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The Newell-Whitehead-Segel (NWS) equation plays a significant role in nonlinear systems, including mathematical biology, plasma physics, solid-state physics, optics, quantum mechanics, cosmology, fluid dynamics, and many others. In this work, we proposed the quintic B-spline collocation method to find the numerical solution of the nonlinear NWS-type equation. Crank-Nicolson finite difference method (FDM) is used to discretize the equation in time space, and quasi-linearization is employed to linearize the nonlinear term. The stability analysis has been discussed using the Von Neumann Method, and stability conditions have been obtained. The numerical results are compared with existing techniques, which demonstrate the effectiveness and applicability of the proposed technique. The proposed method has been applied to four numerical test problems at various time levels and mesh sizes to demonstrate the effectiveness, which involves quadratic, cubic, and quartic order nonlinear terms. The comparison shows good agreement with the exact solution, as demonstrated by absolute error tables and graphs. Moreover, the proposed method is easy to implement and produces good results.
Thube et al. (2026) studied this question.