<div class="csl-bib-body">
<div class="csl-entry">Achleitner, F., Akagi, G., Kühn, C., Melenk, J. M., Rademacher, J., Soresina, C., & Yang, J. (2024). Fractional Dissipative PDEs. In P. Kevrekidis & J. Cuevas-Maraver (Eds.), <i>Fractional Dispersive Models and Applications: Recent Developments and Future Perspectives</i> (Vol. 37, pp. 53–122). Springer. https://doi.org/10.1007/978-3-031-54978-6_3</div>
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/196824
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dc.description.abstract
In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also provide an outlook on closely related classes of PDEs. To simplify the exposition, we only discuss the cases of fractional time derivatives and fractional space derivatives in the PDE separately. As our main tools, we describe analytical as well as numerical methods, which are generically necessary to study nonlinear dynamics. We start with the analytical study of steady states and local linear stability for fractional time derivatives. Then we extend this view to a global perspective and consider time-fractional PDEs and gradient flows. Next, we continue to steady states, linear stability analysis and bifurcations for space-fractional PDEs. As a final analytical consideration we discuss existence and stability of traveling waves for space-fractional PDEs. In the last parts, we provide numerical discretization schemes for fractional (dissipative) PDEs and we utilize these techniques within numerical continuation in applied examples of fractional reaction-diffusion PDEs. We conclude with a brief summary and outlook on open questions in the field.
en
dc.language.iso
en
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dc.subject
fractional PDEs
en
dc.subject
fractional diffusion
en
dc.subject
stability of nonlinear systems
en
dc.title
Fractional Dissipative PDEs
en
dc.type
Book Contribution
en
dc.type
Buchbeitrag
de
dc.contributor.affiliation
Tohoku University, Japan
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dc.contributor.affiliation
Technical University of Munich, Germany
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dc.contributor.affiliation
Universität Hamburg, Germany
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dc.contributor.affiliation
University of Trento, Italy
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dc.contributor.affiliation
Harbin Engineering University, China
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dc.relation.isbn
978-3-031-54977-9
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dc.relation.doi
10.1007/978-3-031-54978-6
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dc.relation.issn
2195-9994
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dc.description.startpage
53
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dc.description.endpage
122
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dc.type.category
Edited Volume Contribution
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dc.relation.eissn
2196-0003
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tuw.booktitle
Fractional Dispersive Models and Applications: Recent Developments and Future Perspectives