROM AUTHOR]

In this paper, the higher-order type 2 Daehee polynomials are introduced and some of their relations and properties are derived. Then, some p-adic integral representations of not only higher-order type 2 Daehee polynomials and numbers but also type 2 Daehee polynomials and numbers are acquired. Several identities and relations related to both central factorial numbers of the second kind and Stirling numbers of the first and second kinds are investigated. Moreover, the conjugate higher-order type 2 Daehee polynomials are considered and some correlations covering the type 2 Daehee polynomials of order β and the conjugate higher-order type 2 Daehee polynomials are attained. [ABSTRACT FROM AUTHOR]

In recent years, using the idea of analytic and bi-univalent functions, many ideas have been developed by different well-known authors, but the using Gegenbauer polynomials along with certain bi-univalent functions is very rare in the literature. We are essentially motivated by this recent research going on, here in our present investigation, we make use of certain q-derivative operator and Gegenbauer polynomials and define a new subclass of analytic and bi-univalent functions. We then obtain certain coefficient bounds, the Fekete–Szegö inequalities and upper bounds for the second-order Hankel determinant for the defined functions class. [ABSTRACT FROM AUTHOR]

We extend several fractional Laplace transform results to matrix-valued functions in this paper, which we will utilize to obtain some useful and valuable theorems, like results on the set of the piecewise continuous functions with conformable exponential order and the low and conditions to obtain the fractional Laplace transform of matrix-valued functions. As an application, we apply the obtained theorems to solve certain fractional initial value problems for vector-valued and matrix-valued functions and the solution will be victor or matrix and it will exact. Also, we establish a fractional system transfer matrix and the conformable fractional Laplace transform of the conformable exponential matrix function, which will give us directly the exact solution for a precisely type of fractional initial value problems. [ABSTRACT FROM AUTHOR]

Dendrimers are spherical three-dimensional molecules with a repetitively branching core. They are normally symmetric around the core. Bismuth (III) iodide has the formula B i I 3 and is an inorganic chemical. The reaction between bismuth and iodine produces this gray-black solid, which was of great interest in qualitative inorganic analysis. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. One of these tools is a graphical representation of chemical molecules, known as the molecular graph of a chemical substance. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we will calculate a new topological index, namely, Sombor index, multiplicative Sombor index, and its reduced version for bismuth (III) iodide and dendrimers. We also plot our computed results of Sombor index, multiplicative Sombor index, and reduced Sombor index to examine how they were affected by the parameters involved. [ABSTRACT FROM AUTHOR]

In this study, we investigate the effectiveness of a robust sampled-data H ∞ load frequency control (LFC) scheme for power systems with randomly occurring time-varying delays. By using the input-delay technique, the sampled-data LFC model is reformulated as a continuous time-delay representation. Then, Bernoulli-distributed white noise sequences are used to describe randomly occurring time-varying delays in the sampled-data LFC model. Some less conservative conditions are achieved by utilizing the Lyapunov–Krasovskii functional (LKF) and employing Jensen inequality and reciprocal convex combination lemma to ensure the considered power system has mean-square asymptotic stability under the designed control scheme. The derived results are based on linear matrix inequalities (LMIs) that can readily be solved using the MATLAB LMI toolbox. The criteria obtained are used to analyze the upper bounds for time delays, and a comparison study to validate the efficacy of the presented method is presented. [ABSTRACT FROM AUTHOR]

In this work, an algorithm for finding numerical solutions of linear fractional delay-integro-differential equations (LFDIDEs) of variable-order (VO) is introduced. The operational matrices are used as discretization technique based on shifted Chebyshev polynomials (SCPs) of the first kind with the spectral collocation method. The proposed VO-LFDIDEs have multiterms of integer, fractional-order derivatives for delayed or nondelayed and mixed Volterra-Fredholm integral terms. The introduced model is a more general form of linear fractional VO pantograph, neutral, and mixed Fredholm–Volterra integro-differential equations with delay arguments. Caputo's VO fractional derivative operator is used to generate the matrices of the derivative. Operational matrices are presented for all terms. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, some examples are included to improve the validity and applicability of the techniques. Finally, comparisons between the proposed method and other methods were used to solve this kind of equation. [ABSTRACT FROM AUTHOR]

The main objective of this article is to establish generalized fractional Hermite–Hadamard and related type integral inequalities for h -Godunova–Levin convexity and h -Godunova–Levin preinvexity with extended Wright generalized Bessel function acting as kernel. Moreover, Hermite–Hadamard-type and trapezoid-type inequalities for several known convexities including Godunova–Levin function, classical convex, s -Godunova–Levin function, P -function, and s -convex function are deduced as corollaries. These obtained results are analyzed in the form of generalization of fractional inequalities. [ABSTRACT FROM AUTHOR]

Several (generalized) hypergeometric functions and a variety of their extensions have been presented and investigated in the literature by many authors. In the present paper, we investigate four new hypergeometric functions in four variables and then establish several recursion formulas for these new functions. Also, some interesting particular cases and consequences of our results are discussed. [ABSTRACT FROM AUTHOR]

In this article, we design a novel fourth-order and derivative free root-finding algorithm. We construct this algorithm by applying the finite difference scheme on the well-known Ostrowski's method. The convergence analysis shows that the newly designed algorithm possesses fourth-order convergence. To demonstrate the applicability of the designed algorithm, we consider five real-life engineering problems in the form of nonlinear scalar functions and then solve them via computer tools. The numerical results show that the new algorithm outperforms the other fourth-order comparable algorithms in the literature in terms of performance, applicability, and efficiency. Finally, we present the dynamics of the designed algorithm via computer tools by examining certain complex polynomials that depict the convergence and other graphical features of the designed algorithm. [ABSTRACT FROM AUTHOR]

Nowadays, the use of computers is becoming very important in various fields of mathematics and engineering sciences. Many complex statistics can be sorted out easily with the help of different computer programs in seconds, especially in computational and applied Mathematics. With the help of different computer tools and languages, a variety of iterative algorithms can be operated in computers for solving different nonlinear problems. The most important factor of an iterative algorithm is its efficiency that relies upon the convergence rate and computational cost per iteration. Taking these facts into account, this article aims to design a new iterative algorithm that is derivative-free and performs better. We construct this algorithm by applying the forward- and finite-difference schemes on Golbabai–Javidi's method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its quartic-order convergence. To analyze it numerically, we consider nine different types of numerical test examples and solve them for demonstrating its accuracy, validity, and applicability. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similar algorithms in the literature. For the graphical analysis, we consider some different degrees' complex polynomials and draw the polynomiographs of the designed quartic-order algorithm and compare it with the other similar existing methods with the help of a computer program. The graphical results reveal the better convergence speed and the other graphical characteristics of the designed algorithm over the other comparable ones. [ABSTRACT FROM AUTHOR]

Recently, hypergeometric functions of four variables are investigated by Bin-Saad and Younis. In this manuscript, our goal is to initiate a new quadruple hypergeometric function denoted by X 84 4 , and then, we ensure the existence of solutions of systems of partial differential equations for this function. We also establish some integral representations involving the quadruple hypergeometric function X 84 4 . [ABSTRACT FROM AUTHOR]

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, by using certain inverse pairs of symbolic operators introduced by Choi and Hasanov in 2011, we establish several decomposition formulas associated with the Gaussian triple hypergeometric functions. Some transformation formulas for these functions have also been obtained. [ABSTRACT FROM AUTHOR]

By using the reciprocally inverse operators of the Burchnall and Chaundy type introduced by Choi and Hasanov in 2011, we aim at deriving certain decomposition formulas for some Gauss's triple hypergeometric functions. We also present some transformation formulas by using our decomposition formulas. [ABSTRACT FROM AUTHOR]

The authors establish a set of fifteen new integral representations of Euler-type for the Sharma and Parihar hypergeometric function in four variables F(4)20; whose kernels include the quadruple functions K1;K10; F(4)14 and X(4)8; the Exton hypergeometric functions of three variables X4; the Lauricella functions of three variable FE; FN and FR; the Appell's series of two variables F2; F3 and F4; and the Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]

The authors establish a set of fifteen new integral representations of Euler-type for the Sharma and Parihar hypergeometric function in four variables F(4)20; whose kernels include the quadruple functions K1;K10; F(4)14 and X(4)8; the Exton hypergeometric functions of three variables X4; the Lauricella functions of three variable FE; FN and FR; the Appell's series of two variables F2; F3 and F4; and the Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]