This book aims to establish a foundation for fractional derivatives and
fractional differential equations. The theory of fractional derivatives
enables considering any positive order of differentiation. The history
of research in this field is very long, with its origins dating back to
Leibniz. Since then, many great mathematicians, such as Abel, have made
contributions that cover not only theoretical aspects but also physical
applications of fractional calculus.
The fractional partial differential equations govern phenomena depending
both on spatial and time variables and require more subtle treatments.
Moreover, fractional partial differential equations are highly demanded
model equations for solving real-world problems such as the anomalous
diffusion in heterogeneous media.
The studies of fractional partial differential equations have continued
to expand explosively. However we observe that available mathematical
theory for fractional partial differential equations is not still
complete. In particular, operator-theoretical approaches are
indispensable for some generalized categories of solutions such as weak
solutions, but feasible operator-theoretic foundations for wide
applications are not available in monographs.
To make this monograph more readable, we are restricting it to a few
fundamental types of time-fractional partial differential equations,
forgoing many other important and exciting topics such as stability for
nonlinear problems. However, we believe that this book works well as an
introduction to mathematical research in such vast fields.
