NUMERICAL INVESTIGATION OF THE STATIC PHI-4 EQUATION USING A HYBRID CUBIC B-SPLINE TECHNIQUE

Abstract

The present work focuses on solving the one-dimensional static Phi-4 (ϕ⁴) nonlinear differential equation using the Hybrid Cubic B-Spline (HCS) method. This nonlinear field equation arises in various physical contexts, including quantum field theory, phase transition models, and nonlinear optics. Traditional methods, such as finite difference or spectral techniques, face limitations in handling sharp transitions or ensuring smoothness across the domain. The HCS approach, a fusion of finite element and spline-based methods, offers high accuracy with smooth approximation. The mathematical derivation of the method, stiffness matrix construction, and implementation of boundary conditions are discussed in detail. Results are validated against the exact analytical kink solution, and the error analysis shows excellent agreement, confirming the efficacy of the method. This work emphasizes the stability, accuracy, and potential of HCS for solving nonlinear boundary value problems in mathematical physics.