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

• Published in: Physica A Vol. 417; pp. 193 - 201
• Main Authors: Guo, Gang, Li, Kun, Wang, Yuhui
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2015
• Subjects: Anomalous diffusion; Finite domain; Fox [formula omitted] function; Modified fractional diffusion equation; Semi-infinite domain; Semi-infinite domain; Fox H function; Finite domain; Modified fractional diffusion equation; Anomalous diffusion
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2014.09.050

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 H function. We show that the semi-infinite solution can be expressed using an infinite series of Fox H functions similar to the infinite case, while the finite solution requires double infinite series including both Fox H 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. •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.