Mathematical and Computational Applications

238 KiB

Open AccessArticle

Conservation Laws and Conserved Quantities for Laminar Radial Jets with Swirl

by R. Naz, I. Naeem and F.M. Mahomed

Math. Comput. Appl. 2010, 15(4), 742-761; https://doi.org/10.3390/mca15040742 - 01 Dec 2010

Cited by 1 | Viewed by 1074

Conserved quantities play a central role in the solution of jet flow problems. A systematic way of deriving conserved quantities for the radial jets with swirl is presented. The multiplier approach is used to derive the conservation laws for the system of three [...] Read more.

Conserved quantities play a central role in the solution of jet flow problems. A systematic way of deriving conserved quantities for the radial jets with swirl is presented. The multiplier approach is used to derive the conservation laws for the system of three boundary layer equations for the velocity components and for the system of two partial differential equations for the stream function. When the swirl is zero or at a large distance from the orifice (at infinity), the boundary layer equations for the radial jets with swirl reduce to those of the purely radial jets. The conserved quantities for the radial liquid, free and wall jets with swirl are derived by integrating the conservation laws across the jets. Full article

214 KiB

Open AccessArticle

On the Redefinition of the Variational and `Partial' Variational Conservation Laws in a Class of Nonlinear PDEs with Mixed Derivatives

by R. Narain and A. H. Kara

Math. Comput. Appl. 2010, 15(4), 732-741; https://doi.org/10.3390/mca15040732 - 01 Dec 2010

Cited by 9 | Viewed by 906

Abstract

The construction of conserved vectors using Noether’s and partial Noether’s theorems are carried out for high order PDEs with mixed derivatives. The resultant conserved flows display some interesting ‘divergence properties’ owing to the existence of the mixed derivatives. These are spelled out for [...] Read more.

The construction of conserved vectors using Noether’s and partial Noether’s theorems are carried out for high order PDEs with mixed derivatives. The resultant conserved flows display some interesting ‘divergence properties’ owing to the existence of the mixed derivatives. These are spelled out for various equations from mathematical physics. Full article

217 KiB

Open AccessArticle

Approximate First Integrals for a System of Two Coupled Van Der Pol Oscillators with Linear Diffusive Coupling

by I. Naeem and F. M. Mahomed

Math. Comput. Appl. 2010, 15(4), 720-731; https://doi.org/10.3390/mca15040720 - 01 Dec 2010

Cited by 8 | Viewed by 990

Abstract

The approximate partial Noether operators for a system of two coupled van der Pol oscillators with linear diffusive coupling are presented via a partial Lagrangian approach. The underlying system of two equations, in general, do not admit a standard Lagrangian. However, the approximate [...] Read more.

The approximate partial Noether operators for a system of two coupled van der Pol oscillators with linear diffusive coupling are presented via a partial Lagrangian approach. The underlying system of two equations, in general, do not admit a standard Lagrangian. However, the approximate first integrals are constructed by utilization of the partial Noether’s theorem with the help of approximate partial Noether operators associated with a partial Lagrangian. These approximate partial Noether operators are not approximate symmetries of the system under study and they do not form an approximate Lie algebra. Moreover, we show how approximate first integrals can be constructed for perturbed ordinary differential equations (ODEs) without making use of a standard Lagrangian. Full article

219 KiB

Open AccessArticle

Symmetry Reduction and Numerical Solution of a Third-Order ODE from Thin Film Flow

by E. Momoniat and F.M. Mahomed

Math. Comput. Appl. 2010, 15(4), 709-719; https://doi.org/10.3390/mca15040709 - 01 Dec 2010

Cited by 13 | Viewed by 1103

Abstract

A new approach to solving high-order ordinary differential equations numerically is presented. Instead of the usual approach of writing a high-order ordinary differential equation as a system of first-order ordinary differential equations, we write the high-order ordinary differential equation in terms of its [...] Read more.

A new approach to solving high-order ordinary differential equations numerically is presented. Instead of the usual approach of writing a high-order ordinary differential equation as a system of first-order ordinary differential equations, we write the high-order ordinary differential equation in terms of its differential invariants. The third-order ordinary differential equation y′′′ = y^−k for constant k is used to illustrate this approach for the cases k = 2 and k = 3. Full article

225 KiB

Open AccessArticle

Group Classification of Coupled Diffusion System with Applications in Soil Science

by M. Molati and F.M. Mahomed

Math. Comput. Appl. 2010, 15(4), 697-708; https://doi.org/10.3390/mca15040697 - 01 Dec 2010

Cited by 1 | Viewed by 1035

Abstract

We perform a complete group classification of a coupled system of diffusion equations with applications in soil science. The canonical forms of the lowdimensional Lie algebras and the Lie algebras of higher dimension provide a means to specify the diffusion coefficients completely. Full article

259 KiB

Open AccessArticle

Symmetry Reductions of a Flow with Power Law Fluid and Contaminant-Modified Viscosity

by R. J. Moitsheki, S. Abelman and T. Hayat

Math. Comput. Appl. 2010, 15(4), 685-696; https://doi.org/10.3390/mca15040685 - 01 Dec 2010

Cited by 3 | Viewed by 1019

Abstract

In this study symmetry analysis is carried out for a system dealing with nonreactive pollutant transport along a single channel. Constitutive equations obeying a power law fluid are used in the description of the mathematical problem. We obtain forms of the source term [...] Read more.

In this study symmetry analysis is carried out for a system dealing with nonreactive pollutant transport along a single channel. Constitutive equations obeying a power law fluid are used in the description of the mathematical problem. We obtain forms of the source term for which the governing system admits extra point symmetries. Invariant solutions which satisfy physical boundary conditions are constructed. In some cases we resort to numerical methods to obtain approximate solutions. Full article

317 KiB

Open AccessArticle

Two-Dimensional Flow in a Deformable Channel with Porous Medium and Variable Magnetic Field

by B. T. Matebese, A. R. Adem, C. M. Khalique and T. Hayat

Math. Comput. Appl. 2010, 15(4), 674-684; https://doi.org/10.3390/mca15040674 - 01 Dec 2010

Cited by 9 | Viewed by 1213

Abstract

This article is concerned with the analytic solution for a nonlinear flow problem of an incompressible viscous fluid. The fluid is taken in a channel having two weakly permeable moving porous walls. An incompressible fluid fills the porous space inside the channel. The [...] Read more.

This article is concerned with the analytic solution for a nonlinear flow problem of an incompressible viscous fluid. The fluid is taken in a channel having two weakly permeable moving porous walls. An incompressible fluid fills the porous space inside the channel. The fluid is magnetohydrodynamic in the presence of a time-dependent magnetic field. Lie group method is applied in the derivation of analytic solution. The effects of the magnetic field, porous medium, permeation Reynolds number and wall dilation rate on the axial velocity are shown and discussed. Full article

212 KiB

Open AccessArticle

A Class of Charged Relativistic Spheres

by K. Komathiraj and S.D. Maharaj

Math. Comput. Appl. 2010, 15(4), 665-673; https://doi.org/10.3390/mca15040665 - 01 Dec 2010

Cited by 6 | Viewed by 996

Abstract

We find a new class of exact solutions to the Einstein-Maxwell equations which can be used to model the interior of charged relativistic objects. These solutions can be written in terms of special functions in general; for particular parameter values it is possible [...] Read more.

We find a new class of exact solutions to the Einstein-Maxwell equations which can be used to model the interior of charged relativistic objects. These solutions can be written in terms of special functions in general; for particular parameter values it is possible to find solutions in terms of elementary functions. Our results contain models found previously for uncharged neutron stars and charged isotropic spheres. Full article

210 KiB

Open AccessArticle

An Analysis of the Symmetries and Conservation Laws of the Class of Zakharov-Kuznetsov Equations

by A. H. Kara

Math. Comput. Appl. 2010, 15(4), 658-664; https://doi.org/10.3390/mca15040658 - 01 Dec 2010

Cited by 10 | Viewed by 1087

Abstract

In this paper, we study and classify the conservation laws of the Zakharov- Kuznetsov equations. It is shown that these can be obtained by studying the interplay between symmetry generators and ‘multipliers’. This is, particularly, useful for the higher-order multipliers. As a final [...] Read more.

In this paper, we study and classify the conservation laws of the Zakharov- Kuznetsov equations. It is shown that these can be obtained by studying the interplay between symmetry generators and ‘multipliers’. This is, particularly, useful for the higher-order multipliers. As a final note, we include Drinfeld-Sokolov-Wilson system to demonstrate the usefulness of the approach to systems of pdes. Full article

391 KiB

Open AccessArticle

Magnetic Field and Endoscope Influences on Peristaltic Transport: An Exact Solution

by T. Hayat, S. Abelman, E. Momoniat and F. M. Mahomed

Math. Comput. Appl. 2010, 15(4), 638-657; https://doi.org/10.3390/mca15040638 - 01 Dec 2010

Cited by 4 | Viewed by 907

Abstract

The effect of magnetic field on peristaltic flow through the gap between uniform tubes is studied under the assumption of long wavelength at low Reynolds number. The inner tube is rigid and the outer tube has a sinusoidal wave travelling down its wall. [...] Read more.

The effect of magnetic field on peristaltic flow through the gap between uniform tubes is studied under the assumption of long wavelength at low Reynolds number. The inner tube is rigid and the outer tube has a sinusoidal wave travelling down its wall. The flow is investigated in a wave frame of reference moving with the velocity of the wave. The analytical solution for velocities and pressure gradient is derived. The effects of magnetic field and an endoscope on the velocities, pressure gradient, pressure rise and frictional forces on the inner and outer tubes are examined. Full article

359 KiB

Open AccessArticle

Peristaltic Mechanism in an Asymmetric Channel with Heat Transfer

by T. Hayat and F.M. Abbasi

Math. Comput. Appl. 2010, 15(4), 621-637; https://doi.org/10.3390/mca15040621 - 01 Dec 2010

Cited by 14 | Viewed by 1015

Abstract

This study reports the effects of velocity and thermal slip parameters on the peristaltic motion of variable viscosity and magnetohydrodynamic (MHD) fluid in an asymmetric channel. Heat transfer coefficient and temperature are given due attention with respect to embedded parameters in the problem. [...] Read more.

This study reports the effects of velocity and thermal slip parameters on the peristaltic motion of variable viscosity and magnetohydrodynamic (MHD) fluid in an asymmetric channel. Heat transfer coefficient and temperature are given due attention with respect to embedded parameters in the problem. Full article

241 KiB

Open AccessArticle

Efficient Boundary Value Problem Solution for a Lane-Emden Equation

by C. Harley and E. Momoniat

Math. Comput. Appl. 2010, 15(4), 613-620; https://doi.org/10.3390/mca15040613 - 01 Dec 2010

Cited by 6 | Viewed by 1044

Abstract

An efficient method for determining an initial guess to an iterative solution to the boundary value problem y" + (k/x)y′ + δe^y = 0 solved subject to y′(0) = 0 and y(1) = 0 is proposed. This initial guess overcomes the instability [...] Read more.

An efficient method for determining an initial guess to an iterative solution to the boundary value problem y" + (k/x)y′ + δe^y = 0 solved subject to y′(0) = 0 and y(1) = 0 is proposed. This initial guess overcomes the instability that occurs at the boundary y′ (0) = 0 for k ≤ 1. When the iterative method becomes singular we can conclude that the maximum value of the critical parameter β has been determined. Full article

239 KiB

Open AccessArticle

Lie Infinitesimal Conserved Quantities for Itô Stochastic ODEs

by E. Fredericks, F.M. Mahomed and K. Masike

Math. Comput. Appl. 2010, 15(4), 601-612; https://doi.org/10.3390/mca15040601 - 01 Dec 2010

Viewed by 940

Abstract

A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itô integral context is pursued. The basis of this construction relies on Lie bracket relations on both the instantaneous drift and diffusion of an Itˆo stochastic ordinary differential equation (SODE). Full article

229 KiB

Open AccessArticle

Integration of Systems of ODEs via Nonlocal Symmetry-Like Operators

by M. U. Farooq, F. M. Mahomed and M. A. Rashid

Math. Comput. Appl. 2010, 15(4), 585-600; https://doi.org/10.3390/mca15040585 - 01 Dec 2010

Cited by 3 | Viewed by 910

Abstract

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex [...] Read more.

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of two second-order ordinary differential equations that admits four Lie-like operators with certain structure is obtainable from a restricted complex ordinary differential equation possessing a connected two-dimensional complex Lie algebra. A direct method of integration for a system of two first and second-order equations which possess nonlocal symmetry-like operators are given. Moreover, we present the use of nonlocal Noether-like operators to effect double reduction of order of systems of two second-order equations that arise from the corresponding scalar complex Euler- Lagrange equations which admit nonlocal Noether symmetry. Full article

642 KiB

Open AccessArticle

Coupling Drift-Flux Models with Unequal Sonic Speeds

by Mapundi K. Banda, Michael Herty and Jean-Medard T. Ngnotchouye

Math. Comput. Appl. 2010, 15(4), 574-584; https://doi.org/10.3390/mca15040574 - 01 Dec 2010

Cited by 6 | Viewed by 1060

Abstract

The well-posedness of a Riemann problem at a junction in a pipeline network is discussed. In addition computational results on the dynamics of the flow of a multi-component gas at such network junctions are presented. The work presented here is a generalisation of [...] Read more.

The well-posedness of a Riemann problem at a junction in a pipeline network is discussed. In addition computational results on the dynamics of the flow of a multi-component gas at such network junctions are presented. The work presented here is a generalisation of [M. K. Banda, M. Herty, and J. M. T. Ngnotchouye, Towards a mathematical analysis of multiphase drift-flux model in networks, SIAM J. Sci. Comput., 31(6): 4633 – 4653, 2010] to models in which the equation of state has different compressibility factors or sonic speeds. Full article

351 KiB

Open AccessArticle

by E.W. Mureithi and D.P. Mason

Math. Comput. Appl. 2010, 15(4), 558-573; https://doi.org/10.3390/mca15040558 - 01 Dec 2010

Cited by 13 | Viewed by 1213

Abstract

The boundary layer flow over a horizontal plate with power law variations in the freestream velocity and wall temperature of the form U_e ∼ xⁿ and T_w − T_∞∼ x^mand with viscous dissipation, is studied. The [...] Read more.

The boundary layer flow over a horizontal plate with power law variations in the freestream velocity and wall temperature of the form U_e ∼ xⁿ and T_w − T_∞∼ x^mand with viscous dissipation, is studied. The boundary layer equations are transformed to a dimensionless system of equations using a non–similarity variable ξ(x) and a pseudo-similarity variable η (x, y). The effects of the various parameters of the flow on velocity and temperature distribution in the boundary layer, on the local skin friction and local heat transfer coefficients and on the non–similar terms, are investigated. Full article

335 KiB

Open AccessArticle

Streamlines and Detached Wakes in Steady Flow past a Spherical Liquid Drop

by G.M. Moremedi and D.P. Mason

Math. Comput. Appl. 2010, 15(4), 543-557; https://doi.org/10.3390/mca15040543 - 01 Dec 2010

Cited by 3 | Viewed by 938

Abstract

The flow interior and exterior to a viscous liquid drop in steady motion in an unbounded quiescent fluid is investigated using the perturbation solution of Taylor and Acrivos (1964) to first order in the Reynolds number. New analytical results are derived for the [...] Read more.

The flow interior and exterior to a viscous liquid drop in steady motion in an unbounded quiescent fluid is investigated using the perturbation solution of Taylor and Acrivos (1964) to first order in the Reynolds number. New analytical results are derived for the detached wake behind the drop. It is found that as the viscosity of the drop tends to infinity the wake becomes attached to the surface of the drop and the results of Proudman and Pearson (1957) for a solid sphere are rederived. Full article

222 KiB

Open AccessArticle

Conservation Laws and Invariant Solutions in the Fanno Model for Turbulent Compressible Flow

by M. Anthonyrajah and DP Mason

Math. Comput. Appl. 2010, 15(4), 529-542; https://doi.org/10.3390/mca15040529 - 01 Dec 2010

Cited by 18 | Viewed by 1059

Abstract

Asymptotic reductions of the Fanno model for one-dimensional turbulent compressible flow of a perfect gas in a long tube are investigated. Conservation laws are derived using the multiplier method for a nonlinear wave equation and a nonlinear diffusion equation for the mean velocity [...] Read more.

Asymptotic reductions of the Fanno model for one-dimensional turbulent compressible flow of a perfect gas in a long tube are investigated. Conservation laws are derived using the multiplier method for a nonlinear wave equation and a nonlinear diffusion equation for the mean velocity and a nonlinear diffusion equation for the mean pressure. Two conserved quantities for the mean velocity are obtained from the conservation laws and boundary conditions. An invariant solution is derived for the mean velocity using the Lie point symmetries associated with the conserved vector which generated the conserved quantity for the boundary value problem. Full article

622 KiB

Open AccessArticle

On a New Class of Models in Elasticity

by K. R. Rajagopal

Math. Comput. Appl. 2010, 15(4), 506-528; https://doi.org/10.3390/mca15040506 - 01 Dec 2010

Cited by 49 | Viewed by 1212

Abstract

Recently, Rajagopal and co-workers have shown (see Rajagopal [1], Rajagopal and Srinivasa [2],[3], Bustamante and Rajagopal[4], Rajagopal and Saccomandi [5]) that if by an elastic body one means a body that is incapable of dissipation, then the class of such bodies is far [...] Read more.

Recently, Rajagopal and co-workers have shown (see Rajagopal [1], Rajagopal and Srinivasa [2],[3], Bustamante and Rajagopal[4], Rajagopal and Saccomandi [5]) that if by an elastic body one means a body that is incapable of dissipation, then the class of such bodies is far larger than either Green elastic or for that matter Cauchy elastic bodies as one could model elastic bodies using implicit constitutive relations between the Cauchy stress and the deformation gradient or implicit constitutive relations that are rate equations involving the Piola-Kirchhoff stress and the Green-St.Venant Strain (see Rajagopal and Srinivasa [2]). Such a generalized framework allows one to develop models whose linearization with regard to the smallness of the displacement gradient allows one to obtain models that have limited linearized strains even while the stresses are very large. Such a possibility has important consequences to problems which, within the context of the classical linearized theory, leads to singularities. In this short paper, we illustrate the implications of such models by considering simple problems within the context of a specific model belonging to the general class, wherein the strains remain small as the stresses tend to very large values. Full article

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Math. Comput. Appl., Volume 15, Issue 4 (December 2010) – 19 articles , Pages 506-761

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