Time-Fractional Differential Equations
A Theoretical Introduction
Series: SpringerBriefs in Mathematics;
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Product details:
- Edition number 1st ed. 2020
- Publisher Springer Nature Singapore
- Date of Publication 30 November 2020
- Number of Volumes 1 pieces, Book
- ISBN 9789811590658
- Binding Paperback
- No. of pages134 pages
- Size 235x155 mm
- Weight 454 g
- Language English
- Illustrations X, 134 p. 4 illus. Illustrations, black & white 116
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Long description:
"
This book aims to establish a foundation for fractional derivatives and fractional differential equations. The theory of fractional derivatives enables considering any positive order of differentiation. The history of research in this field is very long, with its origins dating back to Leibniz. Since then, many great mathematicians, such as Abel, have made contributions that cover not only theoretical aspects but also physical applications of fractional calculus.
The fractional partial differential equations govern phenomena depending both on spatial and time variables and require more subtle treatments. Moreover, fractional partial differential equations are highly demanded model equations for solving real-world problems such as the anomalous diffusion in heterogeneous media.
The studies of fractional partial differential equations have continued to expand explosively. However we observe that available mathematical theory for fractional partial differential equations is not still complete. In particular, operator-theoretical approaches are indispensable for some generalized categories of solutions such as weak solutions, but feasible operator-theoretic foundations for wide applications are not available in monographs.
To make this monograph more readable, we are restricting it to a few fundamental types of time-fractional partial differential equations, forgoing many other important and exciting topics such as stability for nonlinear problems. However, we believe that this book works well as an introduction to mathematical research in such vast fields.
Table of Contents:
Chapter 1: Basics on fractional differentiation and integration.- Chapter 2: Definition of fractional derivatives in Sobolev spaces and properties.- Chapter 3: Fractional ordinary differential equations.- Chapter 4: Initial boundary value problems for time-fractional diffusion equations.- Chapter 5: Decay rate as t →∞.- Chapter 6: Concluding remarks on future works.
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