Ordinary Differential Equation Boundary Value Research Articles

Significance of Cattaneo-Christov double diffusion and induced magnetic field on Maxwell ternary nanofluid flow with magnetic response boundary

The present study delves into the significance of Cattaneo-Christov double diffusion and the induced magnetic field (IMF) on a stagnation-point flow of Maxwell ternary nanofluids. A new boundary condition with the magnetic response is designed to study superior heat and mass transfer of Maxwell ternary nanofluid combined with double diffusion and IMF. The governing equations are formulated by mathematically modeling of the current flow in a Cartesian coordinate system and solved using an improved shooting method and the Runge-Kutta method. Through the application of similarity transformations, the governing equations are transformed into a system of initial boundary value ordinary differential equations. These equations are further transformed into linear equations with initial value problems using the shooting method and subsequently solved using the Runge-Kutta method and Newton's iterative techniques. The numerical results and correlation analysis vividly demonstrate the profound influence of double diffusion and IMF on thermal and concentration patterns via graphical representations. It is found that the double diffusion and the induced magnetic field always helps to achieve lower temperature and concentration. The magnetic response boundary leads to higher heat and mass transfer efficiency for the suction case. The magnetically responsive boundary is verified to be effective for regulating the heat and mass transfer of ternary nanofluid.

Design of Sturm global attractors 1: Meanders with three noses, and reversibility.

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) ut=uxx+f(x,u,ux) on the unit interval 0<x<1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions ut=0. Specifically, we address meanders with only three "noses," each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity f=f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v1⇝v2 between equilibrium vertices v1,v2 of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold Wu(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d-1)-sphere boundary of the global attractor.

Computation of reactive thermosolutal micropolar nanofluid Sakiadis convection flow with gold/silver metallic nanoparticles

In the present study, a mathematical model is developed by combining the Tiwari-Das nanofluid formulation with the Eringen micro-morphic model to simulate the thermo-solutal natural convection chemically reacting micropolar nanofluid flow from a permeable stretching surface with non-uniform heat source/sink effects. The transformed ordinary differential equation boundary value problem features linear momentum, angular momentum, energy and species conservation boundary layer equations with appropriate boundary conditions. This ninth order nonlinear system is solved with Runge-Kutta 45 Fehlberg method. The influence of the effect of the emerging parameters on the flow characteristics are visualized and tabulated. It is observed that increasing volume fraction decreases velocity whereas it elevates microrotation, temperature and nanoparticle concentration. Nanoparticle concentration are elevated for stronger destructive chemical reaction effect whereas they are suppressed with constructive chemical reaction. With greater micropolar boundary condition parameter, the velocity is elevated, microrotation but reduces temperature and thermal boundary layer thickness. Increasing nanoparticle volume fraction enhances both skin friction and couple stress but marginally reduces the Nusselt number. Finally, Au-water micropolar nanofluids achieve the highest skin friction and couple stress magnitudes, then Ag-water and finally Cu-water. Validation of solutions with earlier non-reactive studies in the absence of nanoparticle mass transfer are included.

Consumption and Portfolio Selection with Recursive Utility, Stochastic Income, and Liquidity Constraints

This paper studies a continuous-time optimal consumption and portfolio selection problem when an economic agent with recursive utility has stochastic income and liquidity constraints. To tackle this problem, we introduce a transform of the Hamilton-Jacobi-Bellman equation into a free boundary value ODE (ordinary differential equation). This transform is designed to simplify the procedure required for performing a duality method. Then, we characterize the optimal policies by deriving an integral equation with a fixed interval from the free boundary ODE. The integral equation is used to prove the verification theorem and to generate a stable numerical scheme. It is notable that the optimal portfolio depends on the elasticity of intertemporal substitution (EIS) due to liquidity constraints even if the investment opportunity is constant. Furthermore, we perform various analyses to investigate the impact of the borrowing constraints on the optimal policies and the marginal propensity to consume.

A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

The Shape Effect of Gold Nanoparticles on Squeezing Nanofluid Flow and Heat Transfer between Parallel Plates

The focus of the present paper is to analyze the shape effect of gold (Au) nanoparticles on squeezing nanofluid flow and heat transfer between parallel plates. The different shapes of nanoparticles, namely, column, sphere, hexahedron, tetrahedron, and lamina, have been examined using water as base fluid. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by suitable transformations. As a result, nonlinear boundary value ordinary differential equations are tackled analytically using the homotopy analysis method (HAM) and convergence of the series solution is ensured. The effects of various parameters such as solid volume fraction, thermal radiation, Reynolds number, magnetic field, Eckert number, suction parameter, and shape factor on velocity and temperature profiles are plotted in graphical form. For various values of involved parameters, Nusselt number is analyzed in graphical form. The obtained results demonstrate that the rate of heat transfer is maximum for lamina shape nanoparticles and the sphere shape of nanoparticles has performed a considerable role in temperature distribution as compared to other shapes of nanoparticles.

Buoyancy-Induced MHD Stagnation Point Flow of Williamson Fluid with Thermal Radiation

Flow of fluids subjected to thermal radiation has enormous application in polymer processing, glass blowing, cooling of nuclear reactant and harvesting solar energy. This paper considers the MHD stagnation point flow of non-Newtonian pseudoplastic Williamson fluid induced by buoyancy in the presence of thermal radiation. A system of nonlinear partial differential equations suitable to describe the MHD stagnation point flow of Williamson fluid over a stretching sheet is formulated and then transformed using similarity variables to boundary value ordinary differential equations. The graphs depicting the effect of thermal radiation parameter, buoyancy and electromagnetic force on the fluid velocity and temperature of the stagnation point flow are given and the results revealed that increase in buoyancy leads to an increase in the overall velocity of the flow but a decrease in the temperature of the flow.

Numerical Calculations for a Boundary Layer Flow past a Moving Vertical Porous Plate with Suction/Injection and Thermal Radiation

The work presented herein investigates the velocity, heat transfer, Nusselt number and skin friction profiles involved in boundary layer flow past a moving vertical porous plate. Similarity transformations are employed to convert the governing nonlinear unsteady momentum and energy equations from their partial differential equation forms to boundary value ordinary differential equations. The resulting equations are then solved numerically by the Runge-Kutta fourth order method with the help of a shooting technique. Several features of the flow and heat transfer characteristics for different values of problem parameters are analyzed and discussed. These include the effects of the radiation parameter (R), suction and injection parameter (c), Grashof (Gr) and Prandtl (Pr) numbers on the flow and heat profiles. Numerical results show the impact of blowing and sucking as well as radation on boundary layer flows of this type. Both the skin frictions as well as the heat transfer rate are also significantly related to the radiation parameter. For all these cases; the numerical results are found to be in agreement with the physics of the problem.

Michael Arthur Stephens, 1927–2019

Michael Arthur Stephens, 1927–2019

Mixed convection boundary layer flow and heat transfer over a vertical plate embedded in a porous medium filled with a suspension of nano-encapsulated phase change materials

The flow and thermal behavior of nano-encapsulated phase change materials (NEPCMs) dispersed in a liquid is investigated over a vertical flat plate. The NEPCM particles are consist of an encapsulation shell and a core made of phase change materials (PCMs). When the temperature of the liquid around the NEPCM particle is higher than the fusion temperature of the PCM core, the PCM core absorbs a notable amount of heat (the latent heat of the phase change) from the host fluid and melts. The vertical plate is embedded in a porous medium, and the porous medium is saturated with the suspension of NEPCM. There is a uniform flow over the plate, which can aid or oppose the natural convection flow over the plate. The partial differential governing equations for the flow and heat transfer of NEPCM suspension are introduced and transformed into a high order boundary value ordinary differential equation. The finite difference method with a colocation technique as the grid adaptation and automatic error control is utilized to integrate the governing equations. Validations are performed by comparing the results with the literature. It is found that there are two solution branches in the case of high opposing flows. The results indicate that the presence of NEPCM particles enhances the heat transfer over the plate. The decrease in the fusion temperature of NEPCM cores enhances the heat transfer.

Performance analysis of error-control B-spline Gaussian collocation software for PDEs

B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and 1D partial differential equations (PDEs) for several decades. Such packages represent the numerical solution in terms of a piecewise polynomial (B-spline) basis with basis coefficients determined through the use of Gaussian collocation. The software package, BACOL, developed over a decade ago, was the first 1D PDE package of this type to provide both temporal and spatial error control. A recently developed package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages (particularly the spatial and temporal error estimation and control algorithms) implies that extensive testing and analysis of the test results is an essential factor in the ongoing development of these packages In this paper, we investigate the performance of BACOL and BACOLI with respect to several important machine independent algorithmic measures, examine the effectiveness of the new spatial error estimation and control strategies, and investigate the influence of the choice of the degree of the B-spline basis on the performance of the solvers. These results will provide new insights into how to improve BACOLI, potentially lead to improvements in Gaussian collocation BVODE solvers, and guide further development of B-spline Gaussian collocation software with error control for 2D PDEs.

Generalized Lagrangian Jacobi Gauss collocation method for solving unsteady isothermal gas through a micro-nano porous medium

In the present paper, a new method based on the Generalized Lagrangian Jacobi Gauss (GLJG) collocation method is proposed. The nonlinear Kidder equation, which explains unsteady isothermal gas through a micro-nano porous medium, is a second-order two-point boundary value ordinary differential equation on the unbounded interval $ [0, \infty)$ . Firstly, using the quasilinearization method, the equation is converted to a sequence of linear ordinary differential equations. Then, by using the GLJG collocation method, the problem is reduced to solving a system of algebraic equations. It must be mentioned that this equation is solved without domain truncation and variable changing. A comparison with some numerical solutions made and the obtained results indicate that the presented solution is highly accurate. The important value of the initial slope, $ y'(0)$ , is obtained as $ -1.191790649719421734122828603800159364$ for $\eta = 0.5$ . Comparing to the best result obtained so far, it is accurate up to 36 decimal places.

Scrutinization of thermal stratification, nonlinear thermal radiation and quartic autocatalytic chemical reaction effects on the flow of three-dimensional Eyring-Powell alumina-water nanofluid

PurposeThe purpose of this paper is to scrutinize the effects of nonlinear thermal radiation and thermal stratification effects on the flow of three-dimensional Eyring-Powell 36 nm alumina-water nanofluid within the thin boundary layer in the presence of quartic autocatalytic kind of chemical reaction effects, and to unravel the effects of a magnetic field parameter, random motion of the tiny nanoparticles and volume fraction on the flow.Design/methodology/approachThe chemical reaction between homogeneous (Eyring-Powell 36 nm alumina-water) bulk fluid and heterogeneous (three molecules of the catalyst at the surface) in the flow of magnetohydrodynamic three-dimensional flow is modeled as a quartic autocatalytic kind of chemical reaction. The electromagnetic radiation which occurs within the boundary layer is treated as the nonlinear form due to the fact that Taylor series expansion may not give full details of such effects within the boundary layer. With the aid of appropriate similarity variables, the nonlinear coupled system of partial differential equation which models the flow was reduced to ordinary differential equation boundary value problem.FindingsA favorable agreement of the present results is obtained by comparing it for a limiting case with the published results; hence, reliable results are presented. The concentration of homogeneous bulk fluid (Eyring-Powell nanofluid) increases and decreases with ϕ and Pr, respectively. The increase in the value of magnetic field parameter causes vertical and horizontal velocities of the flow within the boundary layer to decrease significantly. The decrease in the vertical and horizontal velocities of Eyring-Powell nanofluid flow within the boundary layer is guaranteed due to an increase in the value of M. Concentration of homogeneous fluid increases, while the concentration of the heterogeneous catalyst at the wall decreases with M.Originality/valueConsidering the industrial applications of thermal stratification in solar engineering and polymer processing where the behavior of the flow possesses attributes of Eyring-Powell 36 nm alumina-water, this paper presents the solution of the flow problem considering 36 nm alumina nanoparticles, thermophoresis, stratification of thermal energy, Brownian motion and nonlinear thermal radiation. In addition, the aim and objectives of this paper fill such vacuum in the industry.

An Accurate Numerical Method for Solving Unsteady Isothermal Flow of a Gas Through a Semi-Infinite Porous Medium

The Kidder equation, y″(x)+2xy′(x)/1−βy(x)=0, x∈[0,∞), β∈[0,1] with y(0)=1, and y(∞)=0, is a second-order nonlinear two-point boundary value ordinary differential equation (ODE) on the semi-infinite domain, with a boundary condition in the infinite that describes the unsteady isothermal flow of a gas through a semi-infinite micro–nano porous medium and has widely used in the chemical industries. In this paper, a hybrid numerical method is introduced for solving this equation. First, by using the method of quasi-linearization, the equation is converted to a sequence of linear ODEs. Then these linear ODEs are solved by using the rational Legendre functions (RLFs) collocation method. By using 200 collocation points, we obtain a very good approximation solution and the value of the initial slope y′(0)=−1.19179064971942173412282860380015936403 for β=0.50, highly accurate to 38 decimal places. The convergence of numerical results is shown by decreasing the residual errors when the number of collocation points increases.

Application of Meshfree Method Based on Compactly Supported Radial Basis Function for Solving Unsteady Isothermal Gas Through a Micro–Nano Porous Medium

In this paper, we have applied the Meshless method based compactly supported radial basis function collocation for obtaining the numerical solution of unsteady gas equation. The unsteady gas equation is a second order non-linear two-point boundary value ordinary differential equation on the semi-infinite domain, with a boundary condition in the infinite. The compactly supported radial basis function collocation method reduces the solution of the equation to the solution of a system of algebraic equation. also, we compare the results of this work with some results. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work

New numerical solutions for solving Kidder equation by using the rational Jacobi functions

In this paper, a new method based on rational Jacobi functions (RJ) is proposed that utilizes quasilinearization method to solve non-linear singular Kidder equation on unbounded interval. The Kidder equation is a second order non-linear two-point boundary value ordinary differential equation on unbounded interval \([0,\infty )\). The equation is solved without domain truncation and variable changing. First, the quasilinearization method is used to convert the equation to sequence of linear ordinary differential equations. Then, by using RJ collocation method equations are solved. For the evaluation, comparison with some numerical solutions shows that the proposed solution is highly accurate. Using 200 collocation points, the value of initial slope that is important is calculated as \(-1.1917906497194217341228284 \) for \(\kappa =0.5\).

Convergence Analysis of Finite Difference Method for Differential Equation

In this paper, convergence analysis of a finite difference method for the linear second order boundary value ordinary differential equation is determined by investigating basic key concepts such as consistency and stability by using the maximum norm

Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation

The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into partial differential equations by introducing a time-marching scheme. A Crank-Nicolson Newton-Richtmeyer scheme is employed to discretize the resulting equations. Initial guesses are made for the dependent variables and the solution advanced in time until temporal variations of the scalar profile are diminished and the steady-state solutions satisfy the similarity equations. A variation of the heat flux at one of the boundaries produced noticeable variations in the temperature field that can be related to the magnitude of the Prandtl number and velocity ratio parameter.

Calculus of Variations and Nonlinear Optimization Based Algorithm for Optimal Control of Hybrid Systems with Controlled Switching

This paper investigates the optimal control problem of a particular class of hybrid dynamical systems with controlled switching. Given a prespecified sequence of active subsystems, the objective is to seek both the continuous control input and the discrete switching instants that minimize a performance index over a finite time horizon. Based on the use of the calculus of variations, necessary conditions for optimality are derived. An efficient algorithm, based on nonlinear optimization techniques and numerical methods, is proposed to solve the boundary-value ordinary differential equations. In the case of linear quadratic problems, the two-point boundary-value problems can be avoided which reduces the computational effort. Illustrative examples are provided and stress the relevance of the proposed nonlinear optimization algorithm.