Generalized Bernoulli–Laguerre Polynomials: Applications in Coupled Nonlinear System of Variable-Order Fractional PDEs

In this paper, we introduce a general class of coupled nonlinear systems of variable-order fractional partial differential equations (GCNSV-FPDEs) with initial and boundary conditions. We propose a hybrid method based on new generalized Bernoulli–Laguerre polynomials (GB-LPs) for solving GCNSV-FPDEs. The concept of variable-order fractional derivatives (V-FDs) is employed in the Caputo type. We extract the operational matrices (OMs) of classical and V-FDs of GB-LPs. By utilizing GB-LPs, OMs, and the Lagrange multipliers method, we transform the given GCNSV-FPDE into a system of algebraic equations to be solved. The proposed method yields satisfactory results even with a small number of GB-LPs. We provide a full verification of the method’s convergence, and two examples are included to demonstrate its validity and applicability.