A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis

In this paper, a new matrix approach for solving second order linear partial differential equations (PDEs) under given initial conditions has been proposed. The basic idea includes integrating from the considered PDEs and transforming them to the associated integro-differential equations with partial derivatives. Therefore, Bernoulli operational matrices of differentiation and integration together with the completeness of Bernoulli polynomials can be used for transforming integro-differential equations to the corresponding algebraic equations. A rigorous error analysis in the infinity norm is given provided that the known functions and the exact solution are sufficiently smooth and bounded. A numerical example is included to demonstrate the validity and the applicability of the technique. The results confirm the theoretical prediction.