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In this paper a nonlinear backward parabolic problem in one dimensional space is considered. Using a suitable iterative algorithm, the problem is converted to a linear backward parabolic problem. For the corresponding problem, the backward finite differences method with suitable grid size is applied. It is shown that if the coefficients satisfy some special conditions, this algorithm not only is convergent, but also is conditionally stable. Moreover, it is proved that the estimated values converge to the exact solution of the problem. All these approaches examined in some numerical examples. corresponding theorems for the convergency and stability of the solution are studied. [ABSTRACT FROM AUTHOR]