Some New Results on the Generalized Double Laplace Transform and Its Properties
Keywords:
Two-dimensional Laplace transform, Generalized Laplace transform, Convolution theorem, Integral transforms, Partial differential equationsAbstract
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 with respect to 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 brings together multiple existing transforms while extending their applicability to a broader range of partial differential equations in mathematics and physics.
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