This study is related with a new numerical method for solving the singularly perturbed of reaction-diffusion problem. Firstly, explicit boundaries for the solution of the problem are constructed. Then, the difference scheme in the uniform mesh is founded. This problem is discreted applying the finite element method and solved using the Thomas algorithm. The uniform convergence of these difference schemes has been proven with respect to the perturbation parameter ε at the discrete maximum norm. The numerical results supporting the theoretical methods are given and the effectiveness of the method is shown on examples.