Numerical Analysis

Archive for October, 2010

Mathematica NDSolve of a third order nonlinear BVP

(i accidentally posted this on sci.math.symbolic. sorry for dupes.)

Hey all–
I’m extremely new to Mathematica, but an desperately trying to
numerically solve a 3rd order, non linear differential equation:

(h[x]/hf)^3*h”’[x] == (1 – (h[x]/hf)^3)*ohmegasqr*hf/w – s”’[x]
h[x1] == h[x2] == hf, h’[x1] == 0

x1, x2, hf, ohmegasqr, and w are defined values.
s[x] is a defined function.

Unfortunately, I keep getting a "Infinite expression 1/0.^3
encountered." and then "NDSolve::ndnum: Encountered non-numerical
value for a derivative at x == -0.0508. >>" error. I’ve traced the
problem to the initial (h[x]/hf)^3 term, but don’t really know how to
fix it.

Any ideas? I’ve pasted the content of the notebook below, and the
actual Notebook file below that.

Thanks,
David

(* hf=film thickness.Likely,thickness of SU8*)
hf = 4*10^-6;(* meters *)
(* w=width/length of feature along spinning direction (+x). Likely, \
length of funnel *)
w = 100*10^-6;(* meters *)
(* ro=radial position of feature on wafer *)
ro = 2.5*10^-2;(* meters *)
(* rf=radius of wafer,for normalization of x for fcn s *)
rf = 0.0508;(* meters *)
(* precision for step *)
xstep = 0.001;
(* d=feature step height *)
d = 1.05*10^-6;
(* solving bounds *)
x1 = -rf;
x2 = rf;
s[x_] := -d/\[Pi] (ArcTan[(x/w - 1/2)/xstep] +
     ArcTan[(-(x/w) - 1/2)/xstep]);
Plot[s[x], {x, -w, w}, PlotRange -> {0, hf}, AxesLabel -> {"x", "s"}]
(* rot=rotational speed. Likely,SU8 spinning speed *)
rot = (2 \[Pi])/
  60*4100;(* radians/second *)
(* rho=fluid density.Density *)
rho = 1000;(* kg/m^3 *)
(* gamma=surface tension of fluid *)
gamma = 0.03;(* N/m *)
ohmegasqr = (rho*w^3*rot^2*ro)/(hf*gamma)
solution =
  NDSolve[{(h[x]/hf)^3*h”’[x] == (1 – (h[x]/hf)^3)*ohmegasqr*hf/w –
      s”’[x], h[x1] == h[x2] == hf, h’[x1] == 0}, h, {x, x1, x2}];
Plot[h[x] /. solution, {x, x1, x2}]
—-
(* Content-type: application/mathematica *)
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(* http://www.wolfram.com/nb *)
(* CreatedBy=’Mathematica 6.0′ *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
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NotebookDataPosition[       145,          7]
NotebookDataLength[      9811,        259]
NotebookOptionsPosition[      9564,        246]
NotebookOutlinePosition[      9899,        261]
CellTagsIndexPosition[      9856,        258]
WindowFrame->Normal
ContainsDynamic->False*)
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3.413114543058976*^9}, {

read more »

.
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Are you want the detailed astronomical physics

Are you want the detailed astronomical physics
know about the astronomical physics(about NASA articles)
want to share the knowledge of astronomical physics…
here know about the astronomy

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Textbook Key

Hi,

I am looking for the key to Introduction to Chemical Processes:
Principles, Analysis, Synthesis By Regina Murphy.  Please contact me
at kawaee878…@yahoo.com.  Thanks!

posted by admin in Uncategorized and have No Comments

SIGNS OF PROPHETHOOD

Please forgive us for any disturbance, but we have an important
subject
to address to you regarding FAITH, and we Don’t intend to overload
your email with unnecessary messages…

I hope you don’t mind receiving my letter. It is short but full of
information.

SIGNS OF PROPHETHOOD

Since Prophethood is a means by which the noblest of all knowledge can
be acquired and the greatest and most honourable deeds can be carried
out; it is of

Allaah’s mercy to give these Prophets the signs by which they could be
known and through which they could be identified, though, every one of
those who claimed to have a mission has some signs and conditions that
could indicate his truthfulness if he is truthful or otherwise if he
is a liar.  Theses signs are many, the most important of which are:
1.                 The Messenger should call to the worship of Allaah
alone and to the abandonment of worshipping others besides Him. For
this is the purpose for which Allaah created the creation.
2.                 The Messenger should call people to believe in him,
and put his message into practice. Allaah commands His messenger,
Muhammad to say, "O mankind! Verily, I am sent to you all as the
Messenger of Allaah."[1]3.                 Allaah strengthens him with
various kinds of signs of prophethood. Among these signs are the
verses that the Prophet Muhammad brought from Allaah and which his
people could not repudiate or produce its like. Another example is the
sign of Prophet Moosaa when his rod turned to a snake; the sign of
Prophet Eesaa who healed the blind and the leper by the permission of
Allaah, and the sign of Muhammad which is the Great Qur’aan, in spite
of the fact that he was an illiterate who could neither read nor write
among many other signs of the Prophets.Among these signs is the clear
and vivid truth which the Prophets and Messengers brought and which
their opponents could not disprove or deny. Rather, their opponents
knew that what the Prophets had brought was the truth that could not
be resisted. Among the signs also are perfect conditions, beautiful
traits and magnanimous conduct with which Allaah distinguished the
Prophets.4.                 His message should agree in fundamentals
with the fundamentals to which all Messengers and Prophets called.
[2]5. He should not call others to the worship of himself or to direct
any act of worship to him. He should also not call to the
glorification of his tribe or clan. Allaah commands His Prophet
Muhammad to tell people, "Say: I do not tell you that with me is the
treasure of Allaah, nor (that) I know the Unseen; nor I tell you that
I am an angel. I but follow what is revealed to me."[3]6. He should
not ask from people any of the things of this world as a wage for his
mission. Allaah says, informing us about His Prophets Noah, Hood,
Saalih, Loot and Shuaib, that they told their people, "No reward do I
ask of you for it (my message of Islaamic Monotheism); my reward is
only from the Lord of all that exists."[4]Prophet Muhammad also told
his people, "No wage do I ask of you for this (the Qur’aan), nor am I
of those who pretend and fabricate things."[5]These Prophets and
Messengers whom I have told you something of their qualities and signs
of their prophethood, are many. Allaah says, "And verily, We have sent
among every nation a Messenger (proclaiming): Worship Allaah (alone)
and avoid all false deities."[6]
These Messengers have made mankind happy; the history recorded their
stories and the injunctions of their religion were repeatedly
transmitted, that they are the truth and just. Likewise, repeatedly
narrated is the victory with which Allaah honoured them and
destruction of their enemies, like the destruction of the people of
Noah with flood, drowning of Pharaoh, punishment for the people of
Loot; and the victory of Muhammad over his enemies and the spread of
his religion. Whoever knows this will know certainly that they brought
for mankind good and guidance; that they directed them to all that
could benefit them and warn them against all that could be harmful.
The first of them is Adam and the last is Muhammad.

 ByDr. Muhammad bin Abdullaah As-SuhaymTranslated byAbdur-Raafi
Adewale Imaam

——————————————————————————–

[1] Al-A’raaf 7:158.
[2] See: Majmoo’ Fataawa Ibn Taymiyyah Vol. 4  p. 212-213.
[3] Al-An’aam 6:50.
[4] Ash-Shu’araa 26:109, 127, 145, 164, 180.
[5] Saad 38:86.
[6] An-Nahl 16:36.

For more information about Islam
www.imanway1.com

http://www.freewebs.com/mu7ammad/1.htm

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Differential Equation

Is there a known function that satisfies the differential equation

            f”(x) – (a+bx)f(x) = 0 ,   a><0, b><0

?

D. Baruth


Posted via a free Usenet account from http://www.teranews.com

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Naive question about the Lanczos biorthogonalization procedure.

Hello.

Is there a class of matrices of non symmetric matrices for which the
Lanczos biorthogonalization procedure is garanteed not to break down
prematurely? It does not matter if the restrictions are very tight, like
  A normal, negative definite or even more.

Thank you

Carl Christian K. Mikkelsen

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generating 3D Gaussians question (for a BD simulation)

Hi,

I am new to this group and I have probably relatively simple question-
how does one go about generating 3D Gaussian vectors with say a mean
of say C_1 and a variance of C_2?

And what would I need to do if I wanted to generate two of these which
are correlated? i.e. have a covariance <R.V> = C_3.

Thanks in advance,

Carter.

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Perturbation Theory Question

Hi,

I am curious how one validates that a perturbation treatment for the
solution of an algebraic eqn actually works. This problem has arisen
in the context of a simple problem that has a somewhat peculiar twist,
which I’ll describe below:

I am solving an eqn that has the general form:

a = y*(2-y)^2 / { (2-y) + b }

where a is being treated as a constant and b is being treated as a
parameter which can have values on [0,+Inf)

When b=0, the solution y=2 can be tossed out (on physical grounds
related to some particular aspects of this specific problem), and I
can solve the remaining quadratic eqn for y quite simply. Due to the
nature of the problem, the smallest positive value of y is chosen as
our solution (which in this case, corresponds to the solution with the
negative root). Let me denote the solution chosen when b=0 as y0.

When b is not zero, it is expected (due to the physics of the problem)
that the solution for y will increase with b. Hence, it seems
*plausible* that it should be possible to construct a power series
solution for y in terms of b having the form:
y = y0 + a1*b + a2*b^2 +….

This sort of approach is typically used in perturbation analysis,
however, I have two serious concerns in this case that I need to
address:
#1) b is not necessarily small (relative to the coeffs of the other
terms in the cubic eqn)
#2) The "perturbation" (if you wanted to call b that) occurs in the
constant term. I’ve never seen a perturbation analysis of solutions
for algebraic equations where this type of case is handled.

I am going to test this type of solution and see if it works for some
examples, but I would like to know in general if it is valid. Is there
a test of some type I can use for a problem like this?

MTIA,

Matt B.

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stability criterion for implicit integration method

I want to know what does the mean term ‘unconditionaly stable’ (when
wee talk about some method for integration differential eq in time
direction) and what are the stability criterions for such methods.

I’m also interested in procedures to set appropriate time step for
these method (especialy in FEM)

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