Time- and space-fractional partial differential equations

The fundamental solution for time- and space-fractional partial differential operator Dtλ+a2(−▵)γ∕2(λ,γ>0) is given in terms of the Fox’s H-function. Here the time-fractional derivative in the sense of generalized functions (distributions) Dtλ is defined by the convolution Dtλf(t)=Φ−λ(t)*f(t), where Φλ(t)=t+λ−1∕Γ(λ) and f(t)≡0 as t<0, and the fractional n-dimensional Laplace operator (−▵)γ∕2 is defined by its Fourier transform with respect to spatial variable F[(−▵)γ∕2g(x)]=∣ω∣γF[g(x)]. The solutions for initial value problems for time- and space-fractional partial differential equation in the sense of Caputo and Riemann–Liouville time-fractional derivatives, respectively, are obtained by the fundamental solution.