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This book describes state-of-the-art advances and applications of
the unified transform and its relation to the boundary element
method. The authors present the solution of boundary value problems
from several different perspectives, in particular the type of
problems modeled by partial differential equations (PDEs). They
discuss recent applications of the unified transform to the
analysis and numerical modeling of boundary value problems for
linear and integrable nonlinear PDEs and the closely related
boundary element method, a well-established numerical approach for
solving linear elliptic PDEs. The text is divided into three parts.
Part I contains new theoretical results on linear and nonlinear
evolutionary and elliptic problems. New explicit solution
representations for several classes of boundary value problems are
constructed and rigorously analyzed. Part II is a detailed overview
of variational formulations for elliptic problems. It places the
unified transform approach in a classic context alongside the
boundary element method and stresses its novelty. Part III presents
recent numerical applications based on the boundary element method
and on the unified transform.
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