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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 $\cal {F}[(-\Delta)^{\gamma/2} g(x)] = |\omega|^{\gamma}\cal {F}[g(x)]$. 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.

26A33Fractional derivatives and integrals (real functions)
35S10Initial value problems for pseudodifferential operators
44A30Multiple transforms
45K05Integro-partial differential equations
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