On the double Laplace transform with respect to another function

This paper explores the generalized double Laplace transform (GDLT) and its applications in fractional calculus. We begin by establishing essential lemmas and definitions that form the foundation of our findings. The core properties of the GDLT are thoroughly examined, providing a comprehensive understanding of its characteristics. We present novel results related to fractional and classical partial derivatives, as well as the double convolution theorem. Additionally, we calculate the double generalized Laplace transform for various bivariate Mittag-Leffler functions. The practical utility of this new double integral transform is demonstrated through its application in solving a range of fractional partial differential equations, highlighting its significance in applied mathematics.