www.iaset.us editor@iaset.us THE TRIPLE LAPLACE TRANSFORMS AND THEIR PROPERTIES A. K.Thakur, Avinash Kumar & Hetram Suryavanshi Dr. C. V. Raman University, India ABSTRACT This paper deals with the triple Laplace transforms and their properties with examples and applications to functional, integral and partial differential equations. Several simple theorems dealing with general properties of the triple Laplace transform are proved. The convolution, its properties and convolution theorem with a proof are discussed in some detail. The main focus of this paper is to develop the method of the triple Laplace transform to solve initial and boundary value problems in applied mathematics, and mathematical physics. KEYWORDS: Triple Laplace Transform, Double Laplace Transform, Single Laplace Transform, Convolution Article History Received: 17 May 2018 | Revised: 22 May 2018 | Accepted: 26 May 2018 1.1. INTRODUCTION P. S. Laplace (1749–1827) introduced the idea of the Laplace transform in 1782, In his celebrated study of probability theory and celestial mechanics. Laplace’s classic treatise on LaTh´eorie Analytique des Probabilities (Analytical Theory of Probability) contained some basic results of the Laplace transform which is one of the oldest and most commonly used linear integral transforms available in the mathematical literature. This has effectively been used in finding the solutions of linear differential, difference and integral equations. On the other hand, Joseph Fourier’s (1768–1830) monumental treatise on La Th´eorieAnalytique de la Chaleur (The Analytical Theory of Heat) provided the modern mathematical theory of heat conduction, Fourier series, and Fourier integrals with applications. In his treatise, he discovered a double integral representation of a non-periodic function f(x) for all real x which is universally known as the Fourier Integral Theorem in the form ( 29 ( 29 1 2 ikx ik f x e f e d dk ξ ξ ξ π ∞ ∞ - -∞ -∞   =     ∫ ∫ The deep significance of this theorem has been recognized by mathematicians and mathematical physicists of the nineteenth and twentieth centuries. Indeed, this theorem is regarded as one of the most fundamental representation theorems of modern mathematical analysis and has widespread mathematical, physical and engineering applications. According to Lord Kelvin (1824–1907) and Peter Guthrie Tait (1831–1901) once said: “Fourier’s Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly recondite question in modern physics”. Another remarkable fact is that the Fourier integral theorem was used by Fourier to introduce the Fourier transform and the inverse Fourier transform. This celebrated work of Fourier International Journal of Applied Mathematics Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 7, Issue 4, Jun - Jul 2018; 33-44 © IASET