Foundation of the fractional calculus in generalized function algebras

Abstract

We introduce an approach to fractional derivatives involving singularities based on the theory of algebras of generalized functions in the sense of Colombeau. We are interested in solving fractional nonlinear ODEs and PDEs with singularities with a lack of solutions in the space of classical functions or distributions. For these purposes, we embed different forms of fractional derivatives into space of Colombeau special algebra of generalized functions using appropriate techniques such as the regularization with delta sequences and multiplication with different cut-off functions. Finally, we present an example for application of the ideas presented in paper to confirm the reason of introducing fractional derivatives into Colombeau algebra of generalized functions. © 2012 World Scientific Publishing Company.