Fractional derivative of Abel type on a half-line
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- by Elena I. Kaikina PDF
- Trans. Amer. Math. Soc. 364 (2012), 5149-5172 Request permission
Abstract:
We consider the initial-boundary value problem on a half-line for an evolution equation \[ \left ( \partial _{t}+\left \vert \partial _{x}\right \vert ^{\alpha }\right ) u(x,t)=f(x,t),t>0,\ x>0, \] with a fractional derivative of Abel type \begin{equation*} \text {(0.1)\qquad \qquad \qquad \qquad \qquad }\left \vert \partial _{x}\right \vert ^{\alpha }u=\mathcal {R}^{1-\alpha +\left [ \alpha \right ] }\partial _{x}^{\left [ \alpha \right ] +1}u, \text {\qquad \qquad \qquad \qquad \quad }\end{equation*} where $\left [ \alpha \right ]$ denotes the integer part of number $\alpha >0,\alpha$ is not equal to an integer, and\[ \mathcal {R}^{\alpha }u=\frac {1}{2\Gamma (\alpha )\sin (\frac {\pi }{2}\alpha )}\int _{0}^{+\infty }\frac {\mathrm {sign}(x-y)u(y)}{\left \vert x-y\right \vert ^{1-\alpha }}dy \] 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.References
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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
- Received by editor(s): June 11, 2010
- Received by editor(s) in revised form: August 20, 2010
- Published electronically: May 7, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5149-5172
- MSC (2010): Primary 35Q35
- DOI: https://doi.org/10.1090/S0002-9947-2012-05447-X
- MathSciNet review: 2931325