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amin


Total Posts: 259
Joined: Aug 2005
 
Posted: 2017-02-24 13:00
Here I do the complete analysis as I promised yesterday.

Equation(1)

When we are given initial values t0 and y(t0), it is obvious we can easily solve for dy/dt(y(t0),t0).
Let us take a total derivative of Equation(1), it reads


Equation(2)

please notice that I missed a term yesterday when I wrote the equation (2). We make the following observation that time dependent partial derivatives in third and fourth term of equation(2) can be combined to give the expression below.

Equation(3)
Now we can write the third and fourth term in equation (2) to give us the
Equation(4)

using the above expression in equation 2, we get
Equation(5)

which can be solved to give us a second order ODE given below.
Equation(6)

which can easily be solved by the method of iterated integrals if we can integrate the expressions in corresponding integrals.

We notice that at any given time tn, equation (1) has to be satisfied as

which means that everywhere on the surface F, the following equation would have to be true as well
Equation(7)
which makes our second order ODE in equation (6) to be solved as
Equation(8)

amin


Total Posts: 259
Joined: Aug 2005
 
Posted: 2017-03-11 16:39
Here is cross-reference to another related thread in off-topic for applications of the research in this technical forum thread. https://defence.pk/pdf/threads/proposal-for-a-nanotechnology-open-knowledge-research-effort.482885/
I thought some friends would be interested.
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