Fractional and Stochastic Differential Equations in Mathematics

Dear Colleagues,

The origin of fractional calculus goes back to the 17th century, in connection with the names of Leibniz and L'Hospital, and was then developed by Riemann, Liouville and many others. The interest of the mathematical community in fractional calculus has grown considerably since then and a variety of applications to physical problems have flourished.

Few theoretical issues are still to be addressed within the field of nonlocal operators and in particular of fractional calculus. Fractional derivatives of distributed or variable order are now of special interest. Developing methods to approximate solutions is mandatory to obtain quantitative information for applications. Few numerical methods exist to handle operators of fractional order, most of them being unsatisfactory due to instability and exceeding computational cost. From a probabilistic point of view, generalizations of well-known stochastic processes naturally emerge from the study of fractional differential equations. The memory effect introduced by fractional derivatives is related to semi-Markov processes. When potential sources of uncertainty must be taken into account, it is desirable to produce probability distributions for describing physical phenomena. Stochastic PDEs are able to provide mathematical representations of complex systems when noise or random effects are involved, with applications to oceanography, finance, physics, biology, meteorology, environmental sciences, and others. In the recent literature, fractional–stochastic partial differential equations have been proposed, thus combining the flexibility of fractional calculus with the applicability of stochastic PDEs to model uncertainty.

Dr. Clemente Cesarano
Guest Editor

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