Triple Laplace Transform in Bicomplex Space with Application

Mahesh Puri Goswami,Naveen Jha

doi:10.13189/ms.2020.080411

Abstract

In this article, we investigate bicomplex triple Laplace transform in the framework of bicomplexified frequency domain with Region of Convergence (ROC), which is generalization of complex triple Laplace transform. Bicomplex numbers are pairs of complex numbers with commutative ring with unity and zero-divisors, which describe physical interpretation in four dimensional spaces and provide large class of frequency domain. Also, we derive some basic properties and inversion theorem of triple Laplace transform in bicomplex space. In this technique, we use idempotent representation methodology of bicomplex numbers, which play vital role in proving our results. Consequently, the obtained results can be highly applicable in the fields of Quantum Mechanics, Signal Processing, Electric Circuit Theory, Control Engineering, and solving differential equations. Application of bicomplex triple Laplace transform has been discussed in finding the solution of third-order partial differential equation of bicomplex-valued function.

Highlights

Enormous efforts have been done in past few years in the applications of bicomplex functions and fine research has been developed
The concept of bicomplex numbers was introduced by Segre [6], in order to compactly describe physical interpretation in four-dimensional space
Set of bicomplex numbers is a commutative ring with unity and zero divisors which contains the commutative ring of hyperbolic numbers and the field of complex numbers

Summary

Introduction

Enormous efforts have been done in past few years in the applications of bicomplex functions and fine research has been developed. Enormous efforts have been done to expand the theory of integral transforms in bicomplex space and studied their applications by Agarwal et al [19, 20, 21, 23]. For solving the large class of partial differential equations of bicomplex-valued function, we require integral transforms defined for large class. In this procedure we derive triple Laplace transform in bicomplex space with ROC that can be competent the transferring signals from real-valued (x, y, z) domain to bicomplexified frequency (ξ, η, γ) domain.

Preliminaries on Bicomplex Numbers

Bicomplex Triple Laplace Transform

Properties of Bicomplex Triple Laplace Transform

Application

Conclusion