Approximate symmetry analysis of differential equations is a very active area of research in modern group analysis. The researcher describes and utilises three recent developments in the area of approximate symmetry analysis, viz., approximate conditional symmetries, approximate potential symmetries and Noether symmetries and their corresponding conservation laws of a perturbed equation.
The researcher calculates the potential symmetries, with respect to the usual conservation law, of the porous medium equation in one spatial dimension. Furthermore, the researcher finds the approximate conditional symmetries of the perturbed porous medium equation and uses these new symmetries to find new approximate group-invariant solutions of this equation.
The researcher then considers the nonlinear diffusion equation with a perturbed convection term. The potential symmetries for the exact equation with respect to the second conservation law are classified. It is found that these exist only in the linear case. Then approximate potential symmetries with respect to the usual conservation law for an interesting case are reviewed and an approximate group-invariant solution presented. This is done for comparative purposes. It is further shown that no nontrivial approximate potential symmetries of order one exists for the perturbed equation with respect to the other conservation law.
The researcher determines the approximate conditional symmetries for two forms of the perturbed nonlinear heat equation. The researcher utilises these new symmetries to find new approximate group-invariant solutions of the equation. Finally the researcher investigates two forms of the perturbed nonlinear wave equation. The researcher computes Lagrangians and from this the researcher calculates the Noether symmetries, with respect to one of the Lagrangians, and the corresponding conservation laws. This complements the results in the recent literature.
|Degree Type||Doctoral degree|
|Degree Description||PhD (Science)|