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Time- and space-fractional partial differential equations. (English) Zbl 1076.26006
Summary: The fundamental solution for time- and space-fractional partial differential operator D t λ +a 2 (-Δ) 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) D t λ is defined by the convolution D t λ f(t)=-Φ λ (t) * f(t), where Φ λ (t)=t + λ-1 /Γ(λ) and f(t)0 as t<0, and the fractional n-dimensional Laplace operator (-Δ) γ/2 is defined by its Fourier transform with respect to spatial variable [(-Δ) γ/2 g(x)]=|ω| γ [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.
MSC:
26A33Fractional derivatives and integrals (real functions)
35S10Initial value problems for pseudodifferential operators
44A30Multiple transforms
45K05Integro-partial differential equations