Moving mesh partial differential equations have been widely used in the last two decade for solving differential equations exhibiting large solution variations such as shock waves and boundary layers. In this paper, we will briefly review the moving mesh techniques for one and two dimensional problems with finite difference. we have applied a dynamic adaptive finite difference method for solving time-dependent differential equations . The mesh velocities are governed by an equation in which a relaxation time is employed to move nodes in such a way that they remain concentrated in regions of rapid variation of the solution. A numerical example involving a blow-up problem shows the advantage of using a variable relaxation time over a fixed one.

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