An iterative method based on coupled closed-form coefficients expansions for recovering the pollutant source and initial pollution profile

In this paper we develop a weak-form integral equation method for an advection-diffusion equation with an unknown pollutant source and unknown initial pollution profile by using Green's second identity in terms of boundary conditions on the whole space-time boundary. The numerical algorithm is developed to recover the time-dependent pollutant source under an extra final time condition. The iterative algorithm based on coupled closed-form coefficients expansions method can recover the unknown space-dependent pollutant source and initial pollution profile resorting to two extra conditions at different times, which is convergent very fast. Several numerical examples of the inverse pollutant source problem and initial pollution profile problem demonstrate that the present methods are effective and stable against large noise.