@rlopez your first approach was the one I was looking for in the particular instants. The resulting value for "b" is accurate. As for as the infinite numbers are concern, I am happy with the value I got with ur first approach. (I don’t know how to connect the test for the positivity of f with proc but for individual value of b I can check it.)

For the rest of the two points, here is my explanation

1. I have to admit my mistake it's f'(b) not f(b). The reason for this plot (f'(b) vs S) is to find the variations at the surface with the changing S.
2. From physical point of view f''(0) illustrates the resistance factor, So, for analysis it's important to know the effects of S on f''(0) while locating the value of b for each S in the background. 

For all these cases, the best and easiest way will be to have a line plot instead of surface plot. With surface plot it's very hard for me to interpret the output of the ODE.

I hope this make sense.

Thanks

@Carl Love any luck improving the convergence?

@Carl Love I didn't knew that. Thanks

If you still are persistent about the method then here is an example which I copied from internet?

The code is written by Douglas Wilhelm Harder.

"The IVP y⁽¹⁾(t = t y(t) - t² + 1 with y(0) = 3, to approximate y(1), we can do:

t[0] := 0;
t[1] := 1;
y[0] := 3;
f := (t, y) -> t*y - t^2 + 1;
h := t[1] - t[0];
K[0] := f(t[0],       y[0]           );
K[1] := f(t[0] + h/2, y[0] + h/2*K[0]);
K[2] := f(t[0] + h/2, y[0] + h/2*K[1]);
K[3] := f(t[0] + h  , y[0] +   h*K[2]);
y[1] := y[0] + h*(K[0] + 2*K[1] + 2*K[2] + K[3])/6;"

I hope this will help you

THanks

you need to have a look here 

http://www.maplesoft.com/support/help/MapleSim/view.aspx?path=dsolve/numeric

thx

Thanks @rlopez for such a robust approach. As you said with this we can find a value of b which we can then use to find the solution. But suppose, if

• we dont want to fix b and then try to Plot f(b) vs some parameter (say S) ? and
• want to plot f"(0) vs S, what value b will take here?

Eq:= diff(f(y),y$3)+f(y)*diff(f(y),y$2)-diff(f(y),y$1)^2-S*(diff(f(y),y$1)+y/2*diff(f(y),y$2))=0;

f(b)=S*b/2

THx @Carl Love 

On my PC it take long time to execute and shows this Warning all the time

Warning, The use of global variables in numerical ODE problems is deprecated, and will be

removed in a future release. Use the 'parameters' argument instead (see ?dsolve,numeric,

parameters)

 Moreover, I am unable to even get close to the desired results

plots:-implicitplot(F, 0..2, 0..4);

@Carl Love 

I am unable to get the desired output.

The output should look like this

THanks

@Carl Love 

I have send you the paper and maple sheet. Please check your email.

Thanks

@Carl Love 

Here it is.

pdeplot.mw

@Carl Love 

Dear,

WHat I did was to introduce a new variable to make the process easy enough to plot the desired results.

If you look to the original Eqs, 1,2,3 there is no Nux(eta,tau). What I did is that, I defined it as diff(theta(eta, tau), eta) = Nux(eta, tau) and replaced whereever it should be. This way it was easy to plot Nux bcoz, I want to plot -diff(theta(eta, tau), eta) at eta=0. Now by defining Nux, I just need to plot -Nux(0,tau).

Hope this makes thing more clear.

http://www.mapleprimes.com/questions/148997-Positioning-Of-The-Various-Terms-In-Pdsolve

Aside from this, why I am unable to see the old comments on this post, any idea Admin?

Thanks

@Carl Love 

I am stuck with this for long time now. Do you have any idea how to handle this?

THanks

@Carl Love, I already uploaded the maple sheet with contains my code and the expected output as an image.

But here it is once again, if you are unable to download it?

pdeplot.mw

Thanks 

It seems that there is some confusion about my post. 

Here, I will give a simple example which might help to clarify my question

From equation 44 its easy to plot R(a) bcoz we have a mathematical expression. But in my question, we are trying to find numerical solution and then to plot R(a).

Hope this make things more clear. 

Dear @Preben Alsholm , I tried to find those values but no such values exist for which we

cannot find stable solution using maple.

Anyway, I am still interested to know how we can plot parameter domain, considering the

above example as a test case.