Ashfaque H. Bokhari

Ricci collineations of static spherically symmetric spacetimes

The Ricci collineations of static spherically symmetric spacetimes are classified and their relationship with isometries is discussed. A general theorem about this relationship is stated and its extension to all spherically symmetric spacetimes is discussed.

Collineations of the Ricci tensor

Ricci collineations for the Ricci tensor which is constructed from a general spherically symmetric and static metric are classified for all possibilities of Rab(r) (such that Rab≠0 for a=b). It turns out that the only collineations admitted by this tensor can be ten, six, or four and there does not appear any case in between.

A note on a symmetry analysis and exact solutions of a nonlinear fin equation

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.

Curvature collineations of some static spherically symmetric space–times

Curvature collineations of some static spherically symmetric space–times are derived and compared with isometries and Ricci collineations for corresponding space–times.

Symmetries and integrability of a fourth-order Euler–Bernoulli beam equation

The complete symmetry group classification of the fourth-order Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solut...

Adomian Decomposition Method for a Nonlinear Heat Equation with Temperature Dependent Thermal Properties

The solutions of nonlinear heat equation with temperature dependent diffusivity are investigated using the modified Adomian decomposition method. Analysis of the method and examples are given to show that the Adomian series solution gives an excellent approximation to the exact solution. This accuracy can be increased by increasing the number of terms in the series expansion. The Adomian solutions are presented in some situations of interest.

Wave equation on spherically symmetric Lorentzian metrics

Wave equation on a general spherically symmetric spacetime metric is constructed. Noether symmetries of the equation in terms of explicit functions of θ and ϕ are derived subject to certain differential constraints. By restricting the metric to flat Friedman case the Noether symmetries of the wave equation are presented. Invertible transformations are constructed from a specific subalgebra of these Noether symmetries to convert the wave equation with variable coefficients to the one with constant coefficients.

Classification of spherically symmetric static space–times by their curvature collineations

A complete classification of all spherically symmetric static space–times according to their curvature collineations is presented and compared with Ricci collineations of corresponding space–times.

Killing vectors of static spherically symmetric metrics

An error in a previous theorem [J. Math. Phys. 28, 1019 (1987); 29, 525 (1988)] is corrected and the theorem is extended.

Invariant boundary value problems for a fourth-order dynamic Euler-Bernoulli beam equation

We obtain the complete Lie symmetry group classification of the dynamic fourth-order Euler-Bernoulli partial differential equation, where the elastic modulus, the area moment of inertia are constants and the applied load is a nonlinear function. In the Lie analysis, the principal Lie algebra which is two-dimensional extends in three cases, viz., the linear, the exponential, and the general power law. For each of the nontrivial cases, we determine symmetry reductions to ordinary differential equations which are of order four. In only one case related to the power law we are able to have a compatible initial-boundary value problem for a clamped end and a free beam. For these cases we deduce the corresponding fourth-order ordinary differential equations with appropriate boundary conditions. We provide an asymptotic solution for the reduced fourth-order ordinary differential equation corresponding to a clamped or free beam.