The essential value of the double transform is how it can take the partial differential equation (PDE) that describes the physical process occurring in a complex differential domain and transform the PDE to a much simpler algebraic form. This procedure converts the partial derivatives of the PDE into algebraic operations and provides an efficient means for the simultaneous coupling of both the spatial and temporal parameters (variables) involved. On the other hand, the single transform has significant limitations; it only partially simplifies the PDE into ODE form and still requires a complex analytical solution to an ODE. Also, the single transform has great difficulty combining both the boundary and initial conditions into one unified analytical framework (solution), but the double transform allows both conditions to be contained within the solution of the algebraic equation, which significantly reduces the potential for analytical error. The present study introduces a new extension of the Double ZZ Shehu Transform that employs a special complex kernel. This extension was analytically proven to exist and have a unique solution as well as possessing significant analytical properties that are different from the double ZZ Shehu Transform. To showcase the real world application of this extension, the new transform was applied to simulate the flow of fluid through cylindrical conduits. Specifically, hydraulic scenarios that were investigated consist of fluid flow in a pipe with a stationary versus a moving pipe wall.
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