Time–space fractional Euler–Poisson–Darboux equation with Bessel fractional derivative in infinite and finite domains

Abstract

In this paper, we study the time-fractional Euler–Poisson–Darboux equation with the Bessel fractional derivative. The Laplacian operator of this equation is considered in the ordinary and fractional derivatives and also in different coordinates. For the multi-dimensional Euler–Poisson–Darboux equation in the infinite domain (the whole space), we use the joint modified Meijer–Fourier transforms and establish a complex inversion formula for deriving the fundamental solution. The fractional moment of this solution is also presented in different dimensions. For studying the time-fractional Euler–Poisson–Darboux equation by the numerical methods in finite domain, we sketch the semi- and fully-discrete methods along with the matrix transfer technique to analyze the equation with fractional Laplacian operators in the cartesian, polar and spherical coordinates. The associated error and convergence theorems are also discussed. The illustrative examples are finally presented to verify our results in different coordinates.