Hermite wavelet-based numerical study on (1 + 1)-dimensional Allen–Cahn and phi-four equations

In recent years, finding precise numerical solutions to nonlinear partial differential equations (NLPDEs) has become a significant research topic. This paper addresses the two well-known nonlinear models, which are [Formula: see text]-dimensional Allen–Cahn (AC) and phi-four (PF) equations, and they have applications in various real-world scenarios. These two models are solved by a numerical technique called Hermite wavelet collocation scheme (HWCS). The operational matrices of integration were developed using the Hermite wavelet basis. The proposed method is based on the collocation process. By employing appropriate grid points, this method converts the nonlinear model into a set of nonlinear algebraic equations. We obtain a solution by employing the Newton–Raphson scheme to solve these nonlinear algebraic equations. Tables and figures demonstrate that the proposed method yields superior results. The proposed examples demonstrate the efficacy of the proposed idea, and the results affirm the practicality of the proposed approach. We compared the present method with the non-polynomial spline method and trigonometric B-spline collocation technique to check the potential of the projected method.