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Showing posts from November, 2013

Microwave-Generated Tropical Typhoons Unraveled

So, after a bunch of typhoons have struck the Philippines over the years, typhoon Maiyan being the latest and strongest, I've been hearing and seeing over some news media talks about a secret microwave weapon for generating tropical cyclones used by a rival neighboring country to wreak havoc on the Philippines. Well, it does seem logical. If microwaves can heat food, then it can be used to accelerate the heating of Pacific Ocean waters. Now let us not jump to that conclusion too fast. Before we make such a conclusion, let us first analyze the requirements and conditions for such a mechanism to work.
     Microwave, as any electronics engineer would know, is an electromagnetic wave with a frequency ranging from 1 Ghz to around 110 Ghz (L, S, C, X, K, Ku, Ka, V, W bands) that is typically propagated directionally if it were to reach a significant distance (noting that the free space loss = (4*pi*f*D/c)^2 -- the square relationship making its value very high at high frequencie…

Numerical Methods for Partial Differential Equations: A Simplified Unravelling

Liebmann Method


This is simply the Gauss-Seidel method applied for elliptic PDEs (i.e. B^2-4*A*C<0, usually steady-state).

A good example would be applications of the Laplace PDE to heat conduction.
(Working with Maths symbols is hard with a keyboard so I'll just do it on paper)



General steps in using numerical methods for solving PDEs:

Step 1: Transform the continuous PDE into a discrete representation.

Step 2: Isolate the present state variable (or the variable asked by the problem).

Step 3: Use overrelaxation if necessary to fasten convergence.












Explicit Method


This method is used for parabolic PDEs with a little problem in stability. (i.e. B^2-4*A*C=0).

A good example would be applications of the Fourier's Law of Heat Conduction.




                    The problem above differs from the plate problem due to the fact that we are dealing with unbounded sides. Also, the explicit method is only stable when the "constant" (that is, 0.02087 we've used in the problem) is less …