Numerical solutions for nonlinear ordinary and fractional Newell–Whitehead–Segel equation using shifted Schröder polynomials

Abstract This paper introduces a collocation approach for treating the ordinary and fractional Newell–Whitehead–Segel equation (NWSE). The integer and fractional derivatives of shifted Schröder polynomials (SPs), as well as some new theoretical results of these polynomials, are presented and used in conjunction with the collocation method to convert the equation with its underlying conditions into a system of equations that can be treated using a suitable numerical solver. A thorough error analysis is performed to evaluate the accuracy and dependability of our suggested method. Some numerical examples show that our suggested strategy is effective and accurate. The numerical results demonstrate that the suggested collocation approach yields accurate solutions using shifted SPs as basis functions.