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The aim of this work was to study the enhancement of thermal transportation in Carreau Yasuda liquid passed over a vertical surface in the presence of magnetic dipole. A mixture of tri-hybrid nanoparticles (Al2O3,MoS3, TiO3){\text{(Al}}_{2}{\text{O}}_{3}\text{,}\hspace{.25em}{\text{MoS}}_{3}{\text{, TiO}}_{3}\text{)} is inserted into the Carreau Yasuda liquid. The transport phenomenon of heat is derived in the presence of heat source/sink contribution. The concept boundary layer theory is engaged to derive the mathematical expression for momentum and energy in the form of coupled partial differential equations. The derivations are transformed into a set of coupled nonlinear ordinary differential equations (ODEs) with the help of suitable similarity transformation. These converted ODEs have been handled numerically via finite element method. The grid-independent analysis is established for 300 elements. The impact of numerous involved parameters on temperature and velocity solution is plotted and their contribution is recorded. Temperature profile is inclined versus the higher values of heat generation and viscous dissipation numbers while thermal layers are also increasing the behavior. A vital role of magnetic dipole is examined to raise the production of thermal layers but declination is noticed in flow profile.

This research is conducted to investigate heat and mass transport past over a stretched surface having pores in a pseudo-plastic model. To study porosity effect, Darcy Forchheimer relation is used. Thermal and mass transport expressions are derived by engaging the double diffusion theories as extensively used by researchers proposed by Cattaneo and Christov. Furthermore, the thermal performance is studied by mixing the tri-hybrid nanoparticles in a pseudo-plastic material. The phenomenon of boundary layer is used to derive the complex model. The correlation for tri-hybrid nanoparticles is used to convert the model partial differential equations into ordinary differential equations (ODE) along with appropriate similarity transformation. The transfigured ODEs are coupled nonlinear in nature, and the exact solution is not possible. To approximate the solution numerically, finite element scheme (FES) is used and code is developed in MAPLE 18.0 for the graphical results, grid independent survey, and tabular results. The obtained results are compared with the published findings that confirm the accuracy and authenticity of the solution and engaged scheme. From the performed analysis, it is concluded that FES can be applied to complex engineering problems. Furthermore, it is monitored that nanoparticles are essential to boost the thermal performance and higher estimation of Schmidt number control the mass diffusion.

This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model’s parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term. [ABSTRACT FROM AUTHOR]

This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor. [ABSTRACT FROM AUTHOR]

This study's main objective is to analyze the heat and velocity transfer rate with various effects such as the suction and buoyancy effects on a stretching sheet with ternary hybrid nanofluid flow via a porous medium. Ternary hybrid nanofluid is Case‐1 Ag‐TiO2 with base fluid H2O and Case‐2 CNT + Graphene + Al2O3 with base fluid H2O. Hybrid nanofluids have been used to speed up the heat transfer process, following Lie group transformations are used to transform non‐linear Partial Differential Equations (PDEs) into Ordinary Differential Equations (ODEs). The boundary value problem solver was used in conjunction with the ODE45 numerical approach to solve the resulting Ordinary differential equations. On a stretchy surface, the overall relationship between temperature, velocity, shear stress, and heat transfer rate are shown for a range of values of the significant factors. The temperature profiles have been rising with the impact of Da and fw. Utilizing streamlines and a machine learning technique to examine the flow pattern of a fluid. Artificial neurons or nodes make up a neural network, which is known as a neural network in the sense that the term is used today. A network or circuit of biological neurons is referred to as a neural network. [ABSTRACT FROM AUTHOR]

In this article, the fractional third‐order dispersive partial differential equations were investigated by using Shehu decomposition and variational iteration transform methods. The well known Riemann‐Liouville fraction integral, Caputo's fractional‐order derivative, Shehu transform for fractional‐order derivatives and Mittag‐Leffler function were used as the major basis of the methodology. The graphs and table show the solution behavior for various fractional order values. The comparison provided the signed agreement of the solutions to each other as well as with the exact result. The accurateness and efficiency of the suggested techniques are examined using two numerical experiments. The reliability and validity tests show that for various fractional‐order values, the newly proposed method is reliable, accurate, and efficient. [ABSTRACT FROM AUTHOR]

This work presents a reliable algorithm to obtain approximate analytical solutions for a strongly coupled system of singularly perturbed convection–diffusion problems, which exhibit a boundary layer at one end. The proposed method involves constructing a zero-order asymptotic approximate solution for the original system. This approximation results in the formation of two systems: a boundary layer system with a known analytical solution and a reduced terminal value system, which is solved analytically using an improved residual power series approach. This approach combines the residual power series method with Padé approximation and Laplace transformation, resulting in an approximate analytical solution with higher accuracy compared to the conventional residual power series method. In addition, error estimates are extracted, and illustrative examples are provided to demonstrate the accuracy and effectiveness of the method. [ABSTRACT FROM AUTHOR]

The advancement of cutting tool components and design is presently promoting innovative developments in numerous different machining-related industries. The characteristics of nanofluid are important for machining activities such as the drilling process, grinding, rotating, milling, and cutting. Various machining procedures require distinct lubricating oils and nanofluids for cutting-edge innovations. The significant contribution of the current mechanism is to explore the fluctuating heat and mass flux of Darcy–Forchheimer chemically reactive nanofluid along a buoyancy-driven porous plate under solar radiation region. Flow through a Darcy medium has a wide range of applications such as the use of oil in various hydrothermal transfer control, radioactive nuclear disposal systems, water improvement, and filtration of water. The dimensional model is transformed into non-dimension for scaling factors. The primitive-based transformation is applied on steady and oscillatory parts for smooth algorithm in FORTRAN language machine by using an implicit finite difference method. The numerical and graphical results of velocity, temperature, and concentration are executed by the Gaussian elimination method. To enhance the frequency and wavelength, the impact of solar radiations is applied on periodic nanoparticles with Darcy–Forchheimer relation. The novelty of this proposal is to explore the wave oscillations, amplitude, and phase angle of thermal and concentration boundary layer of Darcy–Forchheimer nanofluid flow under chemical reaction and solar radiation region. It is noticed that the prominent wavelength and frequency in thermal and concentration boundary layers is generated under porous and solar radiation region. The significance of temperature variation increases as solar radiation, chemical reaction, Brownian motion, and thermophoresis increase. It is found that minimum oscillation in heat transport is observed as Pr decreases but maximum oscillation in heat transfer is sketched as Pr enhances. [ABSTRACT FROM AUTHOR]

A nonlinear equation f (x) = 0 is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme x n + 1 = x n − f (x n) / [ a + b f (x n) ] , which requires one evaluation of the given function per iteration. A local convergence analysis is adopted to determine the optimal values of a and b. Moreover, upon combining the fractional iterative scheme to the generalized quadrature methods, the fourth-order optimal iterative schemes are derived. The finite differences based on three data are used to estimate the optimal values of a and b. We recast the Newton iterative method to two types of derivative-free iterative schemes by using the finite difference technique. A three-point generalized Hermite interpolation technique is developed, which includes the weight functions with certain constraints. Inserting the derived interpolation formulas into the triple Newton method, the eighth-order optimal iterative schemes are constructed, of which four evaluations of functions per iteration are required. [ABSTRACT FROM AUTHOR]

In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method. [ABSTRACT FROM AUTHOR]

This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of Riemann–Liouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve non-homogeneous fractional differential systems containing 2 n periodic terms. The exact solutions are determined explicitly in a straightforward manner. The solutions are expressed in terms of entire functions with fractional order arguments. Features of the current solutions are discussed and analyzed. In addition, the existing solutions in the literature are recovered as special cases of our results. [ABSTRACT FROM AUTHOR]