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The fundamental solution of the space-time fractional diffusion equation. (English) Zbl 1054.35156

The authors study the Cauchy problem for the space-time fractional diffusion equation

x D θ α u(x,t)= t D * β u(x,t),x,t + ,(1)
u(x,0)=φ(x),x,u(±,t)=0,t>0,(2)

where φL c () is a sufficiently well-behaved function, x D θ α is the Riesz-Feller space-fractional derivative of the order α and the skewness θ, and t D * β is the Caputo time-fractional derivative of the order β. If 1<β2, then the condition (2) is supplemented by the additional condition

u t (x,0)=0·(3)

An analogon of the fundamental solution G α,β θ to the problem (1)–(2) (or (1)–(3)) is itroduced and determined via Fourier-Laplace transform:

G α,β θ ˜ ^(κ,s)=s β-1 s β +ψ α θ (κ),(4)

where

ψ α θ (κ)=|κ| α e i(signκ)θπ/2 ·

A scaling property as well as the similarity relation are obtained for G α,β θ . It is found also the connection of the fundamental solution to the Mittag-Leffler function and to Mellin-Barnes integrals. Some particular cases are considered, namely space-fractional diffusion (0<α2,β=1), time-fractional diffusion (α=2, 0<β2) and neutral diffusion (0<α=β2). A composition rule for G α,β θ is established in the case 0<β1 which ensures its probabilistic interpretation at its range. A general representation of the Green function in terms of Mellin-Barnes integrals is obtained. On its base explicit formulas for G α,β θ as well as asymptotics of the Green function for different values of the parameters are found. Qualitative remarks concerning the solvability of the space-fractional diffusion equation are made illustrated by plots describing the behaviour of the Green function and the fundamental solution to (1).

MSC:
35S10Initial value problems for pseudodifferential operators
26A33Fractional derivatives and integrals (real functions)
33E12Mittag-Leffler functions and generalizations
44A10Laplace transform
35K05Heat equation