Galerkin–Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation

In this paper, we consider the Galerkin–Legendre spectral method for solving the two-dimensional distributed-order time fractional fourth-order partial differential equation. By utilizing the composite Simpson formula to discretize the distributed-order integral, we transform the considered equation into a multi-term time fractional sub-diffusion equation. Then the L2-1σ formula is used to approximate the multi-term Caputo fractional derivatives and the Legendre spectral method is employed for the spatial discretization. The scheme is proved to be unconditionally stable and convergent in both L2- and L∞-norms with fourth-order accuracy in distributed order, second-order accuracy in time and spectral accuracy in space. Finally, some numerical tests are performed to verify the theoretical results.