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xii Foreword

Elsewhere, as with the identification and emergence of the study of inverse problems,
new analytical approaches have stimulated the development of numerical techniques
for the solution of this major class of practical problems. Such work divides naturally
into two parts, the first being the identification and formulation of inverse problems,
the theory of ill-posed problems and the class of one-dimensional inverse problems,
and the second being the study and theory of multidimensional inverse problems.
On occasions the development of analytical results and their implementation by
computer have proceeded in parallel, as with the development of the fast boundary
element methods necessary for the numerical solution of partial differential equations
in several dimensions. This work has been stimulated by the study of boundary inte-
gral equations, which in turn has involved the study of boundary elements, collocation
methods, Galerkin methods, iterative methods and others, and then on to their im-
plementation in the case of the Helmholtz equation, the Lamé equations, the Stokes
equations, and various other equations of physical significance.
A major development in the theory of partial differential equations has been the
use of group theoretic methods when seeking solutions, and in the introduction of
the comparatively new method of differential constraints. In addition to the useful
contributions made by such studies to the understanding of the properties of solu-
tions, and to the identification and construction of new analytical solutions for well
established equations, the approach has also been of value when seeking numerical
solutions. This is mainly because of the way in many special cases, as with similarity
solutions, a group theoretic approach can enable the number of dimensions occurring
in a physical problem to be reduced, thereby resulting in a significant simplification
when seeking a numerical solution in several dimensions. Special analytical solutions
found in this way are also of value when testing the accuracy and efficiency of new
numerical schemes.
A different area in which significant analytical advances have been achieved is in
the field of stochastic differential equations. These equations are finding an increasing
number of applications in physical problems involving random phenomena, and oth-
ers that are only now beginning to emerge, as is happening with the current use of
stochastic models in the financial world. The methods used in the study of stochastic
differential equations differ somewhat from those employed in the applications men-
tioned so far, since they depend for their success on the Ito calculus, martingale theory
and the Doob-Meyer decomposition theorem, the details of which are developed as
necessary in the volume on stochastic differential equations.
There are, of course, other topics in addition to those mentioned above that are of
considerable practical importance, and which have experienced significant develop-
ments in recent years, but accounts of these must wait until later.

Alan Jeffrey
University of Newcastle
Newcastle upon Tyne
United Kingdom