In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.
- Riemann-Liouville fractional derivative
- extremum principle for the Riemann-Liouville fractional derivative
- linear and non-linear time-fractional diffusion equations
- maximum principle
- uniqueness and existence of solutions
ASJC Scopus subject areas
- Applied Mathematics