Space-time fractional Cauchy problem in spaces of generalized functions

作者：Lopushans'ka H P^*; Lopushans'kyi A O

来源：Ukrainian Mathematical Journal, 2013, 64(8): 1215-1230.

DOI：10.1007/s11253-013-0711-z

We prove the theorem on existence and uniqueness of solution to the Cauchy problem
u(t)((beta)) + a(2)(-Delta)(alpha/2) u = F(x,t), (x, t) is an element of R-n x (0, T], a = const,
u(x, 0) = u(0) (x), x is an element of R-n,
where u(t)((beta)) is the Riemann-Liouville fractional derivative of order beta is an element of (0, 1) and u(0) and F belong to spaces of generalized functions. A representation of this solution is obtained by using the vector Green function. We also establish the character of singularities of the solution for t = 0 depending on the order of singularity of a given generalized function in the initial condition and the character of power singularities of the function on the right-hand side of the equation. In this case, the fractional n-dimensional Laplace operator is defined by using the Fourier transformation F[-Delta(alpha/2) psi(x)] = vertical bar lambda vertical bar F-alpha[psi(x)].

• 出版日期2013-1