<p>In this paper, the uniform hyperbolic polynomial B-spline based collocation method is applied to find the numerical solution of the nonlinear one-dimensional Burgers’ equation. This equation illustrates various phenomena that arise in scientific and engineering fields, such as traffic flow, sound and shock waves in viscous media, and chemical reaction-diffusion models, including the Brusselator. In this method, the spatial and time derivatives of the Burgers’ equation are discretized using the Crank-Nicolson and forward finite difference methods. The Rubin and Graves technique is employed to linearize the nonlinear term. To evaluate the performance of the method, various numerical examples are solved, and the outcomes are compared with alternative approaches available in the literature. The results are also verified by comparing the numerical and exact solutions, and the accuracy is quantified using error norms. Additionally, the von Neumann method is applied to analyze the scheme’s stability, and the convergence analysis is discussed both theoretically and numerically. The numerical investigations demonstrate that the presented method yields more accurate results than other schemes, and they are computationally efficient.</p>