A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

In this chapter, we present a modified version of the spectral relaxation method for solving singular initial value problems for some Emden-Fowler equations. This study was motivated by the several applications that these equations have in Science. The first step of the method of solution makes use of linearisation to solve the model problem on a small subinterval of the problem domain. This subinterval contains a singularity at the initial instant. The first step is combined with using the spectral relaxation method to recursively solve the model problem on the rest of the problem domain. We make use of examples to demonstrate that the method is reliable, accurate and computationally efficient. The numerical solutions that are obtained in this chapter are in good agreement with other solutions in the literature.

On Spectral Relaxation Approach to Radiating Powell-Eyring Fluid Flow over a Stretching Disk with Newtonian Heating

Defect and Diffusion Forum ◽

10.4028/www.scientific.net/ddf.387.461 ◽

2018 ◽

Vol 387 ◽

pp. 461-473 ◽

Cited By ~ 1

Author(s):

K. Gangadhar ◽

D. Vijaya Kumar ◽

S. Mohammed Ibrahim ◽

Oluwole Daniel Makinde

Keyword(s):

Boundary Layer ◽

Nonlinear Boundary ◽

Numerical Solutions ◽

Relaxation Method ◽

Newtonian Heating ◽

Nonlinear Boundary Value Problems ◽

Local Skin ◽

Boundary Layer Equations ◽

In this study we use a new spectral relaxation method to investigate an axisymmetric law laminar boundary layer flow of a viscous incompressible non-Newtonian Eyring-Powell fluid and heat transfer over a heated disk with thermal radiation and Newtonian heating. The transformed boundary layer equations are solved numerically using the spectral relaxation method that has been proposed for the solution of nonlinear boundary layer equations. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity and temperature profiles. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. We show that the convergence rate of the spectral relaxation method is significantly improved by using method in conjunction with the successive over-relaxation method. It is observed that CPU time is reduced in SOR method compare with SRM method.

Hydromagnetic Stagnation-Point Flow towards a Radially Stretching Convectively Heated Disk

Mathematical Problems in Engineering ◽

10.1155/2013/616947 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 24

Author(s):

S. Shateyi ◽

O. D. Makinde

Keyword(s):

Stagnation Point ◽

Numerical Solutions ◽

Relaxation Method ◽

Disk Surface ◽

Stagnation Point Flow ◽

Temperature Fields ◽

Physical Parameters ◽

Local Skin ◽

The steady stagnation-point flow and heat transfer of an electrically conducted incompressible viscous fluid are extended to the case where the disk surface is convectively heated and radially stretching. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The governing momentum and energy balance equations give rise to nonlinear boundary value problem. Using a spectral relaxation method with a Chebyshev spectral collocation method, the numerical solutions are obtained over the entire range of the physical parameters. Emphasis has been laid to study the effects of viscous dissipation and Joule heating on the thermal boundary layer. Pertinent results on the effects of various thermophysical parameters on the velocity and temperature fields as well as local skin friction and local Nusselt number are discussed in detail and shown graphically and/or in tabular form.

Numerical Analysis of MHD Stagnation Point Flow Towards a Radially Stretching Convectively Heated Disk

International Journal of Energy ◽

10.46300/91010.2020.14.9 ◽

2020 ◽

Vol 14 ◽

Keyword(s):

Stagnation Point ◽

Numerical Solutions ◽

Relaxation Method ◽

Disk Surface ◽

Stagnation Point Flow ◽

Temperature Fields ◽

Physical Parameters ◽

Local Skin ◽

The steady stagnation point flow and heat transferof an electrically conducting incompressible viscous fluid isextended to the case where the disk surface is convectivelyheated and radially stretching. The fluid is subjected to anexternal uniform magnetic field perpendicular to the planeof the disk. The governing momentum and energy balanceequations give rise to non-linear boundary value problem.Using a spectral relaxation method with a Chebyshev spectralcollocation method, the numerical solutions are obtained overthe entire range of the physical parameters. Emphasis hasbeen laid to study the effects of viscous dissipation and Jouleheating on the thermal boundary layer. Pertinent results on theeffects of various thermophysical parameters on the velocityand temperature fields as well as local skin friction and localNusselt number are discussed in detail and shown graphicallyand/or in tabular form.

Numerical Analysis for the Synthesis of Biodiesel Using Spectral Relaxation Method

Mathematical Problems in Engineering ◽

10.1155/2014/601374 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 2

Author(s):

Z. G. Makukula ◽

S. S. Motsa ◽

S. Shateyi

Keyword(s):

Vegetable Oils ◽

Numerical Solutions ◽

Nonlinear Differential Equations ◽

Relaxation Method ◽

Numerical Results ◽

Reaction Rate Constants ◽

Animal Fats ◽

Alternative Diesel Fuel ◽

Biodiesel is an alternative diesel fuel chemically defined as the mono-alkyl esters of long chain fatty acids derived from vegetable oils or animal fat. It is becoming more attractive as an alternative fuel due to the depleting fossil fuel resources. A mathematical model for the synthesis of biodiesel from vegetable oils and animal fats is presented in this study. Numerical solutions of the model are found using a spectral relaxation method. The method, originally developed for boundary value problems, is an iterative scheme based on the Chebyshev spectral collocation method developed by decoupling systems of equations using Gauss-Seidel type of techniques. The effects of the reaction rate constants and initial concentrations of the reactants on the amount of the final product are being investigated. The accuracy of the numerical results is validated by comparison with known analytical results and numerical results obtained usingode45, an efficient explicit 4th and 5th order Runge-Kutta method used to integrate both linear and nonlinear differential equations.

Slip Velocity and Temperature Jump on Dissipative CASSON Fluid with CATTANEO-CHRISTOV Heat Flux Model: Spectral Relaxation Method

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.10.20905 ◽

2018 ◽

Vol 7 (4.10) ◽

pp. 240

Author(s):

K. Gangadhar ◽

K. V. Ramana ◽

T. Kannan ◽

B. Rushi Kumar

Keyword(s):

Heat Flux ◽

Differential Equations ◽

Numerical Solutions ◽

Fluid Velocity ◽

Thermal Relaxation ◽

Relaxation Method ◽

Casson Fluid ◽

Fluid Temperature ◽

A numerical analysis is performed for investigating the slip flow of a viscous dissipative Casson fluid towards a stretching sheet with Cattaneo-Christov heat flux and variable viscosity. The nonlinear partial differential equations are transformed with appropriate similarity variables into a system of nonlinear ordinary differential equations. Numerical solutions are carried out by using efficient Spectral relaxation method. Notable accuracy of the present results has been obtained with previous results in a limiting sense from the literature. It is found that thermal relaxation time has an inverse relationship with the fluid temperature. Interestingly, the fluid velocity is gradually decreasing with higher values of slip factor.

Numerical solutions for the problem of the boundary layer flow of a Powell–Eyring fluid over an exponential sheet using the spectral relaxation method

Indian Journal of Physics ◽

10.1007/s12648-019-01583-8 ◽

2019 ◽

Vol 94 (9) ◽

pp. 1369-1374

Author(s):

M M Khader

Keyword(s):

Boundary Layer ◽

Boundary Layer Flow ◽

Numerical Solutions ◽

Relaxation Method ◽

Layer Flow ◽

Effects of Newtonian heating on the boundary layer flow of non-Newtonian magnetohydrodynamic nanofluid over a stretched plate using spectral relaxation method

International Journal of Ambient Energy ◽

10.1080/01430750.2019.1694585 ◽

2019 ◽

pp. 1-14

Author(s):

K. Gangadhar ◽

D. Vijaya Kumar ◽

M. Venkata Subba Rao ◽

T. Kannan ◽

G. Sakthivel

Keyword(s):

Boundary Layer ◽

Boundary Layer Flow ◽

Relaxation Method ◽

Newtonian Heating ◽

Layer Flow ◽

Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method

Open Physics ◽

10.1515/phys-2017-0036 ◽

2017 ◽

Vol 15 (1) ◽

pp. 323-334 ◽

Cited By ~ 3

Author(s):

Sami M. Ahamed ◽

Sabyasachi Mondal ◽

Precious Sibanda

Keyword(s):

Heat Transfer ◽

Friction Coefficient ◽

Flat Surface ◽

Internal Heat ◽

Relaxation Method ◽

Skin Friction Coefficient ◽

Partial Slip ◽

Flow Through ◽

AbstractAn unsteady, laminar, mixed convective stagnation point nanofluid flow through a permeable stretching flat surface using internal heat source or sink and partial slip is investigated. The effects of thermophoresis and Brownian motion parameters are revised on the traditional model of nanofluid for which nanofluid particle volume fraction is passively controlled on the boundary. Spectral relaxation method is applied here to solve the non-dimensional conservation equations. The results show the illustration of the impact of skin friction coefficient, different physical parameters, and the heat transfer rate. The nanofluid motion is enhanced with increase in the value of the internal heat sink or source. On the other hand, the rate of heat transfer on the stretching sheet and the skin friction coefficient are reduced by an increase in internal heat generation. This study further shows that the velocity slip increases with decrease in the rate of heat transfer. The outcome results are benchmarked with previously published results.