Numerical Methods for Fractional Calculus presents numerical methods for fractional integrals and fractional derivatives, finite difference methods for fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs), and finite element methods for FPDEs.
The book introduces the basic definitions and properties of fractional integrals and derivatives before covering numerical methods for fractional integrals and derivatives. It then discusses finite difference methods for both FODEs and FPDEs, including the Euler and linear multistep methods. The final chapter shows how to solve FPDEs by using the finite element method.
This book provides efficient and reliable numerical methods for solving fractional calculus problems. It offers a primer for readers to further develop cutting-edge research in numerical fractional calculus. MATLAB® functions are available on the book’s CRC Press web page.
Table of Contents
Introduction to Fractional Calculus
Fractional Integrals and Derivatives
Some Other Properties of Fractional Derivatives
Some Other Fractional Derivatives and Extensions
Fractional Initial and Boundary Problems
Numerical Methods for Fractional Integral and Derivatives
Approximations to Fractional Integrals
Approximations to Riemann–Liouville Derivatives
Approximations to Caputo Derivatives
Approximation to Riesz Derivatives
Short Memory Principle
Numerical Methods for Fractional Ordinary Differential Equations
Fractional Linear Multistep Methods
Finite Difference Methods for Fractional Partial Differential Equations
One-Dimensional Time-Fractional Equations
One-Dimensional Space-Fractional Differential Equations
One-Dimensional Time-Space Fractional Differential Equations
Fractional Differential Equations in Two Space Dimensions
Galerkin Finite Element Methods for Fractional Partial Differential Equations
Galerkin FEM for Stationary Fractional Advection Dispersion Equation
Galerkin FEM for Space-Fractional Diffusion Equation
Galerkin FEM for Time-Fractional Differential Equations
Galerkin FEM for Time-Space Fractional Differential Equations
Changpin Li is a full professor at Shanghai University. He earned his Ph.D. in computational mathematics from Shanghai University. Dr. Li’s main research interests include numerical methods and computations for FPDEs and fractional dynamics. He was awarded the Riemann–Liouville Award for Best FDA Paper (theory) in 2012. He is on the editorial board of several journals, including Fractional Calculus and Applied Analysis, International Journal of Bifurcation and Chaos, and International Journal of Computer Mathematics.
Fanhai Zeng is visiting Brown University as a postdoc fellow. He earned his Ph.D. in computational mathematics from Shanghai University. Dr. Zeng’s research interests include numerical methods and computations for FPDEs.
"The book provides a survey of many different methods for the numerical computation of Riemann–Liouville integrals of fractional order and of fractional derivatives of Riemann–Liouville, Caputo, and Weyl type. Algorithms for the solution of associated ordinary differential equations and certain special classes of partial differential equations are presented as well."
—Zentralblatt MATH 1326