The fractional Laplacian

This book is dedicated to the fractional Laplacian, which is a special instance of the fractional second derivative of a function of one variable but lacks a counterpart in higher dimensions. Due to its isotropic and non-local nature, the fractional Laplacian is of interest in science, engineering, and other application areas where long-range interactions and conceptual or physical particle jumps resulting in an irregular diffusive or conductive flux are encountered.

In the book,

• The concept of the fractional Laplacian is clarified for functions of one, two, three, or an arbitrary number of variables defined over the entire space, satisfying periodicity conditions, or restricted to a finite domain.
• The level of the discourse is suitable for a broad audience of scientists and engineers with rudimentary background in ordinary differential and integral calculus.
• Physical and mathematical concepts are presented together, accompanied by detailed mathematical derivations.
• A numerical framework for solving differential equations involving the fractional Laplacian is developed, and specific algorithms accompanied by numerical results are presented in one, two, and three dimensions.
• Topics in viscous flow and physical examples from science and engineering disciplines are discussed.