Particle swarm optimization (PSO) is a primary intelligence algorithm to solve task scheduling problem for workflow systems in cloud computing. The inertia weight is one of the most important parameters to achieve a balance between the global and local search in PSO algorithm. However, traditional adaptive inertia weight-based PSO task scheduling algorithms usually get local optimal and cause longer execution time and higher cost for scheduling plan. The traditional adaptive inertia weight does not comprehensively represent the information of particle position, and then cannot make a suitable balance between global and local search. Hence, a novel computation method for adaptive inertia weight is proposed to improve the computation method of success value of each particle. This method shows the position state of each particle more accurately and then improves the adaptability of inertia weight by comparing the fitness of each particle with the global best particle. Then a new inertia weight-based PSO algorithm is presented to solve task scheduling problem for cloud workflow systems. The novel weight can adjust the particle velocity more correctly so that the algorithm avoids premature convergence by a proper balance between local and global search. Comparing our new adaptive inertia weight with other five traditional inertia weights (viz. constant, index decreasing, linear decreasing, random, and adaptive inertia weight), the results show that our new adaptive inertia weight-based scheduling algorithm can always achieve stable convergence speed, the optimal fitness and execution cost of the scheduling plan (i.e. roughly 18% reduction of execution cost).