A new algorithm is proposed for solving the time-dependent Navier-Stokes equations in a sequential uncoupled manner. The algorithm, known as PISO (Pressure Implicit with Splitting of Operators) is extended to the Smoothed Particle Hydrodynamics (SPH) context (PISO-SPH). The algorithm consists of one prediction and two correction steps, based on a full Navier-Stokes equation, therefore, a modified Poisson equation is derived which makes the algorithm more stable with less pressure fluctuations. The proposed PISO-SPH method is applied to solve a number of benchmark problems including both unsteady and steady state test cases. Comparing the results with analytical solutions and other numerical methods, it is shown that the proposed method is accurate and straightforward for the simulation of incompressible fluid flows.