@Paras31 I see that your graphs of Short-time behavior are noticeably different from the book's.  That may be due to numerical errors that may have crept in.

The following graphs, plotted using Maple's solution that I had noted in my previous reply, seem to be identical to the book's.

You have w0 and w[0].  Are these supposed to be the same thing?

You have w[0](eta,z).  What is w[0]?

You have w(r,z) := sum(p^i*w[i](r, z), i = 0 .. N); 
This is attempting to define w in terms of w.  Makes no sense.

You have D*w[0](0,z).  I am guessing that the D here is supposed to be the differentiation operator. But w[0] is a function of two variables, so you need to say derivative with respect to which variable.

Conclusion: What you are attempting to do is too complex for a first project in Maple.  You should spend some time learning the basics of the language before attempting this.

@C_R I have not seen Maple for Screen Readers because it is specific to Windows, which I don't have.  I will give it a try if/when Maplesoft comes up with a Linux version.

@Math-dashti In my worksheet I had changed the symbol "eta" of your textbook's equation (7.1) to "t" because t is easier to type.  Your modified worksheet does not work because it now mixes up t and eta.

The next image of the textbook's page that you have posted, ends with "... can be simplified to the following nonlinear ODE".  The ODEs, which come in the next page, are not shown.  You need to replace the ODEs in my worksheet with those ODEs.

And as you do that, you need to think about what to do with "x" that enters the equations.  You cannot solve the equations and obtain a plot without knowing what x is.  I don't know the context, so I cannot help you there.

@Math-dashti This may be what you want.

Download phase-portrait.mw

@dharr Considering the frequent requests for Adomian polynomials in this forum, your LAD code is can be a great time-saver.  I will certainly refer the readers to it when the occasion comes up.

@dharr Okay, perhaps I am looking at this through the wrong lens.  I had expected the series solution to provide an alternative to a numerical solution.  It looks like that's not the case.

There may be some information content in the series solution but I have a hard time getting my hands on it.  Suppose I am interested in the porous medium equation with the initial condition u(x,0)=exp(−x^2) for which there is no symbolic solution.  What sort of useful information do I get out of the LAD series?

@TechnicalSupport Does that work for you on Linux?  It doesn't for me.

@dharr Here is what we get when we evaluate the series solution in LAD_Barenblatt.mw at x=0 and plot the result as a function of time:

We see that the series solution rises dramatically over time, while it's supposed to decrease to zero.

The non-differentiability of the solution does not seem to be the issue.  We get the same kind of incorrect behavior out of LAD if we replace the initial condition with the very well-behaved u(x,0)=exp(−x^2).

The remark in the Kybernetes article about the "consitency of boundary conditions" is not relevant here since we are not solving a boundary value problem; there are no boundaries as the spatial dimension is the entire x axis.

@WD0HHU The stop button does nothing in Maple 2025 and Maple 2025.1.  Until that issue is fixed, I am sticking with Maple 2024.

@lucaud Before executing reflect(display...), you need to load the plots and plottools packages, as in

with(plots):
with(plottools):

If that doesn't work, then upload your entire worksheet for further comments.  To upload the worksheet, "reply" to your own original post, and within the toolbar of the editor window that comes up, click on the big fat green arrow.

I have occasionally heard about Adomian's method but have not had the occasion to delve into it in any depth, and to be honest, I have harbored some doubts about the utility of it.

But I will definitely be a convert if it can produce anything resembling the correct solution in the case of the classical porous medium equation.  I tried your proc but did not get anything useful, although it is very well possible that I did not apply it correctly.

Here is the porous medium equation along with an exact solution.  You may be able to do somehing with it.

PS: Perhaps the method will have a better chance of success with the initial condition u(x,0)=exp(-x^2).  There is no symbolic solution then, but pdsolve,numeric shows that the solution qiuckly takes the shape of an expanding semi-ellipse similar to the one in the animation below.

Download pme.mw

@janhardo The code that you have posted is good.  It implements what I called an alternative approach in my yesterday's message.

@vv Before seeing Carl Love's solution, I would have completely agreed with you.  I am impressed by seeing dsolve's solution pointed out by Carl.

@mmcdara Thanks for your analysis which shows how to obtain an approximate solution by smoothing out the transition at the interface.  In the absence of an explicit symbolic solution, that would have been a good alternative.