Query on indexing to determine coefficients

[bugatti79]

Folks,
I am interested to know what the author is doing in the following

$\displaystyle B_{ij}=EL ij (L)^{(i+j-1)} \left[ \frac{(i-1)(j-1)}{i+j-3} -\frac{2(ij-1)}{i+j-2}+\frac{(i+1)(j+1)}{i+j-1}\right]$

he states that this expression is not valid for $B_{ij}$ when $i=1$ and $j=1,2,...N$

...yet he goes on to actually calculate

$B_{11}=4EIL$, $B_{1j}=B_{j1}=2EIL^j$, $(j>1)$

I understand the the numerator in the first 2 terms inside the big brakets are both 0 when i=j=1 but we still yield a value from the third term...
Any insight will be appreciated
Regards

PS:I notice there is some editing problem with the 3 terms inside the big brackets. There should be a minus and plus separating the terms.

 

[Mark44]

Folks,
I am interested to know what the author is doing in the following

$\displaystyle B_{ij}=EL ij (L)^{(i+j-1)} \left[ \frac{(i-1)(j-1)}{i+j-3} -\frac{2(ij-1)}{i+j-2}+\frac{(i+1)(j+1)}{i+j-1}\right]$

he states that this expression is not valid for $B_{ij}$ when $i=1$ and $j=1,2,...N$

That's not at all obvious. The only restrictions I see are that
1. i + j ≠ 3 (would make the first denominator vanish)
2. i + j ≠ 2 (would make the second denominator vanish)
3. i + j ≠ 1 (would make the third denominator vanish)

...yet he goes on to actually calculate

$B_{11}=4EIL$, $B_{1j}=B_{j1}=2EIL^j$, $(j>1)$

I don't see how. With i = 1, j = 1, the second term in the brackets is 0/0.

I understand the the numerator in the first 2 terms inside the big brakets are both 0 when i=j=1 but we still yield a value from the third term...
Any insight will be appreciated
Regards

PS:I notice there is some editing problem with the 3 terms inside the big brackets. There should be a minus and plus separating the terms.

 

[micromass]

Could you give a link to where you found this question?

 

[bugatti79]

I don't see how. With i = 1, j = 1, the second term in the brackets is 0/0.

Are you saying because one of the terms is indeterminate then the whole equation is invalid and thus cannot be usedt o calcualte $B_{ij}$ for i=j=1?

Could you give a link to where you found this question?

See attached jpeg of question. The answer involves converting the DE into a weak form using a weight function w and splitting the differentiation between the weight function and the dependent variable u.
Would the choice of the approximation functions affect the outcome?

regards

 

[CellCoree]

It seems that the author is using indexing to determine the coefficients for the expression provided. This is a common method in mathematics and science to simplify complex equations and make them easier to understand and manipulate. In this case, the author is using the indices i and j to represent different variables in the equation.

However, it is important to note that this expression is not valid for certain values of i and j, specifically when i=1 and j=1,2,...N. This means that the author's calculations for B_{ij} in these cases may not be accurate.

It is possible that the author has made a mistake in their calculations for B_{11} and B_{1j}, as the first two terms inside the big brackets should result in a 0 value when i=j=1. However, the third term may still yield a value and this could be the reason for the discrepancy.

It would be helpful to have more information about the context in which this expression is being used, as well as any assumptions or limitations that the author may have mentioned. It may also be worth double-checking the calculations to ensure their accuracy.

Overall, indexing can be a useful tool for simplifying and organizing equations, but it is important to be aware of its limitations and potential for error.