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Many fixed point iterative methods have been proposed. Their authors provide, theoretically, the rate of convergence which indicates how much the error changes after one iteration step. Of practical importance are the CPU time required by each method and their accuracy. This has never been investigated. In this work we investigate nine iterative processes and compare their CPU time and accuracy. The schemes are the Picard, Mann, Ishikawa, Noor, Argawal, Abbas-Nazir, Thakur, Ullah, and S* methods. By applying a recently proposed fourth order quadrature rule to the integral equation and replacing the analytical operator with the discretized one, the iterative schemes are derived. They are tested on five Hammerstein equations. The results show that (i) all the methods converge to the exact solution at the correct discretization order of convergence, (ii) the Picard scheme is the fastest and also has good accuracy, and (iii) the S* and Abbas-Nazir schemes are the least efficient. It is concluded that for a contractive problem the Picard scheme is sufficient but if the operator is non-expansive, then the Mann scheme is sufficient.

It is usually desirable to approximate the solution of mathemati- cal problems with high-order of accuracy and preferably using com- pact stencils. This work presents an approach for deriving high-order compact discretization of heat equation with source term. The key contribution of this work is the use of Hermite polynomials to reduce second order spatial derivatives to lower order derivatives. This does not involve the use of the given equation, so it is universal. Then, Tay- lor expansion is used to obtain a compact scheme for first derivatives. This leads to a fourth-order approximation in space. Crank-Nicholson scheme is then applied to derive a fully discrete scheme. The result- ing scheme coincides with the fourth-order compact scheme, but our derivation follows a different philosophy which can be adapted for other equations and higher order accuracy. Two numerical experiments are provided to verify the fourth-order accuracy of the approach.

A fixed point method is developed on a mesh for the solution of nonlinear Fredholm equation. First, the problem is collocated at mesh points and a second order quadrature rule is used to approximate the nonlinear integral. Under the assumption of nonexpansivity of self-map, we construct an Ishikawa iteration to linearize the resulting system and approximate the solution at the mesh points. Four numerical examples are given to verify the accuracy and practicability of the method. The results show that indeed the method converges with second order of accuracy. One important lesson from this study is that the results support the claim, in previous studies, that fixed point iterations can provide reliable means of solving several nonlinear problems. It is recommended to extend this work to functional integral equations using higher order quadrature rules.

This study examines the effect of viscosity and magnetic field on a non-isothermal cylindrical channel flow. This work considered a model of convective-thermal-diffusion with constant viscosity and magnetic field. The governing model equations are nondimensionalized using the dimensionless quantities and then solved analytically using power series method of Frobenius type so as to tackle the singularity in the model equations. Furthermore, the analytical solutions are displayed via graphs to show the effects of the flow parameters on the flow velocity, temperature and concentration profiles. The graphical results show that increase in viscosity, magnetic field and solutal Grashof number parameters retard the fluid flow. While increase in thermal Grashof number enhances the flow velocity. Thermal conductivity and solute injection parameters increase the fluid temperature and concentration respectively while the cooling and diffusive parameters decrease the fluid temperature and concentration respectively. Further studies can be carried out for a multi-directional flow as against the unidirectional flow and in a vertical channel in place of horizontal channel as studied in this paper.

This study investigates chemical reaction and thermal radiation effects on hydromagnetic nanofluid flow over an exponentially stretching sheet. The governing partial differential equations were transformed to ordinary differential equations by using similarity transformation and the resulting equations were solved using asymptotic series method. Graphical results showing the influence of the governing parameters on the velocity, temperature and concentration are displayed. Our results indicated that an increase in stretching sheet, thermal Grashof parameters leads to the increase in the rate of fluid flow while it decreases when magnetic field factor is increased. Also, increasing the thermophoresis number brings about increase in temperature and concentration while the reverse is the case as Prandtl number, Schmidt factor and chemical reaction rate increases. Increase in radiation leads to increase in the temperature. Keywords: Thermophoresis, Stretching sheet, MHD (Magnetohydrodynamic), thermal Radiation, Chemical reaction, Nanofluid.

This paper investigates the nonlinear dispersion of a pollutant in a non-isothermal incompressible flow of a temperature-dependent viscosity fluid in a rectangular channel filled with porous materials. The Brinkman-Forch-heimer effects are incorporated and the fluid is assumed to be variably permeable through the porous channel. External pollutant injection, heat sources and nonlinear radiative heat flux of the Rossland approximation are accounted for. The nonlinear system of partial differential equations governing the velocity, temperature and pollutant concentration is presented in non-dimensional form. A convergent numerical algorithm is formulated using an upwind scheme for the convective part and a conservative-type central scheme for the diffusion parts. The convergence of the scheme is discussed and verified by numerical experiments both in the presence and absence of suction. The scheme is then used to investigate the flow and transport in the channel. The results show that the velocity decreases with increasing suction and Forchheimer parameters, but it increases with increasing porosity.

This study proposes analytical solution to the problem of transport in a Newtonian fluid within a cylindrical domain. The flow is assumed to be dominated along the channel axis, and is taken to be axi-symmetric. No-slip boundary condition is considered for velocity while the temperature and concentration have Dirichlet boundary values. The resulting problem is transformed into a set of non-trivial variable coefficient differential equations in a cylindrical geometry. By adopting the series solution method of Frobenius, the closed-form analytical solutions are derived for the flow variables. We conduct an analysis of the derived model, and showed that, indeed, the flow variables are axi-symmetric. We also state and prove another theorem to show that the derived concentration model is positivity preserving – meaning that it yields positive concentration - provided the boundary value is non-negative. Finally, we present graphical results for the flow variables and discuss the effect of the relevant flow parameters. The results showed that (i) an increase in the cooling parameter, reduces the fluid velocity, (ii) the temperature decreases as the cooling parameter increases and (iii) an increase in the injection parameter, leads to increase in the concentration.

The importance of numerical flux solvers (NFS) in constructing conservative methods is well known, but it is not always clear which solvers more suitable for given conservation laws. To elucidate t...

This paper studies the theoretical analysis and applications of a convergent numerical algorithm to the problem of heat transfer in a convective channel flow. The model equations are derived under ...

We extend previous studies of channel flows to porous media flows with combined effects ofboth heat and mass transfer. We consider a temperaturedependent viscosity fluid and a concentrationdependent diffusivity in an unsteady and pressuredriven nonisothermal Brinkman flow. This leads to the governing equations for velocity, concentration and temperature. By lagging nonlinear coefficients, in time, a convergent finite difference scheme is formulated. We adopt the method of manufactured solutions to verify the convergence and second order spatial accuracy of the scheme. The impact of the flow parameters on the flow fields are numerically investigated. The results show that increase in the Darcy number and temperature parameter both increase the velocity while the increase in the pollutant diffusion parameter decreases the pollutant concentration.

Purpose The purpose of this paper is to formulate and analyse a convergent numerical scheme and apply it to investigate the coupled problem of fluid flow with heat and mass transfer in a porous channel with variable transport properties. Design/methodology/approach This paper derives the model by assuming a fully developed Brinkman flow with temperature-dependent viscosity and incorporating viscous dissipation, variable transport properties and nonlinear heat and mass sources. For the numerical formulation, the nonlinear sources are treated in semi-implicit manner, whereas the non-constant transport properties are treated by lagging in time leading to decoupled diagonally dominant systems. The consistency, stability and convergence results are derived. The method of manufactured solutions is adopted to numerically verify the theoretical results. The scheme is then applied to investigate the impact of relevant parameters, such as the viscosity parameter, on the flow. Findings Based on the numerical findings, the proposed scheme was found to be unconditionally stable and convergent with first- and second-order accuracy in time and space, respectively. Physical results showed that the flow parameters have influence on the flow fields, particularly, the flow is enhanced by increasing porosity and viscosity parameters and the concentration decreases with increasing diffusivity, whereas both the temperature and Nusselt number decrease with increasing thermal conductivity. Practical implications Numerically, the proposed numerical scheme can be applied without concerns on time steps size restrictions. Non-physical solutions cannot be computed. Physically, the flow can be increased by increasing the viscosity parameters. Pollutants with higher diffusivity will have their concentration decreased faster than those of lower diffusivity. The fluid temperature would decrease faster if its thermal conductivity is higher. Originality/value A fully coupled fluid flow with heat and mass transfer problem having nonlinear properties and nonlinear fractional sources and sink terms, presumably, has not been investigated in a general form as done in this study. The detailed numerical analysis of this particular scheme for the identified general model has also not been considered in the past, to the best of the author’s knowledge.

Aims/ Objectives: To investigate the influence of a model parameter on the convergence of two finite difference schemes designed for a convection-diffusion-reaction equation governing the pressure-driven flow of a Newtonian fluid in a rectangular channel.Methodology: By assuming a uni-directional and incompressible channel flow with an exponentially time-varying suction velocity, we formulate a variable-coefficient convectiondiffusion- reaction problem. In the spirit of the method of manufactured solutions, we first obtain a benchmark analytic solution via perturbation technique. This leads to a modified problem which is exactly satisfied by the benchmark solution. Then, we formulate central and backward difference schemes for the modified problem. Consistency and convergence results are obtained in detail. We show, theoretically, that the central scheme is convergent only for values of a model parameter up to an upper bound, while the backward scheme remains convergent for all values of the parameter. An estimate of this upper bound, as a function of the mesh size, is derived. We then conducted numerical experiments to verify the theoretical results.Results: Numerical results showed that no numerical oscillations were observed for values of the model parameter less than the theoretically derived bound.Conclusion: We therefore conclude that the theoretical bound is a safe value to guarantee non-oscillatory solutions of the central scheme.

This paper investigates the problem of ground temperature under a vegetative cover with time dependent suction and radiative heat transfer. By assuming that the ground radiates heat from its’ surface and that the radiative heat flux takes a differential form which follows that of Tarkhar et al [1], the problem was formulated. By imposing a regular time dependent perturbation, the problem was solved. The numerical results and graphs were computed using the software, Mathematica. Results revealed that individual and simultaneous increase in flow parameters caused decrease in the temperature.

This paper investigates the problem of MHD free convection and oscillatory flow of an optically thin fluid bounded by two horizontal porous parallel walls under the influence of an external imposed transverse magnetic field in a porous medium. By taking the radiative heat flux in the differential form and imposing an oscillatory time-dependent perturbation the coupled nonlinear problem is solved for the velocity and temperature profiles. The results show that the fluid velocity increased with increase in the magnetic field, M, Grashof number, Gr, and the porosity, χ parameters; whereas the fluid velocity is decreased with increase in the radiation parameter, F. Equally, the skin friction decreased with increases in magnetic and porosity parameters for 2 = Gr and increased for 5 = Gr , while the reverse is the case with increase in radiation. Finally, the rate of heat transfer is increased with increases in the radiation parameter.