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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.
Mathematics of Planet Earth (MPE) was started and continues to be
consolidated as a collaboration of mathematical science
organisations around the world. These organisations work together
to tackle global environmental, social and economic problems using
mathematics.This textbook introduces the fundamental topics of MPE
to advanced undergraduate and graduate students in mathematics,
physics and engineering while explaining their modern usages and
operational connections. In particular, it discusses the links
between partial differential equations, data assimilation,
dynamical systems, mathematical modelling and numerical simulations
and applies them to insightful examples.The text also complements
advanced courses in geophysical fluid dynamics (GFD) for
meteorology, atmospheric science and oceanography. It links the
fundamental scientific topics of GFD with their potential usage in
applications of climate change and weather variability. The
immediacy of examples provides an excellent introduction for
experienced researchers interested in learning the scope and
primary concepts of MPE.
Mathematics of Planet Earth (MPE) was started and continues to be
consolidated as a collaboration of mathematical science
organisations around the world. These organisations work together
to tackle global environmental, social and economic problems using
mathematics.This textbook introduces the fundamental topics of MPE
to advanced undergraduate and graduate students in mathematics,
physics and engineering while explaining their modern usages and
operational connections. In particular, it discusses the links
between partial differential equations, data assimilation,
dynamical systems, mathematical modelling and numerical simulations
and applies them to insightful examples.The text also complements
advanced courses in geophysical fluid dynamics (GFD) for
meteorology, atmospheric science and oceanography. It links the
fundamental scientific topics of GFD with their potential usage in
applications of climate change and weather variability. The
immediacy of examples provides an excellent introduction for
experienced researchers interested in learning the scope and
primary concepts of MPE.
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