Numerical Treatment of 1D and 2D Variable-Order Fractional Nonlinear Cable Equations via Bernoulli Collocation Technique

Abstract This study introduces an innovative numerical method for solving variable-order fractional nonlinear cable (VOFNC) equations in one and two dimensions. The approach combines spectral collocation with Bernoulli polynomials (BPs), enabling precise operational matrix (OM) generation for fractional derivatives. Specialized Bernoulli basis functions facilitate efficient spatial-temporal discretization and boundary condition handling. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. Numerical experiments demonstrate high accuracy and rapid spectral convergence, surpassing existing methods with fewer collocation points. This methodology offers a reliable and computationally efficient tool for analyzing complex fractional models.