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None
science forum addict
Joined: 12 May 2005
Posts: 99
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 Posted: Sun Mar 12, 2006 12:48 pm Post subject:
Solution for some types of nonlinear wave equation
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Dear Colleagues,
We are pleased to draw your attention to our new paper
The method to seek the solution for some types of nonlinear wave
equation
Abstract:
We will show the technique to seek the solution for nonlinear wave
equations like u'' + g(t)u = 0 and u'' + g(u)u = 0. We also
will consider the possibility to broaden the application area of this
technique for other types of nonlinear wave equations.
Enjoy reading full text here:
http://selftrans.narod.ru/v6_1/contents6_1.html#method
We will be pleased to hear your responds.
Best to you all,
Sergey B. Karavashkin
Head Laboratory SELF
187 apt., 38 bldg.
Prospect Gagarina
Kharkov 61140
Ukraine
Phone: +38 (057) 7370624
e-mail: selftrans@yandex.ru , selflab@mail.ru
http://selftrans.narod.ru/SELFlab/index.html |
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thomas.davidson@dla.mil
science forum beginner
Joined: 07 Feb 2006
Posts: 47
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| Posted: Sun Mar 12, 2006 10:14 pm Post subject:
Re: Solution for some types of nonlinear wave equation
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Was there something about the chapter on Greens Functions that you
missed?
http://www.math.lsa.umich.edu/~vbooth/gf.pdf
Tom Davidson
Richmond, VA |
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None
science forum addict
Joined: 12 May 2005
Posts: 99
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| Posted: Wed Mar 15, 2006 4:13 pm Post subject:
Re: Solution for some types of nonlinear wave equation
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tadchem wrote:
Dear Tom, first of all thank you for this paper. Its style is fine and
formulation strong. Just it shows the difference in our approach and
the approach when solving with the Green function. We omitted this
part, doing not wanting to dive into additional explanation, why the
solutions yielded with the Green function are factually erroneous. Yes,
really so, and kindly understand what I mean.
To simplify, let us take the first problem of your paper. It is known
that "the Green function has a simple physical meaning. It is the
solution for a single source
f(x) = delta (x-x0) " (in the symbols of your paper)
[M.L. Krasnov. Integral equations].
The Green function G (x, x0) will satisfy the solution if
L G (x, x0) = delta(x-x0) , (1)
where L is the operator of a homogeneous differential equation related
to the given heterogeneous equation.
This is the equation which the Green function has factually to satisfy.
And what they conventionally do? Both in your paper and usually, the
solution is sought for the equation
L G (x, x0) = 0 (2)
on the right and left from the point x0; then they join the solutions,
basing on the artificial stipulation that the function is continuous at
the very point. And your paper solves, proceeding from it. But (2) does
not correspond to (1) namely at x0. At this point (1) becomes
heterogeneous! And the Green function found with the conventional
joining techniques is unable to account this basic feature, we can
easily make it sure. Take any Green function determined so and
substitute it to (1) at x = x0. You will yield zero in the WHOLE
interval, including the point x0, not the delta-function. This is not a
solution, as, when you affect the solution by the operator of
differential equation L, it turns the Green function to zero, while it
has to give a delta-function; multiplying it into the right-part
function of the equation and integrating, you have to yield the right
part of the initial differential equation. Thus, it only seems that you
seek the solution of heterogeneous differential equation. In fact, you
find some kind of solution of the homogeneous equation. ;-)
So the solutions found by our technique will generally differ from the
numerical solutions yielded with the Green function.
You can see from our material that we even have not this difficulty and
yield analytic solutions without delta-function. And in many cases
yield them even not as a Fourier-expansion but analytically, which is
basically impossible in case of Green function. And this last is shown
in our paper.
But perhaps you are right, we would have to express it in a detailed
proof. Well, we will do the supplement to this paper and then tell you.
I would hope that you will attentively analyse what I said, as I
understand how unexpected and unusual is it for you.
Best,
Sergey |
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Dirk Van de moortel
science forum Guru
Joined: 01 May 2005
Posts: 3019
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| Posted: Wed Mar 15, 2006 4:50 pm Post subject:
Re: Solution for some types of nonlinear wave equation
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"Sergey Karavashkin" <selftrans@yandex.ru> wrote in message news:1142439198.444756.163600@j52g2000cwj.googlegroups.com...
| Quote: |
tadchem wrote:
Was there something about the chapter on Greens Functions that you
missed?
http://www.math.lsa.umich.edu/~vbooth/gf.pdf
Tom Davidson
Richmond, VA
Dear Tom, first of all thank you for this paper. Its style is fine and
formulation strong. Just it shows the difference in our approach and
the approach when solving with the Green function.
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And it doesn't make deliberate mistakes like showing functions
with non-zero curl(grads):
http://selftrans.narod.ru/v4_1/grad/grad02/grad02.html
Dirk Vdm |
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None
science forum addict
Joined: 12 May 2005
Posts: 99
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| Posted: Thu Mar 16, 2006 10:15 am Post subject:
Re: Solution for some types of nonlinear wave equation
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Dirk Van de moortel wrote:
| Quote: | "Sergey Karavashkin" <selftrans@yandex.ru> wrote in message news:1142439198.444756.163600@j52g2000cwj.googlegroups.com...
tadchem wrote:
Was there something about the chapter on Greens Functions that you
missed?
http://www.math.lsa.umich.edu/~vbooth/gf.pdf
Tom Davidson
Richmond, VA
Dear Tom, first of all thank you for this paper. Its style is fine and
formulation strong. Just it shows the difference in our approach and
the approach when solving with the Green function.
And it doesn't make deliberate mistakes like showing functions
with non-zero curl(grads):
http://selftrans.narod.ru/v4_1/grad/grad02/grad02.html
Dirk Vdm
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Your boss also has no answers to these questions. Thank you for
information, Dirk.  And your head was and is full of dust.
Sergey |
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None
science forum addict
Joined: 12 May 2005
Posts: 99
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| Posted: Fri Mar 17, 2006 6:33 pm Post subject:
Re: Solution for some types of nonlinear wave equation
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tadchem wrote:
Dear Tom, I wrote the promised supplement to the paper,
"On the Green function"
http://selftrans.narod.ru/v6_1/nonlinwaveeq/tadchem/tadchem.html
please read this more detailed proof and make sure that I was correct
in my first reply.
Best,
Sergey |
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