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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fractional derivative of Abel type on a half-line


Author: Elena I. Kaikina
Journal: Trans. Amer. Math. Soc. 364 (2012), 5149-5172
MSC (2010): Primary 35Q35
Published electronically: May 7, 2012
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Abstract: We consider the initial-boundary value problem on a half-line for an evolution equation

$\displaystyle \left ( \partial _{t}+\left \vert \partial _{x}\right \vert ^{\alpha }\right ) u(x,t)=f(x,t),t>0,\ x>0, $

with a fractional derivative of Abel type

$\displaystyle \text {(0.1)\qquad \qquad \qquad \qquad \qquad }\left \vert \part... ... _{x}^{\left [ \alpha \right ] +1}u, \text {\qquad \qquad \qquad \qquad \quad }$    

where $ \left [ \alpha \right ] $ denotes the integer part of number $ \alpha >0,\alpha $ is not equal to an integer, and

$\displaystyle \mathcal {R}^{\alpha }u=\frac {1}{2\Gamma (\alpha )\sin (\frac {\... ...y }\frac {\mathrm {sign}(x-y)u(y)}{\left \vert x-y\right \vert ^{1-\alpha }}dy $

is the modified Riesz potential. We study traditionally important problems of a theory of partial differential equations, such as existence and uniqueness of solution. We propose a new method of solution. Also we get a closed form of the solution.

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Additional Information

Elena I. Kaikina
Affiliation: Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico
Email: ekaikina@matmor.unam.mx

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05447-X
PII: S 0002-9947(2012)05447-X
Keywords: Initial-boundary value problem, Green function, fractional derivative
Received by editor(s): June 11, 2010
Received by editor(s) in revised form: August 20, 2010
Published electronically: May 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.