In the present work analytical model is used, which gives series of solutions of the non-linear differential equation, is based on the method of decomposition. The series were used to derive and test simulant solutions that arise using the concept of double decomposition. These expressions permit an accurate forecasting of contaminant propagation under non-linearity in laboratory or field investigations at early or prolonged times after the spill. The practical scenario of an instantaneous spill is studied for situations of non-linear decay, non-linear Freundlich isotherm, and non-linear Langmuir isotherm. The FDM predictions were found to be in excellent agreement with analytical solutions for a wide range of field conditions with regard to dispersion and source definition. The new developed numerical model can be used for the forecasting of contaminant dispersion in laboratory and field under non-linear reactions, or for the quantitative description of the effect of non-linearity in the sorption parameters, on the time-space distribution of the contaminant. The implicit method used here which is unconditionally stable.