Nonlocal models are models that generalize partial differential equations (PDEs). In PDE-based modeling, for a point, the values can be verified without using any values of other points. In nonlocal modeling, local values and nonlocal values become part of the equations. This book introduces nonlocal modeling, provides mathematical details, and gives methods of numerical approximations and coupling with PDE models.

Readers not interested in mathematical details can read the last chapter. This epilogue summarizes the book. Readers interested in learning nonlocal models through examples can read the first chapter. Many problems in physics and engineering use PDE-based models. The concepts of continuum, discretization, and boundaries in such problems are covered by examples of nonlocal models. One of the many examples given in this chapter is peridynamics. The nonlocal formulation of peridynamics models is described with details.

PDE is a foundational subject. When computational aspects are considered, vector calculus is a foundation of PDEs. The third chapter presents vector calculus for nonlocal models as a foundation of nonlocal models. It gives two comparison tables that compare PDEs with nonlocal models. The comparison parameters include vector calculus, operators, flux, Green’s identities, variations, underlying spaces, weak forms, and so on. Each parameter is discussed in detail and explained with examples. The fourth chapter visits some of the methods of numerical approximations of the nonlocal models. First, it gives a survey of these methods. It describes in detail the finite element methods for nonlocals. The discussion mainly covers how the special challenges of nonlocal models are dealt with in various numerical schemes.

The next two chapters are on specialized topics. Chapter 5 is a heterogeneous way of analysis, where both PDEs and nonlocals are used. The nonlocal models can offer better singularities than PDEs in some cases. But numerical approximations can be worse. The coupling of nonlocal models and PDEs can improve the traces. Chapter 6 is on time dynamics. The time always goes in a positive direction. The chapter discusses the effect of this time property for various equations of nonlocal models.

The book is projected as a monograph. Depending on the level of the syllabus, it can be used as a textbook in graduate or undergraduate studies. For readers new to nonlocal modeling, it is a good first book. There is a long list of references at the end. The book assumes a background in PDEs.