Some New Results on the Generalized Double Laplace Transform and Its Properties - Atugade, Diksha P.; Hiwarekar, Anil P. (Communications in Mathematics and Applications, 2025)
Integral transformations play a fundamental role in various scientific and engineering domains, providing powerful tools for solving differential, integral, and functional equations. Among these, the Laplace transform is one of the most widely used, and many other emerging integral transforms are direct generalizations or modifications of it. This paper focuses on an extension of the classical Laplace transform to a new generalized double Laplace transform framework. The proposed transform is defined in terms of arbitrary, strictly increasing kernel functions, enabling greater flexibility in handling problems with non-uniform scaling in multiple variables. Fundamental properties of the transform, including linearity, shifting, change of scale, and convolution theorems, are established. Moreover, the study establishes that the classical double Laplace transform arises as a particular case within the framework of the generalized version. The derived framework broadens applicability to a wider range of partial differential equations (PDEs) in mathematics and physics.