A system of differential equations
Mehran Basti
Dear Sir/Madam:
We have noted that for each polynomial , we can assign a differential
equation. The broad idea is also applicable for solving systems of
differential equations.
These systems will also occur in our studies of solving polynomials,
with associated systems of differential equations with respect to
coefficients a0(t),a1(t),a2(t),etc.
The following is a system of differential equations resulted from a
problem in biological studies , and assigned to me by a faculty member
of a University in Canada, in 1982-83.
Consider a system of differential equations in (x1(t),x2(t)),where
x1(t) ,x2(t), are unknown functions, and beta(t) gamma(t) are all real
functions of t , not complex valued, lambda also a real number:
eq1 := diff(x1(t),t) =
lambda*x1(t)-2*beta(t)*x1(t)*x2(t)+gamma(t)*(x2(t)^2-
x1(t)^2);
eq2 := diff(x2(t),t) =
lambda*x2(t)-2*gamma(t)*x1(t)*x2(t)+beta(t)*(x1(t)^2-
x2(t)^2);
Let us consider the function:
s1 := y(t) = x1(t)+n*x2(t);
such that, y(t) will satisfy in a Bernoulli differential equation (n
is a number):
DE := diff(y(t),t)+P*y(t)+Q*y(t)^2 = 0;
where P,Q are functions of t, independent of x1(t) and x2(t).
We will soon find out that, one set of solutions can be the following:
sol := {P = -lambda, n = (-1)^(1/2), Q =
-(-1)^(1/2)*beta(t)+gamma(t)};
Thus we will have :
DE1 := diff(y(t),t)-lambda*y(t)+(-(-1)^(1/2)*beta(t)+gamma(t))*y(t)^2
= 0;
The above equation has a complex valued solution:
s2 := y(t) =
exp(lambda*t)/(-(-1)^(1/2)*int(exp(lambda*t)*beta(t),t)+int(exp
(lambda*t)*gamma(t),t)+C1+(-1)^(1/2)*C2);
where all of its elements are real numbers, such as C1 , C2, and
lambda, I selected the constant of integration as C=C1+I*C2.
Adjusting to real valued functions x1(t) and x2(t) (as requested ), we
will have:
s3 := x1(t) =
exp(lambda*t)*(int(exp(lambda*t)*gamma(t),t)+C1)/((int(exp(
lambda*t)*gamma(t),t)+C1)^2+(-int(exp(lambda*t)*beta(t),t)+C2)^2);
s4 := x2(t) =
-exp(lambda*t)*(-int(exp(lambda*t)*beta(t),t)+C2)/((int(exp(
lambda*t)*gamma(t),t)+C1)^2+(-int(exp(lambda*t)*beta(t),t)+C2)^2);
Coordination of problems algebra , differential equations, calculus,
is a key issue in our studies.
Sincerely,
Dr.Mehran Basti
David Wilkinson
author. Why can't Dr Basti publish it in the usual way and give us all a
break from it? Other people don't write endless tracts in their special
areas so why should he?
In article <cbuklh...@forum.mathforum.com>, Mehran Basti
<Mehb...@aol.com> writes
--
David Wilkinson
Mehran Basti
Dear Sir/Madam:
Thank you for your comment.
Perhaps you would like to be more patient!.
The reason is already mentioned.
Since 1985 , not only I have not found a career in academic
institutions, but also no one showed interest in my field of research
and after advertising Exact Analysis in many communities of
mathematics, in the world, and demonstrating, in the math-seminars,
the situation remained the same!.
I have noted before , since my work is related to symbolic
computations, this is an avenue to find a public support. And again
thanks to this newsgroup for understanding.
After all I am also demonstrating in this newsgroup that I am
interested to get a support for a new software to compete with other
software in the market.
And this first of all requires intellectual support of this community.
If you do not have interest in my new research area, this does not
mean others feel the same.
I am sure many are curious to know how I am solving polynomials ,
associated Ledendre,etc. differently.
I have had difficulties publishing my Exact Analysis (1985),and
statements got to be publicly mentioned and the community of
mathematicians thus should feel the outcome.
Although my statements here, will help publishing Exact Analysis , in
the form it may be accepted by everyone .
Publishing a revulsionary paper, such as mine, is different than many
others in the journals.
It requires many factors by Universities and editors, and always takes
a very long time,to see something totally different is presented here,
than usual papers in the journals,(it has been sixteen years!).
I am not undermining outstanding works in the journals by others.
Once it is claimed that new ways to solve polynomials are available,
and methods may have some affect in the future of our research in
Galois theory or Newtons methods, it may show some reactions in
Institute of Henry Poincare’ in France or Newtons Institute at
Cambridge (particularly I am a Cambridge Graduate).
It seems time has finally come for me to get some exposure about my
research.
This awareness by this newsgroup is essential for the breakthrough in
my situation, and advancements in the path of scientific progress.
Sincerely,
Dr.Mehran Basti
kenneth Schatten
make some helpful comments.
I think perhaps you are both right, to a degree. Certainly there is room in
this format, considering the wealth of material, to allow suggestions, such
as Dr. Basti's to be aired, rather than taking up space in the journals with
might be repetitive, wrong, or whatever, that is NOT acceptable to journals
in the normal way of publishing. There certainly is a lot of "SPAMMING", and
at least this is not that.
So, although i have not examined how often Dr. Basti has made submissions
such as this, assuming it to be infrequent, it is not a bad way to air his
ideas. However, he might, for his benefit write to the people who are making
criticisms, or listen to what they are saying, to ascertain why they feel
his ideas are worthless, or of little value. Certainly there is, we would
all agree, no point to reinventing the wheel, so even if the math is
"correct" , it may be uninteresting.
best wishes to all.
ken
"Mehran Basti" <Mehb...@aol.com> wrote in message
news:cbuklh...@forum.mathforum.com...
Mehran Basti
Dear Sir/Madam:
Thank you for your very valuable advise. I keep it in mind, to post on
the same topic.
Also thanks for recent note on this topic
Thus I also repeat my comment on this topic.
About my note in systems of differential equations, from your angle it
may be uninteresting, but once you get to know how I solve
polynomials, you will see that , the methods involves many systems of
differential equation with respect to a0(t), a1(t), etc.
This will occur in our first differentiation of the polynomial with
respect to t:
p:=x^5+a4(t) x^3 +a3(t) x^3+ a2(t) x^2 +a1(t) x+ a0(t) =0
and other differentiations.
We will gradually get to know how to handle systems, surprisingly we
will , otherwise I would not be able to solve a polynomial of degree
15 , with 20 parameters ( I am in a such world).
Thanks
Dr.Mehran Basti