In this paper, we investigate the solitary wave solutions of the nonlinear Murray equation by applying a modified extension approach. To address gaps in the existing literature, we employ two analytical techniques: the new auxiliary equation method (NAEM) and the \((G'/G^2)\) -expansion method. These methods yield a variety of solutions expressed in hyperbolic, trigonometric, exponential, and rational forms, including dark, bright, periodic, dark-bright, smooth topological with high peaks, dark-singular, bright singular and periodic singular soliton solutions. The derived solutions provide insight into the behavior of blood vessel dynamics, particularly in simulating blood flow for patients with cardiovascular diseases. Previous studies have explored related analytical solutions; however, this work contributes new findings that fill a crucial gap in the mathematical modeling of biological systems. Solitary wave solutions are vital to bio-mathematics due to their relevance in describing nonlinear phenomena and their wide range of applications. These solutions can further enhance our understanding of complex biological systems and potentially guide the development of innovative medical models and therapies. To analyze the behavior of solitary wave solutions under different parameters, 2D and 3D visualizations are generated and displayed using MATLAB.
