摘要：An implicit free boundary problem is analyzed which is originate from the diffusion of oxygen in a medium which simultaneously consumes the oxygen, with a rate of consumption of oxygen which depends on the time. A combination of numerical and analytical methods are applied to this problem and the results are finally expressed in the form of an approximate polynomial expression. By this way, numerical solutions and approximate analytical of a partial differential equation are obtained, which describe the diffusion of oxygen in an absorbing medium. Essential mathematical difficulties are associated with the presence of a free boundary (which marks the furthest penetration of oxygen into the medium) and also with the necessity of determining the distribution of oxygen through the medium as a function of time. Some properties are deduced from the solution of the original problem and from the free boundary. Through the results, several examples are presented which arise of considering differents expressions for the rate of consumption of oxygen, being carried out in each one of them the corresponding analysis. This problem has a particular application in medical research: the prediction of the pattern can be used in the treatment of the cancer by radiotherapy [1).
