Algebraic Derivation of a Non-Separable Linear PDE with Cascading Feedback

This research presents a novel method for constructing linear partial differential equations through a process of hierarchical nesting. Moving beyond the classical foundations of isotropic heat transfer, this study explores how complex spatial coupling can be engineered by applying differential operators in cascading layers. By organizing spatial variables into a specific depth-based hierarchy—where effects propagate sequentially through a nested chain of coordinates—we derive an evolution equation that is fundamentally non-separable. The resulting framework demonstrates how mixed-order derivatives can be logically integrated to represent directional coupling and cascading feedback. This approach provides a systematic algebraic pathway for developing advanced models in fields such as control theory and anisotropic systems, where variables are linked by a structured, non-independent hierarchy.