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 ${\partial }_t\rho (x,t)={\partial }_x^2\rho (x,t)-\rho (x,t)$ (also known as the Debye-Falkenhagen equation) by replacing the first-order time derivative with the Caputo fractional derivative of order $0<\alpha <1$ , and the second-order space derivative with the Riesz-Feller fractional derivative of order $0<\beta <2$ . 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 $\alpha$ and $\beta$ .