Optimal Collocation Method Research Articles

Semi-analytical solution of MHD free convective flow of a nanofluid over an exponentially stretching porous sheet in the presence of heat source/sink

In this study, the magnetohydrodynamic flow and heat transfer of electrically conducting nanofluid over an exponentially stretching porous sheet is investigated. By introducing similarity variables, the non-linear set of partial differential governing equations, are transformed into a system of ordinary nonlinear differential equations. The equations are solved semi-analytically by the Optimal Collocation Method (OCM) which is a powerful method to solve equations containing infinite boundary conditions. The accuracy of the OCM results is examined by the Richardson method. The effects of the different physical parameters such as magnetic field strength, nanofluid volume fraction, suction strength, and the heat sink/source value are discussed on the flow and heat transfer characteristics of the problem. The obtained results show that the velocity and temperature of nanofluid increase with the increase in volume fraction. Also, when suction increases, the thickness of the boundary layers decreases. As the strength of the magnetic field (Hartmann) increases, the temperature slightly increases but the flow rate of the nanofluid decreases. The skin friction coefficient and Nusselt number increase when the values of suction parameters, volume fraction, magnetic field, and the field angle increase.

On the effect of spatial fractional heat conduction in MHD boundary layer flow using Gr-Fe3O4–H2O hybrid nanofluid

In this article, the magnetohydrodynamic boundary layer flow and spatial fractional heat transport of Gr-Fe3O4–H2O hybrid nanofluid on a moving flat plate is investigated. The diffusion terms in the energy equation are modified by incorporating the spatial fractional derivatives using Fourier's law. Moreover, the effect of the joule heating, heat sink/source and viscous dissipation on the energy equation is also taken into account. The optimal collocation method as a new semi-analytical approach is introduced and the effects of physical parameters on the flow and heat transport specifications are obtained and analyzed. It is seen that changing the order of fractional derivatives from 0.92 (fractional diffusion) to 1 (classical diffusion) leads to a growth of the temperature. Furthermore, for the case λ = 0.7, the Nusselt number of Gr-Fe3O4–H2O augments by 22% when compared to Fe3O4–H2O and by 66% when compared to H2O. The results of Nusselt number and Skin friction coefficient for Prandtl number of 0.71 are validated by comparing with both the experimental and numerical studies in the existing literature, which confirms that the optimal collocation method can be successfully used to predict the spatial-fractional MHD boundary layer flow.

Peristaltic transport of biological graphene-blood nanofluid considering inclined magnetic field and thermal radiation in a porous media

One of the current challenges of clinical techniques for the destruction of cancer cells is to cauterize the desired area with the help of nanoparticles. In this study, the effect of graphene nanoparticles suspended in the blood is enhanced by applying an angled magnetic field and thermal radiation. The vessel is considered a porous medium, and its walls are modeled by peristaltic waves. Governing equations in the presence of Joule heating effects and viscous dissipation simplified and analytically solved by using of Optimal Collocation Method. The highest effect of nanoparticles' volumetric fraction on the temperature is obtained atφ = 0.6%, and the lowest effect is seen atφ = 0%. Also, the temperature profile rapidly decreases when the inclination angle increases from 0 to π/2. Changing the Hartman number from 0(without magnetic field) to 3 (most magnetic field) leads to a growth of the temperature.

A Lagrange-quadratic spline optimal collocation method for the time tempered fractional diffusion equation

In the current paper, for the time fractional diffusion equation with an exponential tempering, we propose a numerical algorithm based on the Lagrange-quadratic spline interpolations and the optimal technique. The discretized linear systems and some properties are investigated in detail. By using these properties, the coefficient matrix and the right-hand term at each time step are given to analyze the computational cost. Theoretical analyses show that this proposed method enjoys both stability and convergence order of O(τ2+h4). Some numerical examples are provided to verify the practical feasibility and efficiency of the proposed scheme.

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.

Three-dimensional and two-phase nanofluid flow and heat transfer analysis over a stretching infinite solar plate

In this work, 3-D and two-phase nanofluid flow and heat transfer is modeled over a stretching infinite solar plate. The governing equations are presented based on previous studies. The infinite boundary condition and shortcoming of traditional analytical collocation method have been overcome in our study by changing the problem into a finite boundary problem with a new analytical method called optimal collocation method. The accuracy of results is examined by fourth order Runge-Kutta numerical method. Effect of some parameters, Prandtl number, Schmidt number, Brownian motion parameter, thermophoresis parameter, ?=b/a (ratio of the stretching rate along y- to x-directions), and power-law index on the velocities, temperature, and nanoparticles concentration functions are discussed. As an important outcome of our 3-D model analysis, it is found that increase in thermophoretic forces can enhance the thickness of both thermal and nanoparticle volume fraction boundary-layers.

Thermal-flow boundary layer analysis of nanofluid over a porous stretching cylinder under the magnetic field effect

In this paper, thermal-flow boundary layer of single-phase Nanofluid over a horizontal cylinder tube under magnetic force is investigated using Optimal Collocation Method (OCM) and numerical method. After presenting the governing equations and solving them by OCM, the accuracy of results is examined by fourth order Runge–Kutta numerical method. The boundary conditions contain infinite condition due to stretching cylinder, thus it needs to a powerful analytical method to solve it. OCM which is firstly introduced by authors is the simple/powerful method for this analysis. Effects of different nanoparticles (TiO2, Cu and Ag) are investigated on the temperature/velocity profiles as well as Nusselt number and boundary layer thickness. Additionally, the influence of relevant parameters such as the magnetic parameter, the solid volume fraction of nanoparticles on the flow, heat transfer, Nusselt number and skin friction coefficient is discussed.

Forced convection of Al2O3–water nanofluid flow over a porous plate under the variable magnetic field effect

In this paper, the effect of variable magnetic field on the forced-convection boundary-layer of MHD Al2O3–water nanofluid flow over a porous nonlinearly stretching flat plate is investigated using Optimal Collocation Method (OCM) and fourth order Runge–Kutta numerical method. This is the first time that CM method is developed for the problems with coupled equations which have infinite boundary condition. Comparison between OCM and numerical method shows that OCM is an exact and high efficient method for solving these kinds of problems. The influence of the nanofluid volume fraction (φ) and magnetic parameter (Mn) on non-dimensional temperature and velocity profiles are investigated. As an important outcome, by increasing Mn number, thermal boundary layer thickness significantly increased but increasing the nanofluid volume fraction hasn’t very sensible effect on it. Also it is understandable that the temperature of the fluid decreases by increasing suction parameter.

Boundary layer flow analysis of a nanofluid past a porous moving semi-infinite flat plate by optimal collocation method

A novel concept of collocation method is introduced and used for the first time to obtain a simple and accurate solution for the boundary layer in unbounded domain. Similar analysis is however not available yet in the literature. The idea is to transform the equations and boundary conditions into another set of variables and also to augment an extra boundary condition. To demonstrate the effectiveness of this ideal, we apply optimal collocation method (OCM) to find the approximate solution for the boundary layer flow of a nanofluid past a porous moving semi-infinite flat plate. The influence of pertinent parameters on the flow field characteristics is studied. The obtained results have been compared with the numerical solutions from fourth order Runge–Kutta method. The solution shows that the results of the present method are in excellent agreement with those of the numerical one. It is important that we applied OCM for the problem in unbounded domain without using Pade approximants, perturbation, linearization, small parameter or auxiliary parameter. The approach used in the present work can also be extended to different nonlinear problems in unbounded domain and the conclusion is considered useful for engineering applications involving infinity domain.

A sixth-order optimal collocation method for elliptic problems

In this paper, we present a collocation method based on biquintic splines for a fourth order elliptic problems. To have a better accuracy, we formulate the standard collocation method by an appropriate perturbation on the original differential equations that leads to an optimal approximating scheme. As a result, computational results confirm that this method is optimal.