Se proporcionan algunas desigualdades tipo Grüss para la integral de Riemann-Stieltjes de integra... more Se proporcionan algunas desigualdades tipo Grüss para la integral de Riemann-Stieltjes de integrandos de valores continuos complejos definidos sobre el circulo unitario complejo C(0, 1) y varias subclases de integradores son dados. Aplicaciones naturales para funciones de operadores unitarios en espacios de Hilbert son proporcionadas.
The spreading of a thin liquid drop under gravity and small surface tension on a slowly dropping ... more The spreading of a thin liquid drop under gravity and small surface tension on a slowly dropping flat plane is investigated. The initial slope of the flat plane is assumed to be small. By considering a straightforward forward perturbation, the fourth-order nonlinear partial differential equation modelling the spreading of the liquid drop reduces to a second-order nonlinear partial differential equation. This resulting equation is solved using the classical Lie group method. The group invariant solution is found to model the long time behaviour of the liquid drop.
Abstract This paper presents a numerical investigation on steady triple diffusive mixed convectio... more Abstract This paper presents a numerical investigation on steady triple diffusive mixed convection boundary layer flow past a vertical plate moving parallel to the free stream in the upward direction. The temperature of the plate is assumed to be hotter compared to the surrounded fluid temperature. Sodium chloride and Sucrose are chosen as solutal components which are added in the flow stream from below with various concentration levels. The concentrations of NaCl-Water and Sucrose-Water are considered to be higher near the wall compared to the concentrations of NaCl-Water and Sucrose-Water within the free stream. The coupled nonlinear partial differential equations are transformed using the non-similarity variables and solved numerically by an implicit finite difference scheme with quasi-linearization technique. The effects of Richardson numbers, velocity ratio parameters, ratio of buoyancy parameters and Schmidt numbers of both the solutal components on the fluid flow, thermal and species concentration fields are investigated. Results indicate that the species concentration boundary layer thickness decreases with the increase of Schmidt numbers and that increases with the ratio of buoyancy parameters for both the species components. Overall, the mass transfer rate is found to increase with Schmidt numbers approximately 4.36% and 64.56% for NaCl and Sucrose, respectively.
The effect of magnetic field on peristaltic flow through the gap between uniform tubes is studied... 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.
We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) par... more We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) parabolic equations in order to perform reductions to the three Lie canonical forms of a bond-pricing model from finance. As a consequence we arrive at new results on bond-pricing equations that admit four nontrivial symmetries. In the second part we draw attention to a new method developed for equations of economics.
International Journal of Non-Linear Mechanics, 1998
... 1.2)can be expressed in terms of the ratio of the characteristic thickness of the thin film t... more ... 1.2)can be expressed in terms of the ratio of the characteristic thickness of the thin film to the thickness of the Ekman boundary layer in a ... In Section 2the thin-film equations for an incompressible viscous fluid with respect to a frame of reference attached to the rotating disk ...
Zeitschrift für angewandte Mathematik und Physik, 2014
An energy method is used to analyze the stability of solutions of a mixed space-time diffusion eq... more An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.
In this paper, a new weighted identity for differentiable functions of two variables defined on a... more In this paper, a new weighted identity for differentiable functions of two variables defined on a rectangle from the plane is established. By using the obtained identity and analysis, some new weighted integral inequalities for the classes of coordinated convex, coordinated wright-convex and coordinated quasi-convex functions on the rectangle from the plane are established which provide weighted generalization of some recent results proved for coordinated convex functions. Some applications of our results to random variables and 2D weighted quadrature formula are given as well. where f : I → R, ∅ ̸ = I ⊆ R a convex function, a, b ∈ I with a < b. The inequalities in (1.1) are reversed if f is a concave function. The inequalities (1.1) have various applications for generalized means, information measures, quadrature rules etc., and there is growing literature providing its new proofs, extensions, refinements and generalizations, see for example [2, 4, 5, 6, 9, 21, 22] and the references therein. Let us consider now a bidimensional interval [a, b] × [c, d] in R 2 with a < b and c < d, a mapping f : [a, b] × [c, d] → R is said to be convex on [a, b] × [c, d] if the inequality f (λx + (1 − λ)z, λy + (1 − λ)w) ≤ λf (x, y) + (1 − λ)f (z, w), holds for all (x, y), (z, w) ∈ [a, b] × [c, d] and λ ∈ [0, 1] .
An energy method is used to analyze the stability of solutions of a mixed space-time diffusion eq... more An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.
We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) par... more We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) parabolic equations in order to perform reductions to the three Lie canonical forms of a bond-pricing model from finance. As a consequence we arrive at new results on bond-pricing equations that admit four nontrivial symmetries. In the second part we draw attention to a new method developed for equations of economics.
ABSTRACT The third-order ODE yny‴=1 obtained by investigating travelling-wave solutions or steady... more ABSTRACT The third-order ODE yny‴=1 obtained by investigating travelling-wave solutions or steady-state solutions of the lubrication equation is considered. The third-order ODE yny‴=1 admits two generators of Lie point symmetries. These generators of Lie point symmetries effect a reduction of the third-order ODE to first order. The problem is to determine the initial values of the second derivative, when the initial height and gradient are specified, for which a solution to yny‴=1 touches the contact line y=0. Phase planes corresponding to different representations of the first-order ODE for the cases n2, n=2 and n&gt;2 are analyzed. For the case n2 we are able to determine the initial values of the second derivative for which the solution touches the contact line. For n≥2 no values of the initial second derivative are obtained for which a solution touches the contact line. A symmetry reduction of autonomous first integrals of the third-order ODE yny‴=1 is then investigated. For the cases n=0, n=5/4 and n=5/2 the third-order ODE admits second-order autonomous first integrals. The case n=5/4 is special because the second-order autonomous first integral admits the same two generators of Lie point symmetries as the original third-order ODE and can hence be reduced to an algebraic equation. Investigations of the phase plane for the case n=5/4 shows that the original third-order ODE satisfies the contact line condition y=0 for initial values of the second derivative y″(0)≤−3.
The equation describing the flow of a thin liquid film is analysed by investigating similarity so... more The equation describing the flow of a thin liquid film is analysed by investigating similarity solutions. The flow is driven by gravity and surface shear. A mass sink/source term is also included. Surface tension effects are neglected in this investigation.
We compare the numerical solutions of three fractional partial differential equations that occur ... more We compare the numerical solutions of three fractional partial differential equations that occur in finance. These fractional partial differential equations fall in the class of Lévy models. They are known as the FMLS (Finite Moment Log Stable), CGMY and KoBol models. Conditions for the convergence of each of these models is obtained.
In this paper, we obtain an approximate implicit solution admitted by the Lane-Emden equation y 0... more In this paper, we obtain an approximate implicit solution admitted by the Lane-Emden equation y 00 + (2/x)y 0 + e y = 0 describing the dimensionless density distribution in an isothermal gas sphere. The new approximate implicit solution has a larger radius of convergence than the power series solution. This is achieved by reducing the Lane-Emden equation to first-order using Lie group analysis and determining a power series solution of the reduced equation. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equation. The approximate implicit solution diverges from the power series solution in the radius of convergence.
Fourier and Bessel function solutions of two mixed derivative equations are investigated. For the... more Fourier and Bessel function solutions of two mixed derivative equations are investigated. For the appropriate sign of the material constants in the derivation of the mixed derivative equation, we obtain both Fourier and Bessel function solutions that tend to the corresponding solutions of the phenomenological diffusion equation. For the opposite sign of the material constants, the solutions diverge.
We compare two finite difference schemes to solve the third-order ordinary differential equation ... more We compare two finite difference schemes to solve the third-order ordinary differential equation y = y −k from thin film flow. The boundary conditions come from Tanner's problem for the surface tension driven flow of a thin film. We show that a central difference approximation to the third derivative in the model equation produces a solution curve with oscillations. A difference scheme based on a combination of forward and backward differences produces a smooth accurate solution curve. Both the 0-stability and von Neumann stability properties of the different finite difference schemes are analyzed. The solution curves obtained from both approaches are presented and discussed.
Se proporcionan algunas desigualdades tipo Grüss para la integral de Riemann-Stieltjes de integra... more Se proporcionan algunas desigualdades tipo Grüss para la integral de Riemann-Stieltjes de integrandos de valores continuos complejos definidos sobre el circulo unitario complejo C(0, 1) y varias subclases de integradores son dados. Aplicaciones naturales para funciones de operadores unitarios en espacios de Hilbert son proporcionadas.
The spreading of a thin liquid drop under gravity and small surface tension on a slowly dropping ... more The spreading of a thin liquid drop under gravity and small surface tension on a slowly dropping flat plane is investigated. The initial slope of the flat plane is assumed to be small. By considering a straightforward forward perturbation, the fourth-order nonlinear partial differential equation modelling the spreading of the liquid drop reduces to a second-order nonlinear partial differential equation. This resulting equation is solved using the classical Lie group method. The group invariant solution is found to model the long time behaviour of the liquid drop.
Abstract This paper presents a numerical investigation on steady triple diffusive mixed convectio... more Abstract This paper presents a numerical investigation on steady triple diffusive mixed convection boundary layer flow past a vertical plate moving parallel to the free stream in the upward direction. The temperature of the plate is assumed to be hotter compared to the surrounded fluid temperature. Sodium chloride and Sucrose are chosen as solutal components which are added in the flow stream from below with various concentration levels. The concentrations of NaCl-Water and Sucrose-Water are considered to be higher near the wall compared to the concentrations of NaCl-Water and Sucrose-Water within the free stream. The coupled nonlinear partial differential equations are transformed using the non-similarity variables and solved numerically by an implicit finite difference scheme with quasi-linearization technique. The effects of Richardson numbers, velocity ratio parameters, ratio of buoyancy parameters and Schmidt numbers of both the solutal components on the fluid flow, thermal and species concentration fields are investigated. Results indicate that the species concentration boundary layer thickness decreases with the increase of Schmidt numbers and that increases with the ratio of buoyancy parameters for both the species components. Overall, the mass transfer rate is found to increase with Schmidt numbers approximately 4.36% and 64.56% for NaCl and Sucrose, respectively.
The effect of magnetic field on peristaltic flow through the gap between uniform tubes is studied... 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.
We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) par... more We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) parabolic equations in order to perform reductions to the three Lie canonical forms of a bond-pricing model from finance. As a consequence we arrive at new results on bond-pricing equations that admit four nontrivial symmetries. In the second part we draw attention to a new method developed for equations of economics.
International Journal of Non-Linear Mechanics, 1998
... 1.2)can be expressed in terms of the ratio of the characteristic thickness of the thin film t... more ... 1.2)can be expressed in terms of the ratio of the characteristic thickness of the thin film to the thickness of the Ekman boundary layer in a ... In Section 2the thin-film equations for an incompressible viscous fluid with respect to a frame of reference attached to the rotating disk ...
Zeitschrift für angewandte Mathematik und Physik, 2014
An energy method is used to analyze the stability of solutions of a mixed space-time diffusion eq... more An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.
In this paper, a new weighted identity for differentiable functions of two variables defined on a... more In this paper, a new weighted identity for differentiable functions of two variables defined on a rectangle from the plane is established. By using the obtained identity and analysis, some new weighted integral inequalities for the classes of coordinated convex, coordinated wright-convex and coordinated quasi-convex functions on the rectangle from the plane are established which provide weighted generalization of some recent results proved for coordinated convex functions. Some applications of our results to random variables and 2D weighted quadrature formula are given as well. where f : I → R, ∅ ̸ = I ⊆ R a convex function, a, b ∈ I with a < b. The inequalities in (1.1) are reversed if f is a concave function. The inequalities (1.1) have various applications for generalized means, information measures, quadrature rules etc., and there is growing literature providing its new proofs, extensions, refinements and generalizations, see for example [2, 4, 5, 6, 9, 21, 22] and the references therein. Let us consider now a bidimensional interval [a, b] × [c, d] in R 2 with a < b and c < d, a mapping f : [a, b] × [c, d] → R is said to be convex on [a, b] × [c, d] if the inequality f (λx + (1 − λ)z, λy + (1 − λ)w) ≤ λf (x, y) + (1 − λ)f (z, w), holds for all (x, y), (z, w) ∈ [a, b] × [c, d] and λ ∈ [0, 1] .
An energy method is used to analyze the stability of solutions of a mixed space-time diffusion eq... more An energy method is used to analyze the stability of solutions of a mixed space-time diffusion equation that has application in the unidirectional flow of a second-grade fluid and the distribution of a compound Poisson process. Solutions to the model equation satisfying Dirichlet boundary conditions are proven to dissipate total energy and are hence stable. The stability of asymptotic solutions satisfying Neumann boundary conditions coincides with the condition for the positivity of numerical solutions of the model equation from a Crank–Nicolson scheme. The Crank–Nicolson scheme is proven to yield stable numerical solutions for both Dirichlet and Neumann boundary conditions for positive values of the critical parameter. Numerical solutions are compared to analytical solutions that are valid on a finite domain.
We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) par... more We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) parabolic equations in order to perform reductions to the three Lie canonical forms of a bond-pricing model from finance. As a consequence we arrive at new results on bond-pricing equations that admit four nontrivial symmetries. In the second part we draw attention to a new method developed for equations of economics.
ABSTRACT The third-order ODE yny‴=1 obtained by investigating travelling-wave solutions or steady... more ABSTRACT The third-order ODE yny‴=1 obtained by investigating travelling-wave solutions or steady-state solutions of the lubrication equation is considered. The third-order ODE yny‴=1 admits two generators of Lie point symmetries. These generators of Lie point symmetries effect a reduction of the third-order ODE to first order. The problem is to determine the initial values of the second derivative, when the initial height and gradient are specified, for which a solution to yny‴=1 touches the contact line y=0. Phase planes corresponding to different representations of the first-order ODE for the cases n2, n=2 and n&gt;2 are analyzed. For the case n2 we are able to determine the initial values of the second derivative for which the solution touches the contact line. For n≥2 no values of the initial second derivative are obtained for which a solution touches the contact line. A symmetry reduction of autonomous first integrals of the third-order ODE yny‴=1 is then investigated. For the cases n=0, n=5/4 and n=5/2 the third-order ODE admits second-order autonomous first integrals. The case n=5/4 is special because the second-order autonomous first integral admits the same two generators of Lie point symmetries as the original third-order ODE and can hence be reduced to an algebraic equation. Investigations of the phase plane for the case n=5/4 shows that the original third-order ODE satisfies the contact line condition y=0 for initial values of the second derivative y″(0)≤−3.
The equation describing the flow of a thin liquid film is analysed by investigating similarity so... more The equation describing the flow of a thin liquid film is analysed by investigating similarity solutions. The flow is driven by gravity and surface shear. A mass sink/source term is also included. Surface tension effects are neglected in this investigation.
We compare the numerical solutions of three fractional partial differential equations that occur ... more We compare the numerical solutions of three fractional partial differential equations that occur in finance. These fractional partial differential equations fall in the class of Lévy models. They are known as the FMLS (Finite Moment Log Stable), CGMY and KoBol models. Conditions for the convergence of each of these models is obtained.
In this paper, we obtain an approximate implicit solution admitted by the Lane-Emden equation y 0... more In this paper, we obtain an approximate implicit solution admitted by the Lane-Emden equation y 00 + (2/x)y 0 + e y = 0 describing the dimensionless density distribution in an isothermal gas sphere. The new approximate implicit solution has a larger radius of convergence than the power series solution. This is achieved by reducing the Lane-Emden equation to first-order using Lie group analysis and determining a power series solution of the reduced equation. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equation. The approximate implicit solution diverges from the power series solution in the radius of convergence.
Fourier and Bessel function solutions of two mixed derivative equations are investigated. For the... more Fourier and Bessel function solutions of two mixed derivative equations are investigated. For the appropriate sign of the material constants in the derivation of the mixed derivative equation, we obtain both Fourier and Bessel function solutions that tend to the corresponding solutions of the phenomenological diffusion equation. For the opposite sign of the material constants, the solutions diverge.
We compare two finite difference schemes to solve the third-order ordinary differential equation ... more We compare two finite difference schemes to solve the third-order ordinary differential equation y = y −k from thin film flow. The boundary conditions come from Tanner's problem for the surface tension driven flow of a thin film. We show that a central difference approximation to the third derivative in the model equation produces a solution curve with oscillations. A difference scheme based on a combination of forward and backward differences produces a smooth accurate solution curve. Both the 0-stability and von Neumann stability properties of the different finite difference schemes are analyzed. The solution curves obtained from both approaches are presented and discussed.
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Papers by E. Momoniat