Darboux problem for proportional partial fractional differential equations

Abstract

In this paper, we introduce and study a novel fractional derivative called the generalized proportional Caputo fractional derivative. Under adequate assumptions on the source term, we prove existence and uniqueness aspects of the Darboux problems involving the generalized proportional fractional operator. A numerical scheme based on finite differences approximation is also investigated, and results of convergence are established. Several numerical tests are performed in illustration.