Bernoulli-Based Operational Matrix Approach for Handling Singular Nonlinear Second-Order Differential Equations

Abstract

     This paper investigates the application of the Bernoulli operational matrices (BOM) method for solving second-order singular nonlinear differential equations. Unlike traditional implementations of operational matrices with Chebyshev, Legendre, or Bernstein polynomials, this work emphasizes the efficiency of the Bernoulli basis in achieving high accuracy with relatively few expansion terms. The proposed approach expands the solution in shifted Bernoulli polynomials and uses corresponding operational matrices of differentiation, integration, and product operations to reduce the problem to a system of algebraic equations.

The method was implemented in Python and tested on benchmark models such as the Lane–Emden equation. Numerical experiments show that with only  basis functions, the BOM achieves a maximum error of order  closely matching reference solutions. The convergence analysis confirms the spectral accuracy of the method, with errors decreasing rapidly as  increases. A comparison with Chebyshev collocation highlights that the Bernoulli approach reaches the same accuracy with fewer basis terms and lower computational effort.

These results establish the Bernoulli operational matrices method as a reliable and computationally efficient tool for singular nonlinear models. Future extensions may include fractional-order systems and higher-dimensional applications.