Symmetry Group Classification and Conservation Laws of the Nonlinear Fractional Diffusion Equation with the Riesz Potential Symmetry properties of a nonlinear twodimensional spacefractional diffusion equation with the Riesz potential of the order     xcexb1 xe2x88x88 ( 0 , 1 )     are studied. Lie point symmetry group classification of this equation is performed with respect to diffusivity function. To construct conservation laws for the considered equation, the concept of nonlinear selfadjointness is adopted to a certain class of spacefractional differential equations with the Riesz potential. It is proved that the equation in question is nonlinearly selfadjoint. An extension of Ibragimovxe2x80x99s constructive algorithm for finding conservation laws is proposed, and the corresponding Noether operators for fractional differential equations with the Riesz potential are presented in an explicit form. To illustrate the proposed approach, conservation laws for the considered nonlinear spacefractional diffusion equation are constructed by using its Lie point symmetries.

The solution theory of the nonlinear qfractional differential equations Abstract   In this paper, we study the solution theory of the nonlinear    q   fractional differential equation:         c      D    q    xcexb1    x   (  t  )     f   (  t  ,  x   (  t  )   )   ,    0    xcexb1  ,    q    1    , with given initial value. We first give the existence theorem under the assumption that       t    xcexb2    f   (  t  ,  x  )      is continuous where     0  xe2x89xa4  xcexb2    xcexb1    . Then, by establishing a    q   analogue Gronwall inequality, we prove that the solution     x   (  t  )      is stable with respect to the initial value if     f   (  t  ,  x  )      satisfies the Lipschitz condition on variable    x   . This stability result also implies the uniqueness of solution.