On theoretical and practical aspects of Duhamel’s integral

Abstract

The paper is a newapproach to the Duhamel integral. It contains an overviewof formulas and applications of Duhamel’s integral as well as a number of new results on the Duhamel integral and principle. Basic definitions are recalled and formulas for Duhamel’s integral are derived via Laplace transformation and Leibniz integral rule. Applications of Duhamel’s integral for solving certain types of differential and integral equations are presented. Moreover, an interpretation of Duhamel’s formula in the theory of operator semigroups is given. Some applications of Duhamel’s formula in control systems analysis are discussed. The work is also devoted to the usage of Duhamel’s integral for differential equations with fractional order derivative.

Keywords

• duhamel’s integral
• duhamel’s principle
• duhamel’s formula
• laplace transformation
• semigroup of operators
• leibniz integral rule
• volterra integral equation
• caputo fractional derivative