zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. (English) Zbl 1181.35322
Summary: We consider a one-dimensional fractional diffusion equation: $\partial_t^\alpha u(x,t)= \frac{\partial}{\partial x}(p(x)\frac{\partial u}{\partial x}(x,t))$, $0<x<\ell$, where $0<\alpha<1$ and $\partial_t^{\alpha} $ denotes the Caputo derivative in time of order $\alpha $. We attach the homogeneous Neumann boundary condition at $x=0$, $\ell$ and the initial value given by the Dirac delta function. We prove that $\alpha $ and $p(x)$, $0<x<\ell$, are uniquely determined by data $u(0,t)$, $0<t<T$. The uniqueness result is a theoretical background in experimentally determining the order $\alpha $ of many anomalous diffusion phenomena which are important, for example, in environmental engineering. The proof is based on the eigenfunction expansion of the weak solution to the initial value/boundary value problem and the Gel’fand-Levitan theory.

35R30Inverse problems for PDE
35R11Fractional partial differential equations
35D30Weak solutions of PDE
33E12Mittag-Leffler functions and generalizations
Full Text: DOI