Fundamental solution of the time-space bi-fractional diffusion equation with a kinetic source term for anomalous transport

The purpose of this paper is to study the fundamental solution of the time-space bi-fractional diffusion equation incorporating an additional kinetic source term in semi-infinite space. The equation is a generalization of the integer-order model ∂ _t ρ (x,t) = ∂ ²_x ρ (x,t) - ρ (x,t) ∂ t ρ ( x , t ) = ∂ x 2 ρ ( x , t ) - ρ ( x , t ) (also known as the Debye–Falkenhagen equation) by replacing the first-order time derivative with the Caputo fractional derivative of order 0<α < 1 0 < α < 1 , and the second-order space derivative with the Riesz-Feller fractional derivative of order 0< β <2 0 < β < 2 . Using the Laplace-Fourier transforms method, it is shown that the parametric solutions are expressed in terms of the Fox’s H -function that we evaluate for different values of α α and β β .