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The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space. (English) Zbl 1468.35231

Summary: This paper intends on obtaining the explicit solution of \(n\)-dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is \(t^{-\alpha}E_{1-\alpha,1-\alpha}(-t^{1-\alpha})\), \(\alpha\in(0,1)\), where \(E_{\alpha,\beta}\) is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained.

MSC:

35R11 Fractional partial differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
35R09 Integro-partial differential equations
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