An Equation for Cascading Feedback in Coupled Variables: Operator-Exponential Closed-Form Solution for a Non-Separable Mixed-Order PDE

This work introduces a novel linear evolution equation designed to model "cascading feedback" in complex, multidimensional systems. It addresses mathematical challenges where variables from different dimensions are intertwined in a non-separable way, preventing the use of standard techniques like separation of variables. By utilizing the Elzaki transform, the derivation provides a robust and fully explicit solution that captures how initial changes in one dimension propagate and amplify through their couplings with others. The derivation provides complete step-by-step transparency, showing that the method is entirely general and applicable to any linear constant-coefficient evolution equation where traditional methods fail. While the specific equation serves as a rigorous pedagogical test case, it acts as a valuable analytical tool and benchmark for real-world applications. These include studying anisotropic wave propagation in geophysics, validating the design of advanced composite materials, and testing the performance of engineered metamaterials in acoustics.