Numerical algorithms for multidimensional time-fractional wave equation of distributed-order with a nonlinear source term

Fractional differential equations (FDEs) of distributed-order are important in depicting the models where the order of differentiation distributes over a certain range. Numerically solving this kind of FDEs requires not only discretizations of the temporal and spatial derivatives, but also approximation of the distributed-order integral, which brings much more difficulty. In this paper, based on the mid-point quadrature rule and composite two-point Gauss–Legendre quadrature rule, two finite difference schemes are established. Different from the previous works, which concerned only one- or two-dimensional problems with linear source terms, time-fractional wave equations of distributed-order whose source term is nonlinear in two and even three dimensions are considered. In addition, to improve the computational efficiency, the technique of alternating direction implicit (ADI) decomposition is also adopted. The unique solvability of the difference scheme is discussed, and the unconditional stability and convergence are analyzed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithms for both the two- and three-dimensional cases.