# Journal of Nonlinear Mathematical Physics

Volume 15, Issue Supplement 1, August 2008

### Euler's Tercentenary

Special issue celebrating the birth of Leonard Euler.

**Research Article**

## 2. Nonlocal Extensions of Similarity Methods

George Bluman

Pages: 1 - 24

Similarity methods include the calculation and use of symmetries and conservation laws for a given partial differential equation (PDE). There exists a variety of software to calculate and use local symmetries and local conservation laws. However, it is often the case that a given PDE admits no useful...

**Research Article**

## 3. Complex Lie Symmetries for Variational Problems

Sajid Ali, Fazal M Mahomed, Asghar Qadir

Pages: 25 - 35

We present the use of complex Lie symmetries in variational problems by defining a complex Lagrangian and considering its Euler-Lagrange ordinary differential equation. This Lagrangian results in two real “Lagrangians” for the corresponding system of partial differential equations, which satisfy Euler-Lagrange...

**Research Article**

## 4. Symmetries and Differential Forms

A.H. Davison, A.H. Kara

Pages: 36 - 43

The method for writing a differential equation or system of differential equations in terms of differential forms and finding their symmetries was devised by Harrison and Estabrook (1971). A modification to the method is demonstrated on a wave equation with variable speed, and the modified method is...

**Research Article**

## 5. A Formal Approach for Handling Lie Point Symmetries of Scalar First-Order Ito Stochastic Ordinary Differential Equations

Ebrahim Fredericks, Fazal M Mahomed

Pages: 44 - 59

Many methods of deriving Lie point symmetries for Itˆo stochastic ordinary differential equations (SODEs) have surfaced. In the Itˆo calculus context both the formal and intuitive understanding of how to construct these symmetries has led to seemingly disparate results. The impact of Lie point symmetries...

**Research Article**

## 6. On the Origins of Symmetries of Partial Differential Equations: the Example of the Korteweg-de Vries Equation

Keshlan S. Govinder, Barbara Abraham-Shrauner

Pages: 60 - 68

Type II hidden symmetries of partial differential equations () are extra symme- tries in addition to the inherited symmetries of the differential equations which arise when the number of independent and dependent variables is reduced by a Lie point symmetry. (Type I hidden symmetries arise in the increase...

**Research Article**

## 7. Alternate Derivation of the Critical Value of the Frank-Kamenetskii Parameter in Cylindrical Geometry

Charis Harely, Ebrahim Momoniat

Pages: 69 - 76

Noether’s theorem is used to determine first integrals admitted by a generalised Lane-Emden equation of the second kind modelling a thermal explosion. These first integrals exist for rectangular and cylindrical geometry. For rectangular geometry the first integrals show the symmetry of the temperature...

**Research Article**

## 8. The Rayleigh Problem for a Third Grade Electrically Conducting Fluid in a Magnetic Field

Tasawar Hayat, Herman Mambili-Mamboundou, Ebrahim Momoniat, Fazal M. Mahomed

Pages: 77 - 90

The influence of a magnetic field on the flow of an incompressible third grade electrically conducting fluid bounded by a rigid plate is investigated. The flow is induced due by the motion of a plate in its own plane with an arbitrary velocity. The solution of the equations of conservation of mass and...

**Research Article**

## 9. Peristaltic MHD Flow of Third Grade Fluid with an Endoscope and Variable Viscosity

T. Hayat, Ebrahim Momoniat, Fazal M. Mahomed

Pages: 91 - 104

This investigation deals with the mechanism of peristaltic transport of a non-Newtonian, incompressible and electrically conducting fluid with variable viscosity and an endoscope effects. The magnetic Reynolds number is taken to be small. The mechanical properties of the material are represented by the...

**Research Article**

## 10. On the Exact Solutions of the Nonlinear Wave and (omega)4-Model Equations

A.H. Kara, A.H. Bokhari, F.D. Zaman

Pages: 105 - 111

The nonlinear wave equation with variable long wave velocity and the Gordon-type equations (in particular, the omega-model equation) display a range of symmetry generators, inter alia, translations, Lorentz rotations and scaling - all of which are related to conservation laws. We do a study of the symmetries...

**Research Article**

## 11. Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations

Fazal M. Mahomed

Pages: 112 - 123

We obtain a complete invariant characterization of scalar linear (1+1) parabolic equations under equivalence transformations for all the four canonical forms. Firstly semi-invariants under changes of independent and dependent variables and the construction of the relevant transformations that relate...

**Research Article**

## 12. Conditional Linearizability Criteria for Third Order Ordinary Differential Equations

Fazal M. Mahomed, Asghar Qadir

Pages: 124 - 133

Using geometric methods for linearizing systems of second order cubically non-linear in the first derivatives ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for conditional linearizability via point transformation with...

**Research Article**

## 13. Group Invariant Solution for a Two-Dimensional Turbulent Free Jet described by Eddy Viscosity

D.P. Mason, D.L. Hill

Pages: 134 - 148

The group invariant solution for the stream function and the effective viscosity of a two-dimensional turbulent free jet are derived. Prandtl’s hypothesis is not imposed. When the eddy viscosity is constant across the jet it is found that the mean velocity profile is the same as that of a laminar jet...

**Research Article**

## 14. Exact Solutions of a Spherically Symmetric Energy Transport Model for Semiconductors

Motlatsi Molati

Pages: 149 - 158

The symmetry classification and reduction of a non–stationary spherically symmetric energy–transport model for semiconductors was investigated by Molati and Wafo Soh (2005). In this work the exact solutions of the reduced model in the stationary case are constructed.

**Research Article**

## 15. A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations

Sibusiso Moyo, P. G. L. Leach

Pages: 159 - 164

We study the integrability properties of the hierarchy of a class of nonlinear ordinary differential equations and point out some of the properties of these equations and their connection to the Ermakov-Pinney equation.

**Research Article**

## 16. Partial Noether Operators and First Integrals for a System with two Degrees of Freedom

I. Naeem, Fazal M. Mahomed

Pages: 165 - 178

We construct all partial Noether operators corresponding to a partial agrangian for a system with two degrees of freedom. Then all the first integrals are obtained explicitly by utilizing a Noether-like theorem with the help of the partial Noether operators. We show how the first integrals can be constructed...

**Research Article**

## 17. Symmetry Solutions of a Third-Order Ordinary Differential Equation which Arises from Prandtl Boundary Layer Equations

R. Naz, Fazal M. Mahomed, David P. Mason

Pages: 179 - 191

The similarity solution to Prandtl’s boundary layer equations for two-dimensional and radial flows with vanishing or constant mainstream velocity gives rise to a third-order ordinary differential equation which depends on a parameter ?. For special values of ? the third-order ordinary differential equation...

**Research Article**

## 18. Transformation Groups Applied to Two-Dimensional Boundary Value Problems in Fluid Mechanics

Kevin Paul Pereira

Pages: 192 - 202

The boundary value problems for the two-dimensional, steady, irrotational flow of a frictionless, incompressible fluid past a wedge and a circular cylinder are considered. It is shown that by considering first the invariance of the boundary condition we are able to obtain a transformation group that...

**Research Article**

## 19. Filtration of a Visco-Elastic Liquid with Relaxation: a Note on Lie Point Symmetries and Reductions

Astri Sjoberg, Ozgul Kartal

Pages: 203 - 210

We present the Lie point symmetries admitted by third order partial differential equations (PDEs) which model the pressure of a visco-elastic liquid with relaxation which filtrates through a porous medium. The symmetries are used to construct reductions of the PDEs to ordinary differential equations...

**Research Article**

## 20. Exact Time Dependent Solutions to (1+1) Fokker-Planck Equation via Linearizing Transformations to the Ito Equations

Gazanfer Unal, C. Masood Khalique

Pages: 211 - 221

It is shown that invertible linearizing transformations of the one-dimensional Ito stochastic differential equations cast the associated Fokker-Planck equation to the heat equation. This leads to the time-dependent exact solutions to the Fokker-Planck equations via inverse transformations. To obtain...