In view of the dense distribution of multiple roots of nonlinear equation systems, the heuristic optimisation algorithm is used to solve the root loss problem of multiple roots in the process of one operation, and a fuzzy neighborhood-based slime mould algorithm (FASMA) is proposed. The fuzzy neighborhood technique is applied to divide the population into sub-populations, so that the algorithm has the ability to locate multiple roots of nonlinear equations simultaneously. At the same time, the slime mould algorithm (SMA) is used to update the individuals in the fuzzy neighborhood. The SMA simulates the positive and negative feedback of the propagation wave generated by the biological oscillator in the process of slime mould foraging in nature, which makes the proposed algorithm have good local search ability. In addition, the archiving mechanism and individual reinitialization are used to enhance the population diversity and improve the ability of the algorithm to jump out of the local optimum. In order to verify the validity of the proposed algorithm, the special cases of nonlinear equation systems with different density roots and 15 dense-root nonlinear equation systems are tested. The results of FASMA are compared with those of FNODE, DREA and NCDE algorithms. The experimental results show that the proposed algorithm FASMA performs better in term of both the root ratio and the success rate.