COMPREHENSIVE REVIEW OF DOUBLE INTEGRAL TRANSFORMS

Integral transform techniques play a fundamental role in obtaining analytical and semi-analytical solutions of differential, integral, and integro-differential equations arising in science and engineering. In recent years, several hybrid (double) integral transforms have been introduced to enhance the efficiency and applicability of classical transform methods. This paper presents a comprehensive comparative study of selected double integral transforms, the study systematically compares the transforms of standard elementary functions and examines their operational properties, including linearity, differentiation, initial and boundary value properties.The analysis highlights the mathematical similarities and distinctive characteristics of each transform, providing a unified framework for evaluating their computational effectiveness. Furthermore, the paper discusses the relative advantages of these transforms with respect to simplifying complex mathematical models, reducing computational effort, and improving the solution process for ordinary and partial differential equations. Their applicability is explored across diverse domains, including mathematical physics, heat and mass transfer, fluid dynamics, control engineering, signal processing, electromagnetics, quantum mechanics, population dynamics, and engineering sciences.