In this paper we are interested in the solution of one-dimensional quasi-linear diffusion-reaction equation. The nonlinear reaction term includes the first derivative in space of the solution. We use the finite difference method to discretize this problem. The modification of a general methodology for investigation of difference schemes approximating non-stationary differential equations is used and the results for the stability and convergence of the numerical solution are proved. The convergence of the discrete derivative of the solution is proved in the maximum norm.
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