Abstract
Transform methods occupy a central position in applied mathematics, converting differential equations into algebraic or simpler functional equations that admit systematic solution. Among these methods, the Laplace transform and the Fourier transform (together with its companion, the Fourier series) are the most widely deployed in science and engineering. Despite their common ancestry in spectral analysis, the two transforms differ fundamentally in their domains of definition, their convergence requirements, the classes of problems to which each is naturally suited, and their analytical and computational trade-offs. This paper presents a rigorous comparative study of the Laplace and Fourier transforms as tools for solving differential equations. Beginning with precise mathematical foundations and convergence theory, the paper develops four canonical case studies spanning ordinary differential equations with initial conditions, the heat equation on a finite interval, the heat equation on the whole real line, and a class of neutral delay differential equations that motivates a modern hybrid Laplace–Fourier approach. A structured comparison table and analytical workflow figures are included. The paper argues that method selection is not merely a matter of convention but must be guided by the geometry of the problem domain, the type of auxiliary conditions imposed, and the required convergence rate of the resulting representation.
KEYWORDS
Laplace Transform, Fourier Transform, Fourier Series, Differential Equations, Initial-Value Problems, Boundary-Value Problems.
Saakshi Tomar1*, Vishal Saxena2
1Research Scholar, Jayoti Vidyapeeth Women’s University, Jaipur-303122, Rajasthan, India
2Professor, Jayoti Vidyapeeth Women’s University, Jaipur-303122, Rajasthan, India