Exact solutions of a modified fractional diffusion equation in the finite and semi-infinite domains

• We investigate the solutions for the modified fractional diffusion equation.
• Exact semi-infinite and finite solutions subject to absorbing boundaries are found.
• The crossover between more and less anomalous behaviour is demonstrated.

We investigate the solutions of a modified fractional diffusion equation which has a secondary fractional time derivative acting on a diffusion operator. We obtain analytical solutions for the modified equation in the finite and semi-infinite domains subject to absorbing boundary conditions. Most of the results have been derived by using the Laplace transform, the Fourier Cosine transform, the Mellin transform and the properties of Fox HH function. We show that the semi-infinite solution can be expressed using an infinite series of Fox HH functions similar to the infinite case, while the finite solution requires double infinite series including both Fox HH functions and trigonometric functions instead of one infinite series. The characteristic crossover between more and less anomalous behaviour as well as the effect of absorbing boundary conditions are clearly demonstrated according to the analytical solutions.