# Boundary value problem for the time-fractional wave equation

## DOI:

https://doi.org/10.31489/2024m2/124-134## Keywords:

fractional derivative, Laplace transform, Fourier transform, Mittag-Leffler function, Wright function## Abstract

In the article, the boundary value problem for the wave equation with a fractional time derivative and with initial conditions specified in the form of a fractional derivative in the Riemann-Liouville sense is solved. The definition domain of the desired function is the upper half-plane (x,t). To solve the problem, the Fourier transform with respect to the spatial variable was applied, then the Laplace transform with respect to the time variable was used. After applying the inverse Laplace transform, the solution to the transformed problem contains a two-parameter Mittag-Leffler function. Using the inverse Fourier transform, a solution to the problem was obtained in explicit form, which contains the Wright function. Next, we consider limiting cases of the fractional derivative’s order which is included in the equation of the problem.