Nonlinear reaction diffusion and telegraph type equations have long served as foundational models for understanding wave propagation in excitable media, biological tissues, electrical transmission systems, and stochastic control processes. From the early conceptualization of bistable transmission lines and neuristor dynamics to modern developments in density dependent diffusion and random differential operational calculus, a continuous theoretical lineage connects electrical engineering, mathematical biology, and applied probability. This article develops an integrated, publication ready theoretical and computational framework that synthesizes the Nagumo type reaction diffusion and telegraph models with deterministic and stochastic differential transformation methodologies. Drawing exclusively from the provided references, the study constructs a unified interpretation of how nonlinear wave propagation, stability, and control can be modeled, approximated, and interpreted in systems subject to both deterministic and random influences.

The paper begins by tracing the historical evolution of the Nagumo and neuristor models from early transmission line theories to biologically inspired nerve impulse representations. Building on Scott and Nagumo and later Buratti and Lindgren, the conceptualization of bistable dynamics and waveforms is reframed as a general theory of excitable systems that can be extended into reaction diffusion and telegraph frameworks. The work of Pickard is incorporated to emphasize how physiological conduction in medullated and unmedullated fibers can be described by nonlinear wave equations, while Ahmed and Abdusalam are used to link these ideas to telegraph reaction diffusion and coupled map lattices in biological systems.

The analytical core of the article draws heavily on Abdusalam, Van Gorder, Vajravelu, Pedersen, Mansour, and others who developed analytic, variational, and numerical perspectives on density dependent and nonlinear Nagumo diffusion equations. Their results are interpreted not merely as isolated solution techniques but as part of a broader epistemological shift toward viewing wave speed, stability, and pattern formation as emergent properties of nonlinear transport under feedback controlled reaction mechanisms.

A second major theoretical pillar is the differential transformation method and its variants, including projected, improved, and random differential transformations as developed by Zhou, Bert and Zeng, Arikoglu and Ozkol, Jang, Ebaid, and Villafuerte and collaborators. These methods are presented as a coherent computational philosophy that transforms complex nonlinear and stochastic differential equations into tractable recursive structures, allowing both deterministic and random effects to be studied in a unified manner.