@Kitonum 

Thank you (Kitonum and Cal Love) very much for your help. It is a good idea. I will try to see how I could substitue all occurrences diff(u,`$`(x,n)) by u[`nx`]. I am quite new to Maple.

@Kitonum 

I want to display derivations by subscripts such as 

f_6x 

for diff(f(x,y),x$6);

Thank you!

@Markiyan Hirnyk 

Thank you (Edgardo, Markiyan, and Carl) for all your help. I am studying the methods that you showed me. The first question is to calculate P=L^{1/2}, where L=(D^2 + u), and u=u(x) is a function of x. Assuming P=D+f_0+f_1D^{-1}+f_2D^{-2}+f_3D^{-3}+..., and equating PP= L. One can calculate by hand that
(D^2 + u)^{1/2}=D+(u/2)D_{-1}-(u_x/4)D_{-2}+((-u^2+u_{xx})/8)D_{-3}+.... In the past ten years pseudo differential operators become a hot topic in mathematics. It is very successful in studying nonlinear PDE. I am hoping that new packages can be added to Maple to deal with pseudo differential operators. 

I tried your first piece of code. It works well!

I tries the second, it works too. However, I am quite new to Maple. I have to study more to understand commands like freeze, thaw etc. Your code helps me a lot. thank you so much! 

I tried your first piece of code. It works well!

I tries the second, it works too. However, I am quite new to Maple. I have to study more to understand commands like freeze, thaw etc. Your code helps me a lot. thank you so much! 

Thank you very much for your help. I am studying the solutions that you provided. The two references “Subscripts as Partial Differentiation Operators” and “programming subscripted partial differentiation operators” Look very helpful

Thank you very much for your help. I am studying the solutions that you provided. The two references “Subscripts as Partial Differentiation Operators” and “programming subscripted partial differentiation operators” Look very helpful