Anisotropic diffusion
Let's proceed step-by-step:


  I need some more explanation on what is gradient? How we calculate gradient? please explain me in a lucid manner I have zero knowledge about gradient.please explain me with easy explanation


a gradient relate to a measure of the derivate, that is, the coefficient of variation. If you think of a measure of position, the derivate is the speed. Now, we do not have a single point, but a field of pixel with values. Now think of it as a landscape, the luminance as the height and the gradient as the slope. It has both a magnitude (how steep it is) and a direction.

In the formulation that you cite, the slope is computed thanks to the divergence operator. In this case it equals to the sum of slope in the directions of the reference frame (that is, in x and y). It is used to allow a more general formulation in non-euclidean spaces.


  This gradient magnitude equation is what I am unable to understand please see the following equation and explain me in detail. I don't Know why they use divergence and gradient 


the basic idea behind this equation is to control diffusion by the contours:


diffusion helps to smooth out data by averaging locally. one drawback is that if you do it too long, you lose all information.
contours is the abstract concept of a boundary. Think of it for instance as the contour of a object in an image or the transient in the square wave that you show. To detect the position of these contours, you may use the gradient, and specifically the amplitude of the gradient (how steep the slope is)
now comes the heuristic of anisotropic diffusion: if you diffuse, but that you control the speed of this diffusion by the borders, then, you obtain a better result.


These equations just lay down these concepts quantitatively.