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Numerical Integration of Space Fractional Partial Differential Equations

Vol 1 - Introduction to Algorithms and Computer Coding in R

Part of the book series: Synthesis Lectures on Mathematics & Statistics (SLMS)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xii
  2. Introduction to Fractional Partial Differential Equations

    • Younes Salehi, William E. Schiesser
    Pages 1-33
  3. Variation in the Order of the Fractional Derivatives

    • Younes Salehi, William E. Schiesser
    Pages 35-63
  4. Dirichlet, Neumann, Robin BCs

    • Younes Salehi, William E. Schiesser
    Pages 65-102
  5. Convection SFPDEs

    • Younes Salehi, William E. Schiesser
    Pages 103-127
  6. Nonlinear SFPDEs

    • Younes Salehi, William E. Schiesser
    Pages 129-164
  7. Back Matter

    Pages 165-188

About this book

Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as:

Vol 1: Introduction to Algorithms and Computer Coding in R

Vol 2: Applications from Classical Integer PDEs.

Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative.

The Caputo derivative is defined as a convolution integral. Thus, rather than being local (with a value at a particular point in space), the Caputo derivative is non-local (it is based on an integration in space), which is one of the reasons that it has properties not shared by integer derivatives.

A principal objective of the two volumes is to provide the reader with a set of documented R routines that are discussed in detail, and can be downloaded and executed without having to first study the details of the relevant numerical analysis and then code a set of routines.

In the first volume, the emphasis is on basic concepts of SFPDEs and the associated numerical algorithms. The presentation is not as formal mathematics, e.g., theorems and proofs. Rather, the presentation is by examples of SFPDEs, including a detailed discussion of the algorithms for computing numerical solutions to SFPDEs and a detailed explanation of the associated source code.

Authors and Affiliations

  • Razi University, Iran

    Younes Salehi

  • Lehigh University, USA

    William E. Schiesser

About the authors

My research focus is applied mathematics broadly. This includes numerical linear algebra, optimization and solving differential equations. My primary research interest concerns the areas of numerical analysis, scientific computing and high performance computing with particular emphasis on the numerical solution of ordinary differential equations (ODEs) and partial differential equations (PDEs). One focus of my work is programming efficient numerical methods for ODEs and PDEs. I have extensive experience in MATLAB, Maple, Mathematica and R programming of transportable numerical method routines, but I am also experienced in programming in C, C++ and C#, and could readily apply these programming systems to numerical ODE/PDEs. Recently, I have become interested in fractional differential equations (FDEs), especially the numerical solution of fractional initial value problems (FIVPs) and space fractional differential equations (SFPDEs).William E. Schiesser is Emeritus McCann Professor of Computational Biomedical Engineering and Chemical and Biomolecular Engineering, and Professor of Mathematics at Lehigh University. His research is directed toward numerical methods and associated software for ordinary, differential-algebraic and partial differential equations (ODE/DAE/PDEs). He is the author, coauthor or coeditor of 18 books, and his ODE/DAE/PDE computer routines have been accessed by some 5,000 colleges and universities, corporations and government agencies.

Bibliographic Information

Buying options

eBook USD 49.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 64.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions