mlog

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These are replies submitted by mlog

@tomleslie 
Dear Tom,
thank you for your answer.

I want to do this in order to reduce the computation time required for numerical solution of nonlinear heat equation with Robin’s condition on one of the boundaries:

du/dt=f1(u)*d(f2(u)*du/dx)/dx
IC u(x,0)=f3(x)
BC1 du(0,t)/dx=f4(u) at x=0
BC2 u(1,t)=Const1 at x=1

I need lower dx in the vicinity of x=0 (where u changes rapidly) than at the rest of the interval (where it changes slowly).

My knowledge is insufficient, so for the moment I do it in a “dumb” way: solve the problem by steps, starting from a smaller interval x=0…L1 with small spacestep, and increase the interval (increase L1) as soon as u(L1/2) starts to vary noticeably.

I believe there must be a better solution included in Maple.

Sincerely, Max

@Carl Love thank you very much carl!

@vv thank you very much!

@Carl Love 

got it. thank you!

@vv 

Thank you for the explanation!
M.

@Markiyan Hirnyk 

Dear Prof. Hirnyk,

I did not want to overcharge this post with details, but, please, find below the requested information.

Denominator of F stands for reciprocal hydraulic permeability of colloidal system: https://en.wikipedia.org/wiki/Permeability_(earth_sciences)

Numerator of F stands for the first derivative of osmotic pressure of colloidal system with respect to its particle volume fraction: https://uk.wikipedia.org/wiki/%D0%9E%D1%81%D0%BC%D0%BE%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D1%82%D0%B8%D1%81%D0%BA

G stands for shear rate in this system under some constant shear stress:
https://en.wikipedia.org/wiki/Shear_rate

At the same value shear stress, shear rate can be quiet high (in diluted “liquid” system) and very low (in concentrated “solid” system):
https://www.sciencedirect.com/science/article/pii/S0009250901002913
https://www.sciencedirect.com/science/article/pii/S0377025798000949

I would like to say in advance that I am not ready to dispute here, whether application of discussed integral is meaningful for treatment of “real life” problems. I just wanted to get some suggestions about math.

I also want to thank you sincerely for your previous posts that helped me a lot in my previous work.

With best regards,
Maksym

@vv 

Dear VV,
thank you very much for your help!
The result is the same as mine “with the coefficient”.
Did I get it right?
Would it be correct to say that the “road map” for this kind of integration is to
(1) use procedures everywhere before plotting/evaluating the final function,
(2) convert all floating-point coefficients to rational numbers
(3) and get the normalization coefficient with the use of simplify(convert(f, rational)),
(4) then normalize the final function before plotting/evaluating with the help of denominator obtained at step (3)?
Sincerely,
M.

@Markiyan Hirnyk 

I’m sorry for the inconvenience.
Denominator of F stands for reciprocal hydraulic permeability of a system of colloidal particles. The units of permeability are m^2.
x denotes particle volume fraction in this system (dimensionless).
Units of b[1], b[2], b[3] are m^-2.
The functional dependence of permeability on particle volume fraction is partially measured in experiment (for high x) and partially deduced from Happel equation (for low x). Then all was fitted by a polymon (that is denominator of F).
Sincerely,
M.

@Markiyan Hirnyk 

Dear Professor Hirnyk,
thank you for your question! I hope it is. I try to describe mass transport, and the integral is a combination of viscosity, permeability, etc. (actually measured or evaluated).
For the moment, I do not know if my approach to description of real mass transport is correct.
I posted my question, because I do not know, which method of the integration yields the correct result. Since two presented results are different, I only can doubt in both of them.

I can reformulate my question: “What is the correct way to compute an integral of a monotonically and strongly decreasing (or increasing) function, which has inflection points on the interval of integration?”

Thank you again,
M.

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