MATLAB: Trying to Find the solution of trigonometric equation

#trigonometric equationeigenvalue equation

i am trying to solve Esi*tan(pi*tox/y)-Eox*cot(pi*tsi/(2*y))=0 for finding the value of y where Esi=11.68*8.85e-12; Eox=3.9*8.85e-12;

i am trying to find y for various tox and tsi combinations let for case 1:tox=1e-9; tsi=10e-9; now matlab is not giving any solution i am attaching my code for this. i think this equation has many solutions for a particular tox,tsi so may be that's why matlab is not giving the answers can any one give me the code by that i can get every or at least two three solutions (values of y)? (tox,tsi should be in nano meter scale)

Best Answer

Rather than using ‘solve’, you should try ‘fzero’ with various different estimated values for y obtained from doing a plot. This will give you a number of precise values for y. There should be an infinite number of possible solutions to this equation as y approaches zero. (Note: It might be more convenient to first find solutions for 1/y and then take their reciprocals.)

Related Solutions

MATLAB: Solution of trigonometric equation

This seems silly. Why are you using a 3 digit approximation to pi (and pi/2) here, then trying to solve for anything?

pi is already well defined in MATLAB.

format long g
pi
ans =
          3.14159265358979

And pi/2 is also well defined. You write it as

pi/2
ans =
           1.5707963267949

Of course, all of your other numbers are equally short, and I am sure, are not exactly values like 3.9 and 11.8.

Next, if you want results to have units of NANOMETERS, then setting tsi and tox to be on the order of 1e-9 is silly!

tox MUST have units of nanometers for your equations to make any sense at all. Therefore, tox is 1. Then x will also have units of nanometers, and it should be on the order of 1 too.

Eox = 3.9;
Esi = 11.8;
tox = 1;
tsi = 10;
x = 1:0.1:50;

So now everything will have units of nanometers. A number means something only to you. You must define the context, or a number is just a number. So if all numbers of interest are in the form of nanometers, then all is fine.

Regardless, it looks like you are trying to solve for all solutions of a problem that probably has infinitely many solutions.

So if we look for small x, then we see this plot:

ezplot(@(x) (tan(pi*tox./x).*tan(pi/2*tsi./(x)))-(3.9/11.8),[.01,2]);
ylim([-5 5])
refline(0,0)

Are you looking for the 4 largest solutions?

So if we instead re-parameterize your problem in terms of u=1/x, now looking for the smallest solutions, of infinitely many solutions, we will see this:

ezplot(@(u) (tan(pi*tox*u).*tan(pi/2*tsi*u))-(3.9/11.8),[0,.75]);
refline(0,0)

So for u=1/x no larger than 0.75 nanometers, we see the first 4 solutions, represented graphically. As x gets smaller, there will be infinitely many more solutions. But those are the possible solutions for the smallest values of u, and therefore, the largest values of x.

The difference is, when we try to plot this relation as a function of x, it gets nasty looking. Instead, things are more well behaved when we look for the smallest solutions in u, thus 1/x.

MATLAB: Solving symbolic trig equation

You can NEVER solve for those variables, at least as the equations are written.

First of all, z never even appears. But since you never try to solve for it, who cares?

But look at x + y. They always appear tegether, in EXACTLY the same way. In fact, we can transform the problem in a very simple way, by replacing (x+y) with z.

Now your problem reduces to:

syms z
eq1 = -sin(z)+cos(z-1.6)== 0
eq2 = cos(z)+cos(z-3.2)== 0

There is no more and no less information content in the problem. Surely you can agree, since x+y ALWAYS appear in exactly the same place, and always together.

solve(eq1,z)
ans =
pi/4 + 4/5
solve(eq2,z)
ans =
 -log(-exp(8i/5)*1i)*1i
  -log(exp(8i/5)*1i)*1i
  
vpa(ans)
ans =
 0.029203673205103380768678308360249
  -3.1123889803846898576939650749193

Solving eq1 for z, we find a simple solution. Solving eq2 for z, however, we get a completely different value for z. It is not really imaginary when we resolve it into a simpler numeric form. But see that it is not the first value we got for z.

So nothing in the universe will ever allow a solution to both equations for z, and therefore solving for x and y is just as impossible. It is very easy to write equations that have no solution. Surprise! You just did it.

Best Answer

Related Solutions

MATLAB: Solution of trigonometric equation

MATLAB: Solving symbolic trig equation

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