79 - Operational Calculus*

Abstract

This chapter focuses on operational calculus applicable to ordinary differential equations and partial differential equations. It is sometimes easier to solve a differential equation in a transformed space. The use of operational calculus yields a reformulation of the original differential equation. The procedure for operational calculus is that an ordinary differential equation is transformed to a field of operators. The equation in that field is then solved, and then transformed back. In this field, ordinary functions, generalized functions, and differential operators are all treated as objects in a single algebraic structure. The operator field that is used has among other elements, an identity operator (I), a differentiation operator (often denoted by D or s) and an integration operator (often denoted by D−1). The operational calculus is also called the Heaviside calculus. The operational calculus, at its simplest level, has a great similarity with Laplace transforms. It is sometimes difficult to justify the formal steps that are employed in using the operation calculus. One solution is to use a more precisely defined operator, such as the primary operator.