The Application of Abstract Algebra in Operational Calculus

This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of Mikusiński has been revised. We restate the concept of Mikusiński’s operator field as a convolutional field and define a new concept of operator field with the differential operator p as the core, effectively overcoming the confusion between the concepts of operators and functions, which represents the limitation of traditional theory. In addition, the classical Laplace transform is integrated into our theory in a homomorphic form, revealing the principle that the Laplace transform is compatible with operational calculus theory.