A Spectral ADI Approach to Two‐Dimensional Euler–Poisson–Darboux Equations With Distributed‐Order Fractional Operators

ABSTRACT This paper presents a novel numerical scheme for solving a two‐dimensional Euler–Poisson–Darboux equation incorporating a distributed‐order space‐fractional operator and a time‐fractional derivative of Bessel type. To approximate the distributed‐order Riesz space derivative with high accuracy, the Gauss quadrature method is employed, outperforming traditional midpoint quadrature techniques. This transforms the original problem into a multiterm fractional differential equation. Spatial discretization is achieved using an alternating direction implicit (ADI) Galerkin–Legendre spectral method, while the Crank–Nicolson scheme is applied for time integration. The stability and convergence of the proposed method are rigorously analyzed and confirmed. Numerical experiments are conducted to validate the theoretical results and demonstrate the efficiency and accuracy of the approach, highlighting its potential for solving complex fractional differential equations arising in mathematical physics.