Vibration of Continuous Systems, 2nd Edition

7Solution Procedure: Integral Transform Methods

7.1 INTRODUCTION

Integral transforms are considered to be operational methods or operational calculus methods that are developed for the efficient solution of differential and integral equations. In these methods, the operations of differentiation and integration are symbolized by algebraic operators. Oliver Heaviside (1850–1925) was the first person to develop and use the operational methods for the solution of the telegraph equation and the second‐order hyperbolic partial differential equations with constant coefficients in 1892 [1]. However, his operational methods were based mostly on intuition and lacked mathematical rigor. Although subsequently, the operational methods have developed into one of most useful mathematical methods, contemporary mathematicians hardly recognized Heaviside's work on operational methods, due to its lack of mathematical rigor.

Subsequently, many mathematicians have tried to interpret and justify Heaviside's work. For example, Bromwich and Wagner tried to justify Heaviside's work on the basis of contour integration [2,3]. Carson attempted to derive the operational method using an infinite integral of the Laplace type [4]. Van der Pol and other mathematicians tried to derive the operational method by employing complex variable theory [5]. All these attempts proved successful in establishing the mathematical validity of the operational method in the early part of the twentieth century. As such, the ...