Formatted abstract

In a paper entitled "An Expansion Method for Singular Perturbation Problems" [1 ], Mahony proposed a method for producing uniformly valid, asymptotic expansions of solutions of singular perturbation problems. This method was applied to a particular problem, and in this case worked well. In order to obtain an assessment of tho method, its application to a range of problems has been considered in this thesis. In addition, certain clarifications of mathematical difficulties noted by Mahony, but not resolved by him, have been achieved.
In order to deal with a singular point* (or line, or surface) in a solution domain, one is usually forced into introducing some type of stretched coordinate description of the problem close to this point. This description usually ceases to be of use away from the singular point because the stretched coordinate system is inappropriate away from this point. In the approach used by Mahony, however, a stretched variable "the boundary layer function" is introduced which is appropriate to the situation close to, and away from, the singular point. This 'boundary layer function" is introduced firstly as an unknown, and a solution is sought as a function of the original independent variables and this boundary layer function. The freedom of mathematical form thus introduced can be utilized to extend the range of validity of the solution away from the singular point. Further boundary layer functions can be introduced if there is more than one singular point until the complete solution domain is covered.
In Part I of this thesis, the method has been applied to the second order, linear, ordinary, differential equation
E^{3}y"  g(x,E)y = 0 ,
where E is a small positive parameter. In recent years this equation has been investigated by many authors and several of the simpler cases have been solved. The methods employed by these authors, however, are not readily extensible to apply to other than ordinary, linear, differential equations. It was found that standard results can be obtained in the cases considered, with far less labour, using the Mahony approach. Some problems associated with the above equation, which have not previously been tackled, have also been considered. The difficulties associated with nonuniqueness, which troubled Mahony, are clarified by these examples.
In Part II, the method has been applied to the partial differential equation
E^{3}V^{2}ψ  g(x) ψ= 0 ,
with g(x) negative only in bounded domains in the finite solution domain, and with the associated boundary condition ψ → 0 as x → 00. This i s , of course, the potential well problem of Quantum Mechanics. The twodimensional case is considered in detail. The application of Mahony's technique leads naturally to a particular class of eigensolutions. The boundary layer function introduced to cope with the situation in this case, determines a coordinate network in terms of which the problem can be solved quite generally. Thus the geometry of a particular problem is absorbed into the geometry of this coordinate net, no mathematical techniques special to a particular geometry being required.
Finally, in Part III , the method has been applied to a nonlinear, ordinary, differential equation case of importance in the theory of shocks. Certain limitations of the method are exhibited in this case, and a combination of the Lagerstrom Kaplun Matching technique [15] and the Mahony approach leads t o suitable results.
The Mahony approach to singular perturbation problems leads to uniformly valid solution representations and thus escapes certain difficulties associated with the Lagerstrom Kaplun Matching technique. Thus, if the Mahony approach can be employed in a particular case it will produce more satisfactory results, sometimes at the expense of more analytic apparatus, but sometimes with rather less labour. It will be shown that the method extends to cases which cannot be satisfactorily attacked using the matching technique. At times, however, (as was found in the nonlinear differential equation case and in isolated portions of the solution domain in the partial differential equation case) it becomes impracticable to cover the complete solution domain with a single Mahony type expansion. Thus, if the solution domain contains more than one singular point it does not seem practicable, in general, to incorporate new "boundary layer coordinates" in the solution expansion to cope with the situation close to all the singular points. It is easier to employ separate matched expansions about each of the singular points in these cases.
