Local Galerkin Method Based on the Moving Least Squares Approximation for Solving Delay Integral Equations Arisen from an Air Pollution Model

Mathematical models for measuring pollutants, by predicting the amount of air quality elements, play an important role to protect the human health. As one of these models, delay Volterra integral equations are applied to simulate a network of sensors with past memory to evaluate the emissions of pollutants in the air. This paper presents a computational method to solve these types of delay integral equations using the discrete Galerkin scheme together with the moving least squares (MLS) approach as basis. The MLS is an effective technique to estimate an unknown function which includes a locally weighted least squares polynomial fitting over a small set of all points. The composite Gauss-Legendre quadrature formula is utilized to compute integrals appearing in the proposed method. Since the scheme is constructed on a local scattered data approximation, its algorithm is attractive and easy to run on a computer with normal features. The error estimation and convergence rate of the method are provided. Finally, numerical examples illustrate the efficiency and accuracy of the new technique and confirm the theoretical results obtained in the error analysis.