A new theory of fractional differential calculus

This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux this is the introduction derivative notion which natural generalization integer order derivatives; it also helps to unify multiple existing definitions and characterize what functions are fractionally differentiable. Various rules including fundamental theorem calculus, product chain rules, integration by parts formulas established for derivatives. Additionally, relationships with classical derivatives detailed characterizations weakly differentiable established. Furthermore, systematically extended general distributions instead only some special distributions. lays down solid theoretical foundation rigorously developing theories Sobolev spaces, variations, PDEs as well their numerical solutions subsequent works.