On the solution of fractional evolution equations

This paper is devoted to the solution of the bi-fractional differential equation ((C)D(t)(alpha)u)(t, x) = lambda((L)D(x)(beta)u) (t, x) (t > 0, -infinity < x < infinity) for real 0 < alpha less than or equal to 1, beta > 0 and lambda not equal 0, with the initial conditions lim(x-->+/-infinity) u(t, x) = 0 u(0+, x) = g(x). Here ((C)D(t)(alpha)u) (t, x) is the partial derivative coinciding with the Caputo fractional derivative for 0 < alpha < 1 and with the usual derivative for alpha = 1, while ((L)D(x)(beta)u) (t, x)) is the Liouville partial fractional derivative ((L)D(t)(beta)u)(t,x)) of order beta > 0. The Laplace and Fourier transforms are applied to solve the above problem in closed form. The fundamental solution of these problems is established and its moments are calculated. The special case alpha = 1/2 and P = 1 is presented, and its application is given to obtain the Dirac-type decomposition for the ordinary diffusion equation.