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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 \(D_t^\lambda+a^2(-\Delta)^{\gamma/2}\) (\(\lambda\), \(\gamma > 0\)) is given in terms of the Fox’s \(H\)-function. Here the time-fractional derivative in the sense of generalized functions (distributions) \(D_t^\lambda\) is defined by the convolution \(D_t^\lambda f(t)=-\Phi_\lambda (t)^*f(t)\), where \(\Phi_\lambda(t)=t_+^{\lambda-1}/\Gamma(\lambda)\) and \(f(t) \equiv 0\) as \(t<0\), and the fractional \(n\)-dimensional Laplace operator \((-\Delta)^{\gamma/2}\) is defined by its Fourier transform with respect to spatial variable \(\mathcal {F}[(-\Delta)^{\gamma/2} g(x)] = |\omega|^{\gamma}\mathcal {F}[g(x)]\). 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:

26A33 Fractional derivatives and integrals
35S10 Initial value problems for PDEs with pseudodifferential operators
44A30 Multiple integral transforms
45K05 Integro-partial differential equations
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