Electron. J. Diff. Equ., Vol. 2013 (2013), No. 106, pp. 1-15.

Existence of solutions to boundary value problems arising from the fractional advection dispersion equation

Lingju Kong

Abstract:
We study the existence of multiple solutions to the boundary value problem
$$\displaylines{ \frac{d}{dt}\Big(\frac12{}_0D_t^{-\beta}(u'(t))+\frac12{}_tD_T^{-\beta}(u'(t)) \Big)+\lambda \nabla F(t,u(t))=0,\quad t\in [0,T],\cr u(0)=u(T)=0, }$$
where $T>0$, $\lambda>0$ is a parameter, $0\leq\beta<1$, ${}_0D_t^{-\beta}$ and ${}_tD_T^{-\beta}$ are, respectively, the left and right Riemann-Liouville fractional integrals of order $\beta$, $F: [0,T]\times\mathbb{R}^N\to\mathbb{R}$ is a given function. Our interest in the above system arises from studying the steady fractional advection dispersion equation. By applying variational methods, we obtain sufficient conditions under which the above equation has at least three solutions. Our results are new even for the special case when $\beta=0$. Examples are provided to illustrate the applicability of our results.

Submitted December 17, 2012. Published April 24, 2013.
Math Subject Classifications: 34B15, 34A08.
Key Words: Three solutions; fractional boundary value problem; variational methods.

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Lingju Kong
Department of Mathematics
the University of Tennessee at Chattanooga
Chattanooga, TN 37403, USA
email: Lingju-Kong@utc.edu

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