Greetings mathematicians and physicists,
As we know, the idea of assuming a higher space dimension than the natural 3 dimension is effective in solving many physical and mathematical problems.
Here, I'll pose some philosophical questions about " time ", and I don't know where they'll lead us when we try to answer them.
I look forward to hearing from you soon. Thank you so much.
Best from Algeria,
Khaldi Said, Phd student in mathematics.
I want to focus only on the last questions. The role of the indipendent variables in a PDE is just like x1,x2,...xN, if you want you can assign as "time" any of those..
The PDE you wrote is a linear first order equation with a production term. Just assume the form Du/Dt=u assuming unitary positive transport velocity.
These questions are tricky in physics but not sure pure maths have limitations:
1. Isn't because time flows only toward the future or past, which makes it stays on one axis, hence, 1 single axis or dimension?
2. If someone's time flows in several time axes, it basically means the person leaves in parallel time realities simultaneously, I have not heard of an experiment proving this, my understanding is that the Einstein general relativity proves time can flow (faster or slower) due to gravity (slower as you approach a black hole for instance, shown in the Schwarzschild's equation), maybe I am wrong.
3. Unless you have specific cases with limited intervals and boundaries, the maths solving methods should apply, for instance, what will be wrong with the Laplace Transform to solve your point 4. equation?
4. Use the Laplace transform.
Thank you so much Prof. Filippo Maria Denaro for take your time, your interest and quick response.
Let me explain my philosophy on 'time duality" as much as possible.
As we know, there are two types of differential equations, the first one is ODE where the unknown depends only on single varaible (time in general), the second type is PDE where the unkonwn depends on more than two varaibels (space, or space-time in general).
Moreover, in PDEs, we distinguish between two types, the first one is stationary (only space varaible) such as Laplace or Poisson equations, while the second one is evolutionarry or nonstationary (both space-time varaibles) such as heat, plate or wave equations.
In this last type, the time varaible plays a great importance from the poit of view of the qualitatives properties of solutions, for example to prove the following notions (global existence, blowup or local existence, decay estimates of energy....and so on).
Because my research field based on the above notions, I tried to imagine "time duality (t,s)" from the point of view of pure mathematics, but I don't know how to imagine it in our world.
In this case, the ODE becomes PDE, because it contains some partial derivatives with respect to our new time varaibels (t,s).
Simply, the solution of the following ODE: U_t(t)+U(t)=0, U(0)=A is U(t)=Ae^-t.
For this reason I ask:
What is the exact solution of PDE stated in (4)?
What is the solution of the following modified heat equation: U_t(t,s,x)+U_s(t,s,x)-\Delta U(t,s,x)=0, U(0,0,x)=u_0(x), t,s>0, x\in R^3 ?
My respects,
Best from Algeria,
Khaldi Said
Thank you so much Prof. Dieudonne Ndong Ovono for take your time, your interest and quick response.
Yes, I agree with you that there are no limits of pure mathematics for solving many pure Cauchy problems. I should emphasize that my new vision of "time duality" is purely mathematical, and I have no idea how to picture it in our real world.
Now, I want to focuse my vision on how to catch the exact solution of the PDE based on "time duality" and look for some qualitative properties of it.
Yes, to do this we can use Laplace and Fourier transforms as you proposed.
Here, "time duality" is a mathematical trick like high space dimensions n=4, 20, 100...
My respects,
Best from Algeria,
Khaldi Said
Dear Khaldi Said
in pure mathematics you don't necessarily think in terms of time and space but only in terms of dependent and independent variables. A PDE is a functional relation involving x1, x2, ...xn independent variables.
PDEs are not classified in terms of stationary or non stationary equations but as hyperbolic, parabolic, elliptic equations.
For example you can have hyperbolic equations for both steady and unsteady "physical" problems. The same happens for other types of equations.
Thus, just think of any of your equation in terms of x1,x2,x3 variables, classify the type of PDE and then you could interpret x1=t, x2=s, x3=x.
If s is your "second time" just see the mathematical solution in a time-time domain. Has it a physical correspondance? I don't know, in theoretical physics there are some theories for hyperspace.
Best regards
Indeed, Laplace or Fourrier transforms according to the spacial domaine. I did not try, but you can also investigate Double Laplace-Sumudu Transform of Derivatives (also, Sumudu-Kamal, Gupta, Laplace-Aboodh, etc)
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