Towards a new triple integral transform (Laplace–ARA–Sumudu) with applications

The main objective of this work is to introduce a novel generalization of double transformations called the triple Laplace–ARA–Sumudu transform (TLARAST). This hybrid transformation extends the concepts of double Laplace–Sumudu, double Laplace–ARA and double ARA–Sumudu transforms into a triple hybrid transform. The article provides the definition of TLARAST and investigates its fundamental properties, including existence, inverses and related theorems. Furthermore, new results concerning TLARAST for partial derivatives and the theorem of multi-convolution are introduced and discussed. The practicality and efficacy of TLARAST are demonstrated by applying it to solve various types of partial differential equations with significant applications in physics and other scientific fields, such as the heat equation, Laplace equation, Poisson equation and wave equation. The solutions are illustrated through figures created using Mathematica software. Overall, this study underscores the usefulness and efficiency of TLARAST in solving partial differential equations involving multiple variables.