A numerical scheme for solving nonlinear backward parabolic problems

‎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. ‎Al these‎ ‎approaches examined in some numerical examples. corresponding‎ ‎theorems for the convergency and stability of the solution are‎ ‎studied‎.