Need some clarification here. Are you looking at

\int_0^{\frac{\pi}{dx} f(k;eb)dk - \int_0^{\frac{\pi}{dx} f(k;ea)dk, where ea, eb are parameters, k is the independent variable and 'dx' is a constant?

Hi,

The question is somewhat confusing.

Are Ea and Eb the independent variables of each function? If yes, then what are xa and xb?

Are the two functions behaving the same (same mathematical shapes)?

Is there any relation between Ea and Eb?

Is k a parameter? If yes, then what is the meaning of dx in the denominator of the limit pi/dx?

It would be helpful if you can give an explicit formulation of the problem.

Hi

Thank you for your reply.

I am looking at the following:

area = \int_0^{\frac{\pi}{dx} f(k;eb;dxa)dk - \int_0^{\frac{\pi}{dx} f(k;ea;dxb)dk,

ea, dxa, eb, and exb are related to the model parameters of a and b. There are the dispersion relations for 2 models.

dx is not a constant per se. The limit of k is defined by pi/dx, so for model A dx = dxa, and for model B dx= dxb. I am trying out a generalized case wherein I do not have to assume a value for dx.

I have an update. I squared the 'area' and differentiated wrt eb and equated it to 0 and now I think I have arrived at an equation that gives me the value of eb for any ea, dxa, dx and dxb values for which the area would be minimum.

However, when I was differentiating 'area' wrt eb without squaring it, I was not getting anythign because the first term which is devoid of eb was just getting cancelled out. What do you think?

Hamed Shakouri G. Hi,

Ea and Eb are independent. They represent the Young's moduli of models A and b, respectively. xa and xb represent the cell discretization lengths of models A and B, respectively.

No relation between Ea and Eb.

Both the curves are a fuction of k as they are plotted in the k-w curve.

pi/dx represents the limit of k. For model A, the limit is k/dxa and for model b it is k/dxb. I did not want to assume any value of dx and wanted to solve it for a generalized case wherein dx can be any value.

Just to clarify once more: are you squaring the integral error, or are you integrating squared error? Assuming that you are squaring the integral error: if your integral was g(alpha)+ f(beta) and you are differentiating with respect to beta, the derivative is f'(beta). However, if you are minimizing square of this quantity, the derivative is 2g(alpha)+2f(beta)f'(beta), setting which to zero might give a value of beta in terms of alpha.

Paresh Date

I tried both, i.e. integrating the squared error and squaring the integral error. I am of the belief that the former is not a correct approach and does not yield appropriate results.

When squaring the integral error I am observing the same pattern as you have mentioned. My question is that is it something normal that we have to do in order to attain minimum/maximum conditions in cases such as these (where the integral is given by g(alpha)+f(beta))? Can we say that we are manipulating the data to go ahead with the task of minimizing? Or, this falls under a known approach/method wherein you have to square the data before minimizing/maximizing?

Thank you

Hello Siddhesh Raorane

Express the integrand as (Ea-Eb)∂f/∂Eb

where f = f(k, Eb, dxb)

Then integrate it wrt k fom 0 to π /dx

Theerafter, differentiate the definite integral wrt Eb and equate to zero.

The solution of the resulting equation in terms of Eb will give the desired solution.