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The explicit formula for solution of wave differential equation with fractional derivatives in the multi-dimensional space
  • Askar Rahmonov,
  • D. K. Durdiev,
  • Elina Shishkina
Askar Rahmonov
Bukhara State University

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D. K. Durdiev
Bukhara Branch of the Institute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan
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Elina Shishkina
Voronezh State University
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This paper devoted to the obtaining the explicit solution of $n$-dimensional wave equation with Gerasimov–Caputo fractional derivative 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 classical homogeneous hyperbolic integro-differential equation with memory in which the kernel is $t^{1-\alpha}E_{2-\alpha, 2-\alpha}(-t^{2-\alpha}), \ \alpha\in(1, 2),$ 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 the solution of the considered problem is obtained.