Generalized solutions of the Cauchy problem involving $\Phi$-Caputo fractional derivatives

Abstract

The main objective of this research paper is to embed $\Phi$-Caputo fractional derivative in the Colombeau algebra of generalized functions and we investigate the existence and uniqueness of the Cauchy problem involving $\Phi$-Caputo fractional derivatives in the extended Colombeau algebras. Finally, we give an example of how the ideas presented in the document can be applied.