Fractional Bernoulli polynomials optimization approach for nonlinear system of time fractional PDEs in (2+1)-dimensional

In recent years, the inherent numerical challenges associated with investigating systems of (2+1)-dimensional fractional differential equations have made them a subject of limited study. This paper marks the first introduction of a general class of nonlinear systems of (2+1)-dimensional fractional differential equations that encompass initial and boundary conditions. The approach begins by expanding the unknown functions using fractional Bernoulli polynomials as basis functions. Subsequently, an optimization problem is formulated utilizing these expansions and the operational matrices of the fractional derivatives. This formulation is then solved using the Lagrange multiplier algorithm, transforming the optimization problem into a system of nonlinear algebraic equations. Furthermore, the convergence analysis of the obtained approximate solution via the proposed technique is explored. Finally, the numerical algorithm's validity is assessed through a numerical example.