Format:
1 Online-Ressource (xii, 189 Seiten)
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Illustrationen
ISBN:
1681732084
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9781681732084
Series Statement:
Synthesis lectures on mathematics and statistics #19
Content:
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
Content:
1. Introduction to fractional partial differential equations -- 1.1 Introduction -- 1.2 Computer routines, example 1 -- 1.2.1 Main program -- 1.2.2 Subordinate ODE/MOL routine -- 1.2.3 Model output -- 1.3 Computer routines, example 2 -- 1.3.1 Main program -- 1.3.2 Subordinate ODE/MOL routine -- 1.3.3 Model output -- 1.3.4 Summary and conclusions -- References --
Content:
2. Variation in the order of the fractional derivatives -- 2.1 Introduction -- 2.2 Computer routines, example 1 -- 2.2.1 Main program -- 2.2.2 Subordinate ODE/MOL routine -- 2.2.3 Model output -- 2.3 Computer routines, example 2 -- 2.3.1 Main program -- 2.3.2 Subordinate ODE/MOL routine -- 2.3.3 Model output -- 2.4 Summary and discussion --
Content:
3. Dirichlet, Neumann, Robin BCs -- 3.1 Introduction -- 3.2 Example 1, Dirichlet BCs -- 3.2.1 Main program -- 3.2.2 Subordinate ODE/MOL routine -- 3.2.3 Model output -- 3.3 Example 2, Dirichlet BCs -- 3.3.1 Main program -- 3.3.2 Subordinate ODE/MOL routine -- 3.3.3 Model output -- 3.4 Example 2, Neumann BCs -- 3.4.1 Main program -- 3.4.2 Subordinate ODE/MOL routine -- 3.4.3 Model output -- 3.5 Example 2, Robin BCs -- 3.5.1 Main program -- 3.5.2 Subordinate ODE/MOL routine -- 3.5.3 Model output -- 3.6 Summary and conclusions --
Content:
4. Convection SFPDEs -- 4.1 Introduction -- 4.2 Integer/fractional convection model -- 4.2.1 Main program -- 4.2.2 Subordinate ODE/MOL routine -- 4.2.3 SFPDE output -- 4.3 Summary and conclusions --
Content:
5. Nonlinear SFPDEs -- 5.1 Introduction -- 5.1.1 Example 1 -- 5.1.2 Main program -- 5.1.3 Subordinate ODE/MOL routine -- 5.1.4 Model output -- 5.2 Example 2 -- 5.2.1 Main program -- 5.2.2 Subordinate ODE/MOL routine -- 5.2.3 Model output -- 5.3 Summary and conclusions --
Content:
A. Analytical Caputo differentiation of selected functions -- B. Derivation of a SFPDE analytical solution -- Introduction -- SFPDE equations -- Main program -- ODE/MOL routine -- Numerical output -- Summary and conclusions -- Authors' Biographies -- Index
Note:
Includes bibliographical references and index
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Part of: Synthesis digital library of engineering and computer science
Additional Edition:
ISBN 9781681732077
Additional Edition:
Erscheint auch als Druck-Ausgabe Numerical integration of space fractional partial differental equations ; vol 1: Introduction to algorithms and comuter coding in R San Rafael, California : Morgan & Claypool Publishers, 2018 ISBN 9781681732077
Language:
English
Keywords:
Electronic books
DOI:
10.1007/978-3-031-02411-5
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