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Solve non linear second order differential equation with initial and boundary condition

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Gravel

Solve non linear second order differential equation with initial and boundary condition

Hi, 

     I am trying to model the sedimentation of spherical colloidal suspensions. For that, I need to solve the differential equation (check image file). I am attaching the mathcad file and the original paper. What I want is to plot volume fraction (phi) as a function of height (x) at a certain time (t) (Something similar to figure 3 in the paper). I would greatly appreciate the help.

 

Regards

 

 

21 REPLIES 21

Re: Solve non linear second order differential equation with initial and boundary condition

Hi,

It should be convenient to install Mathcad 15 since it offers the possibility to solve  partial differential equations, with the "Pdesolve" operator inserted in a solution block (there is an example), and of which Prime is deprived.

Re: Solve non linear second order differential equation with initial and boundary condition

I'm afraid PDEsolve (real Mathcad) will not help.

My results, analysing your equations:

LM_20190223_NonLinearPDE1.png

LM_20190223_NonLinearPDE2.png

LM_20190223_NonLinearPDE3.png

Unfortunately, the coefficient of x in the exponent is positive, and large (about 23 million /m), which means that the function goes steep to infinity...

I don't suppose this accurately models your phenomenon...where did I, or you, go wrong?

 

Success!
Luc

Re: Solve non linear second order differential equation with initial and boundary condition

Hi Luc,

            There is a slight confusion with your solution. The equation 4 that I had written is a modified version of equation 3 and is a boundary condition. Therefore I don't think phi (x,t) is independent of t.  The derivative of the expression may be 0 at the boundaries but not everywhere else. Volume fraction will change along the solution length as sedimentation takes place with time. I am attaching the mathcad file with my comments. Take a look.

 

Regards

          

Re: Solve non linear second order differential equation with initial and boundary condition

I guess you're right, I mistook the "x ∈ {0, xmax}" (= just 0 and xmax) for an "x ∈ [0, xmax]" (=the entire range from 0 to xmax).

 Sorry about that.

Other than that, Werner already explained that PDEsolve is not available in Prime, if you want to use PDEsolve you'll have to use (real) Mathcad.

{ Note that as a licensed user of Prime you're allowed to install and use Mathcad 15 on the same machine. You can use the very same license file that you used for licensing Prime to license the Mathcad 15 application. }

 

Success!
Luc

 

Re: Solve non linear second order differential equation with initial and boundary condition

Hi Luc,

            Thanks. Let me try it  in mathcad 15. I have never really used it. I'll post again if I have issues.

 

Regards

Re: Solve non linear second order differential equation with initial and boundary condition

You'll love the speed (once you've unlearnt the Prime habits).

Be aware that (real) Mathcad does not like units with its solvers, so you'll have to set up the stuff unitless (= you're advised to express every value in standard units).

 

Success!
Luc

 

Re: Solve non linear second order differential equation with initial and boundary condition

It may not all be correct, but this is what I get in Mathcad 11. I wonder if it's how you'd expect it to be:

LM_20190225_NonLinearPDEsolve.png

The file is attached. You should be able to open it in Mathcad 15, and the part that is shown in the picture here should be similar.

The symbolics manipulations may not work in Mathcad 15, but you should be able to make them work with some tweaking.

 

Success!
Luc

Re: Solve non linear second order differential equation with initial and boundary condition

Hi Luc,

             Sorry for the late reply. I am able to open the file that you sent me. But some how I am not able to see any of the evaluation or symbolic results. There is some issue with viewing mathcad 11 file in 15 (see attached). I have written everything in mathcad 15 but it is giving me an error when I do the pdesolve.

 

It says "This value must be a function, but has the form: Unitless." I am not sure why I am getting it. Can you please take a look at it. (File attached)

 

Also is there any way to plot phi vs x for a particular time say after 20 hrs. That will give me a better idea. Thanks for the help.

 

Regards

Re: Solve non linear second order differential equation with initial and boundary condition

As demonstrated by Image # 1, you didn't calculate. Open my Mathcad 11 file, scroll to the bottom, and press the F9 key on your keyboard once.

That should solve most, if not all of the issues.

(I usually work in the mode where the sheet does NOT calculate automatically, too keep manipulations editing quick...)

 

Don't hesitate to come back if there are problems left.

 

Success!
Luc