  This article provides an effective computational algorithm based on Legendre
wavelet (LW) and standard tau approach to approximate the solution of
multi-dimensional distributed order time-space fractional weakly singular
integro-partial differential equation (DOT-SFWSIPDE). To the best of our
understanding, the proposed computational algorithm is new and has not been
previously applied for solving DOT-SFWSIPDE. The matrix representation of
distributed order fractional derivatives, integer order derivatives and weakly
singular kernel associated with the integral based on LWare established to find
the numerical solutions of the proposed DOT-SFWSIPDE. Moreover, the association
of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with
constructed matrix representation of differential and integral operators
diminish DOT-SFWSIPDE into system of linear algebraic equations. Error bounds,
convergence analysis, numerical algorithms and also error estimation of the
DOT-SFWSIPDE are regorously investigated. For the reliability of the proposed
computational algorithm, numerous test examples has been incorporated in the
manuscript to ensure the robustness and theoretical results of proposed
technique.
