Foundations for L-fractional partial differential equations

Develop rigorous theorems for L-fractional partial differential equations analogous to Caputo-based results, including existence, uniqueness, and representation via bivariate fractional power series.

Background

The paper notes a lack of studies on L-fractional PDEs. In the Caputo setting, formal solutions via bivariate fractional series exist, but rigorous theory remains to be developed for L-fractional operators.

References

Finally, what about fractional partial differential equations? There are no studies for the L-fractional derivative. In the Caputo context, formal solutions have been found in terms of bivariate fractional power series, but rigorous theorems are yet to be investigated.

— Theory on linear L-fractional differential equations and a new Mittag-Leffler-type function  (2403.00341 - Jornet, 2024) in Section 8, Open Problems

A systematic study of nonlinear fractional differential equations, fractional PDEs, and solutions with initial weak singularities is left for future work.

— An Exact-Moment Local Legendre Frame Method with Block Convolution for Caputo Fractional Differentiation  (2608.19157 - Zhao et al., 19 Aug 2026) in Section 5.4, subsection “Application to a fractional differential equation”