Abstract
The present paper deals with the Cauchy problem for the multi-term time-space fractional diffusion equation in one dimensional space. The time fractional derivatives are defined as Caputo fractional derivatives and the space fractional derivative is defined in the Riesz sense. Firstly the domain of the fractional Laplacian is extended to a Banach space. Then the analytical solutions are established by using the Luchko theorem and the multivariate Mittag-Leffler function.
Highlights
1 Introduction The fractional calculus has already become a powerful tool which describes many nonlinear complex phenomena arising in fluid mechanics, thermodynamics, plasma dynamics, continuum mechanics, quantum mechanics, electrodynamics and biological systems [, ]
Because of the convenience in handling initial conditions, the Caputo fractional derivative has been more widely used in practice [ ]
In [ ] the authors gave a new definition of the Caputo fractional derivative on a bounded interval in the fractional Sobolev space and proved the maximal regularity of solutions of time fractional diffusion equations
Summary
The fractional calculus has already become a powerful tool which describes many nonlinear complex phenomena arising in fluid mechanics, thermodynamics, plasma dynamics, continuum mechanics, quantum mechanics, electrodynamics and biological systems [ , ]. In [ ] the authors gave a new definition of the Caputo fractional derivative on a bounded interval in the fractional Sobolev space and proved the maximal regularity of solutions of time fractional diffusion equations. In [ ] the authors proved the maximum principle for multi-term time-space Caputo-Riesz fractional diffusion equations and derived the uniqueness and continuous dependence of the solution. The authors of [ ] used the Luchko theorem to obtain the analytical solutions for multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a bounded interval. By extending the domain of the fractional Laplacian to a Banach space and using the multivariate Mittag-Leffler function, the analytical solutions of the multi-term fractional diffusion equation The Caputo fractional derivative Dαu, ≤ α ≤ r, is well defined and the inclusion
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