On the double Laplace-typed integral transform for solving partial differential equations via quantum calculus

In this work, we extend double Laplace-type integral transform (DLTT) and apply the concept of quantum calculus or \(q\)-calculus to establish the definitions of double \(q\)-Laplace-type integral transform (\(q\)-DLTT). We present some main properties and compute some functions of the \(q\)-DLTT. Also, we apply them to establish simple formulas for the general solution to solve the \(q\)-PDEs, such as \(q\)-wave equation, \(q\)-Klein-Gordon equation, \(q\)-Laplace equation, \(q\)-heat equation, and \(q\)-telegraph equation. By using the Mathematica program, the 3D plots, which represent for the \(q\)-PDEs, are provided. Finally, a convergence comparison between the exact solution and \(q\)-calculus of PDEs is also given in this research.