Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

Sequential sampling theory is the dominant theoretical framework for explaining human decision making behavior. In this theory, the decision process is determined by a stochastic process in a bounded domain, with the likelihood function of the model being the first passage-time distribution of this process. However, many sequential sampling models have no analytic formulation for the first passage-time distribution, making these models difficult and computationally taxing to fit. Thus, developing an efficient and general approximation procedure for the first-passage time distribution of sequential sampling models is crucial for the field of mathematical psychology. Here, we have developed a numerical algorithm based on the mesh-free techniques for numerically solving the Fokker-Planck differential equation, which allows us to approximate the first passage-time distribution of some important classes of sequential sampling models, including the urgency signal model, Ornstein-Uhlenbeck model, and collapsing threshold model. Since the considered equation is a singular partial differential equation in a moving boundary domain, the proposed numerical method first transforms the equation to a regular domain, and then the solution is approximated based on the collocation method. However, since satisfying the boundary conditions in the collocation algorithms can be very tricky, we have imposed the boundary conditions in the basis functions. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. The performance of the proposed method is illustrated by solving several examples.