Turbulence, as an effective factor in determining investment risk, can play an important role in investors' decisions. A proper estimate of stock price volatility or option transactions in an investment cycle is a very important starting point in investment risk control. This paper proposes a governing equation for stock market indices using an advanced water drop algorithm. The proposed governing equation can be expressed as an equation by the algorithm with non-constant diffusion coefficient. The governing equation accounts for non-stationary effects and describes the time evolution of the probability distribution function of price returns. A stochastic differential equation models the time evolution of price returns in a different way than the classical Black-Scholes equation. All equations account for a deterministic part or trend and a random part or q-Gaussian noise. q-Gaussian noise can be decomposed into Gaussian noise that is influenced by a standard deviation or fluctuation. The presented model has been validated using the total stock price index data of Tehran Stock Exchange based on daily values from 1390 to the end of 1401. We show that the price return becomes Gaussian, thereby being fixed by normalizing the reduced data set. Data normalization is calculated by subtracting the trend and then dividing by the standard deviation of the reduced price return. Stationary test includes displaying the power spectrum in terms of autocorrelation of time series.