Placement

A three-dimensional hybrid nanofluid flow over a stretching/shrinking sheet is numerically studied. The hybrid nanofluid being considered in this study used water as the base fluid and mixed with two types of solid nanoparticles, namely alumina (Al2O3) and copper (Cu). The main focus of the current study is to examine the effect of magnetic field, Joule heating, and rotating sheet on the velocity, and temperature profiles. In addition, the impact of suction and stretching sheet on the variations of reduced skin friction, , and reduced heat transfer are studied as well. The fluid flow and heat transfer problem presented in this study is governed by a system of nonlinear partial differential equations (PDEs), which is then transformed into the corresponding system of high order nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting system of higher order nonlinear ODEs is solved numerically using a boundary value solver known as bvp4c, which operates on the MATLAB computational platform. Results revealed that dual solutions exist for shrinking sheet while unique solutions are observed for stretching sheet with various values of Cu nanoparticles volume fraction and magnetic parameter. Dual solutions also exist for the value of the suction parameter greater than its critical point with various values of Cu nanoparticles volume fraction. Velocity profile of the hybrid nanofluid increases alongside with the value of magnetic parameter but declination was observed in the profile of and temperature, for both solutions as the value of Cu nanoparticles volume fraction increases. When the value of rotational parameter increases, both velocity and profiles increase for both solutions. This indicates that the momentum boundary layer thickness increases with increasing values of for both solutions, but thermal boundary layer thickness decreases for the first solution and increases for the second solution. Finally, an increment in the value of Eckert number causes the temperature of the hybrid nanofluid to rise as well for both first and second solutions.

This study aims to develop an efficient and accurate numerical method for solving the boundary layer fluid flow over a stretching sheet using a modified spectral quasilinearization method. The governing partial differential equations (PDEs) for momentum and energy are first transformed into a system of nonlinear ordinary differential equations (ODEs) using similarity transformations. The improvement in the method of solution is realized by numerically solving the flow equations defined over a larger semi-infinite domain [0, ∞) using spectral quasilinearization method embedded on overlapping sub-intervals. This method is better than its counterpart on a single domain as it maintains high accuracy and at the same time results in a sparse differentiation matrix that is easily invertible and saves CPU time. The numerical simulations and solution error analysis were performed using MATLAB version 2018a. Convergence analysis demonstrates exponential error decay, with residual errors reducing from the order of 10<sup>−2</sup> to approximately 10<sup>−12</sup> within four iterations, confirming the accuracy and efficiency of the numerical scheme. Additionally, the impact of the Prandtl number on thermal boundary layer thickness is examined, revealing sharper temperature gradients for higher Pr values. This method can be adapted to solve other fluid flow problems represented as systems of nonlinear ODEs.

Thermal analysis of Non-Newtonian visco-inelastic fluid MHD flow between rotating disks

<div class="section abstract"><div class="htmlview paragraph">In this paper, a new solution method is presented to study the effect of wave propagation in engine manifolds, which includes solving one-dimensional models for compressible flow of air. Velocity, pressure, and density profiles are found by solving a system of non-linear Partial Differential Equations (PDEs) in space and time derived from Euler’s equations. The 1D model includes frictional losses, area change, and heat transfer. The solution is traditionally found by utilizing the Method of Characteristics and applying finite difference solutions to the resulting system of ordinary differential equations (ODEs) over a discretized grid. In this work, orthogonal collocation is used to solve the system of ODEs that is defined along the characteristic curves. Orthogonal polynomials are utilized to approximate velocity, pressure, sound speed, and the characteristic curves along which the system of PDEs reduce to a system of ODEs. The approximation polynomials are defined over the whole manifold domain, transforming Euler’s equations into a system of ODEs that can be solved using a generic ODE solver. This reduction is done symbolically using a computer algebra system (Maple). The method results in a system of ODEs that has a higher spatial order along the whole space compared to methods based on finite differences, reducing the number of nodes required to find an acceptable solution that captures the state dynamics at different locations inside the manifold in real time. The proposed model is compared against the Method of Characteristics (MOC) that is used as a reference model; this comparison includes the states at the inlet, outlet, and midpoint. In summary, a high order method that can calculate solutions of the 1D manifold model equations is developed by finding the respective polynomial approximations along the 1D space and solving the resultant system using a generic solver in real time.</div></div>

The current work investigates the exact analytical solution for 2D mixed convective flow and heat transfer of Casson hybrid nanofluid over stretching/shrinking sheet linear velocity in porous medium. The nonlinear system of partial differential equations (PDEs) is formed for the defined flow which is then converted into nonlinear system of ordinary differential equations (ODEs) via similarity transformations. Then the resultant dimensionless equation is solved analytically to obtain the exact solution for velocity profile. The hybrid nanofluid Cu – Al2O3/H2O is used in the analysis. Further the effect of porous medium, hybrid nanofluids and Brinkman ratio on the velocity profile is examined with the help of graphs. It found many applications in the fields in solar energy, heating and cooling systems, regenerative heat exchange, chemical catalytic reactors and oil extraction etc.

SUMMARY The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how "numerical" basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

This manuscript studies the impact of heat transfer in the context of their valuable applications. There has been a lot of interest in using non-Newtonian fluids in biological and engineering disciplines. With such a considerable interest in non-Newtonian fluids, we aim to examine the MHD flow of chemically reactive Casson liquid by a permeable stretching surface by considering the heat source and viscous dissipation effects. The impression of current conductivity as a linear function of the temperature field is subjected to temperature-dependent viscosity fluctuation. A mathematical model simulates the arisen nonlinear partial differential equations (PDEs). By using the suitable transformations, the system of PDEs is then transformed to a nonlinear system of ordinary differential equations (ODEs). The impacts of the pertinent parameters on the velocity profile, energy, and concentration distribution have been discussed. The fundamental dimensionless partial differential flow laws are analyzed using an efficient and validated analytical homotopic (HAM) technique. The ongoing investigation has been converged, according to a stability and convergence analysis. The impression of innumerable dominant physical parameters on the momentum boundary layer, thermal boundary layer, and concentration profile has been made realistically by plotted graphics.

In this research work, we investigate an unsteady flow over a rotating disk. We assign symbols to the selected dependent and independent quantities. Then all physical systems are modeled to mathematical form by applying physical laws for an ionized liquid flow over a rotating disk with nanoparticles from the set of Poisson Nernst–Planck model, Energy equation and Navier–Stokes equations. The set of partial differential equations along with the boundary conditions are transformed to a set of coupled ordinary differential equations for an electro-viscous flow of nanofluid over a rotating disk by using similarity transformations. The unknown physical quantities are investigated through Parametric Continuation Method (PCM). For physical purpose, physical quantities like flow behavior thermal properties, thermal variation, the distribution of ions in the fluid region, skin friction, are analyzed through graphical and tabulated results. As exact solutions are not possible for nonlinear ordinary differential equations (ODEs) system, therefore, such quantities are subjected to numerical calculation following Parametric Continuation Method (PCM) and validated the result through BVP4c package in Matlab.

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations.

The aim of this study is to provide numerical solutions to magnetohydrodynamic (MHD) cross flow with heat and mass transfer over a stretching sheet in a hybrid nanofluid with suction/injection and convective boundary conditions. This will be accomplished by presenting numeric solutions. Nanoparticles of graphene oxide and silver were suspended in kerosene oil as the base fluid taken into consideration is a nanofluid hybrid. After that, the physical flow problem is modeled as a set of partial differential equations, which are then transformed into an equivalent system of nonlinear ordinary differential equations (ODEs) by making use of the relevant similarity transformations. To gain different graphical and numerical results for analyzing the influence of numerous physical restrictions on velocity, heat, and mass profiles, the system of ODEs is solved using the computing power of the Lobatto IIIA technique. The plot of the velocity profile, temperature, concentration, entropy generation and Bejan number with separating magnitude like that power index number ( 0.1 < n < 2.0 0.1\lt n\lt 2.0 ), Weissenberg number ( 0.1 < We < 1.0 0.1\lt {\rm{We}}\lt 1.0 ), Suction/injection ( 0.1 < S < 1.0 0.1\lt S\lt 1.0 ), Magnetic parameter (0.1 < M < 1.0), Biot number ( 0.1 < Bi < 0.3 0.1\lt {\rm{Bi}}\lt 0.3 ), Radiation parameter ( 0.1 < Rd < 2.0 0.1\lt {\rm{Rd}}\lt 2.0 ), Schmidt number ( 1.1 < Sc < 2.0 1.1\lt {\rm{Sc}}\lt 2.0 ), and Chemical reaction parameter ( 1.1 < Sc < 2.0 1.1\lt {\rm{Sc}}\lt 2.0 ). We noted that when the Weissenberg parameter ( We {\rm{We}} ) increases, the flow speed will increase as well. Also, the increasing values of M M slows the flow speed but the opposite effect is observed in the temperature profile. Moreover, the greater values of M M , We, and S reduced in the local skin friction. It is concluded that hybrid nanofluid is better than ordinary nanofluid by percentage values. The results are used in industry and technology in fluid-based systems with shrinkable or flexible materials, MHD flow meters, pumps, and the drawing of plastic strips. Because hybrid nanoparticles speed up the cooling process, they are important in cutting and production.

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

In this thesis we investigate the existence of solutions to boundary value problems (BVPs) for nonlinear systems of first-order ordinary differential equations (ODEs) and difference equations. The thesis is divided into two parts. Part one is devoted to the study of difference equations where two types of problems are studied. First, we derive the existence of solutions to first-order systems of difference equations that arise when one applies the trapezoidal rule to approximate solutions of second-order scalar ODEs. Strict discrete lower solutions are used with maximum principle arguments in the discrete problem to obtain a priori bounds on solutions. The a priori bounds on difference quotients for solutions of the discrete problem are obtained from the Bernstein-Nagumo condition. We use homototopy to find solutions of the discrete amroximations. Second, motivated by Ma [60] we consider existence and uniqueness of solutions to difference equations which arise as discrete approximations to three-point BVPs for systems of first-order ODES after employing the Euler method. Several existence and uniqueness results for three-point BVPs are established, using the contraction mapping theorem and the Brouwer fixed point theorem in Rn. In part two, two types of problems for systems of first-order ODES are analyzed. First, we study the existence of a unique solution to a system of first-order differentia1 equations under multi-point BVPs with nonlinear boundary conditions. We prove the existence and uniqueness of solutions for a system of first-order three-point BVP under which the properties of linear three-point BVP are preserved under small nonlinear perturbations of both the differentia1 equation and the boundary conditions. The method of proof of the existence of solutions using the Banach contraction mapping principle is similar to that used by Rodriguez [75], who studied the discrete analogue of this BVP. Second, motivated by Cronin 120,211 we investigate the existence of solutions to three-point BVPs in perturbed systems of first-order ODEs at resonance where the associated homogeneous linear problem has nontrivial solutions. Based on the second part of the investigation of three-point BVPs in perturbed systems of first-order ODEs at resonance, four different results were obtained: (i) The existence of solutions via a version of the Brouwer Fixed Point Theorem which is due to Miranda. (ii) The existence of solutions to the problem through the application of the Implicit Function Theorem. These two results extend the work of Feng and Webb [34], and Gupta [39] for the problems at resonance in Euclidean 2-space. (iii) Existence results for three-point BVPs at resonance for general BVPs through the application of Brouwer degree theory; we show the degree is non zero through the application of Borsuk's Theorem for one problem and through a result of Cronin involving pairs of polynomials in two variables whose terms of highest order have no common factors. (iv) The existence of solutions to the entrainment of frequency problem in Euclidean 2-space. We will investigate conditions under which entrainment of frequency for three-point as well as two-point BVPs by adapting the work of Cronin in periodic case. We will describe such conditions and then show the degree is non zero through a result of Cronin involving pairs of polynomials in two variables whose terms of lowest order have no common factors.

Heat transport keeps vital impact in field of engineering like heat spacing, air conditioning, refrigeration, and in chemical processing, such as heating of base and final products, heat recovery, tempering of containers, autoclaves, and reactors. This manuscript reveals theoretical and an experimental finding related for heat transport of unsteady cross nanofluid and chemical process with inclined magnetic field over the wedge. Heat transport is scrutinized with the existence of nonuniform heat sink source and radiation. Furthermore, mass transfer is studied with higher order chemical process, thermophoresis, and Brownian motion. Mathematical system produces nonlinear partial differential equations (PDEs) of time‐dependent velocity, energy, and concentration. Moreover, these PDEs changed into the system of nonlinear ordinary differential equations (ODEs). The obtained ODEs are passed out through shooting technique to convert the boundary value problem into initial value problem and further bvp4c took them for numerical solution. Bvp4c is MATLAB procedure, which is most convergent for numerical solution of ODEs based on RK‐45 procedure. Many applicable parameters are inspected graphically and in tabular form. Numerical outcomes related to physical quantities are hashed out in tabular form. Chemical process boosts the heat and mass transport. Higher radiation responses higher temperature and inclined magnetic strength reduces the velocity of cross nanofluid.

A stochastic computing approach is implemented in the present work to solve the nonlinear nanofluidics system that occurs in the model of atomic physics. The process converts the partial differential nanofluidics system with suitable level of similarities transformation into nonlinear systems of differential equations. For the construction of datasets, finite difference scheme (Lobatto IIIA) is applied through different selection of collocation points for nonlinear nanofluidics system having accuracy of order four. Lobatto IIIA has a strong point to tackle extremely nonlinear systems of ordinary differential equations in smooth way. For different scenarios, datasets are well trained through computing scheme to investigate the heat transfer and thermal performance of nanofluidic transportation system of nanofluids and hybrid nanofluids toward stretching surfaces with variation of Biot number, Nusselt number and skin fraction. Furthermore, the reliability, accuracy and efficiency are endorsed through various statistical analysis and graphical illustrations of proposed computing scheme.