## 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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### References:

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