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>> 05.11.04

Literature review (ix)
This is the last part of the literature review, as submitted in part for the annual review and which will eventually form the introduction to my thesis. For more details on this series, please see the first part. Although much research effort has been expended in both the classical and nonclassical formulations of Lie theory concerning B- and IVPs as outlined above, there is still no general, systematic procedure that a user can employ to find symmetries that satisfy conditions (i)-(iii). Instead of this being the fault of the various... {>>}

>> 04.11.04

Literature review (viii)
The remainder of the classical techniques currently available for dealing with BVPs also concentrate on classifying those initial conditions that, through the application of symmetry methods, will solve classes of associated IVPs. As a result, rather prescriptive methods have been developed in which B- and IVPs that can be solved have been classified according to their symmetries, instead of finding and using the symmetries of a given system in order to solve it. (Note that the terms boundary value problem and initial value problem are used synonymously. The relationship between... {>>}

>> 03.11.04

Literature review (vii)
This is the seventh part of the literature review, as submitted in part for the annual review and which will eventually form the introduction to my thesis. For more details on this series, please see the first part. The most successful use of the classical method has been employed in a series of works by Bluman and various co-authors. For a BVP for an nth order ODE, it has been shown that the reduction of order made possible through a multi-parameter solvable Lie group can lead to obtaining qualitative results... {>>}

>> 02.11.04

Literature review (vi)
This is the sixth part of the literature review, as submitted in part for the annual review and which will eventually form the introduction to my thesis. For more details on this series, please see the first part. The area of symmetry analysis is therefore a large and varied one, extending naturally when called upon to do so. Although Lie's approach and its many extensions has had great success in determining and classifying solutions of differential equations, it has proved much less successful in the treatment of boundary- and initial-value... {>>}

>> 01.11.04

Literature review (v)
This is the fifth part of the literature review, as submitted in part for the annual review and which will eventually form the introduction to my thesis. For more details on this series, please see the first part. Generalised symmetries do not act geometrically on the space of variables but are still considered to be symmetries; dissimilarly, there are a class of point transformations that are not at all symmetries but can lead to exact solutions of PDEs. Such "symmetries" are referred to as "nonclassical" symmetries and were introduced by... {>>}