Problem proving two statements by using summation by parts.

[Bede]

Hi everyone. I have two problems I do not manage to solve.
- Problem number 1:
I am supposed to use summation by parts to prove that

V_j^m*(V_j-1^m-2V_j^m+V_j+1^m)=-

(V_j+1-V_j^m)<sup>2

- Problem number 2:
In this problem, we are given the fact that a*b is less or equal to 0.5*(a^2+b^2). This might be used to prove that:</sup>

[(V_j^m+1)+0,5*([(V_j-1^m)²+(V_j+1^m)²]] is less or equal to

[(V_j^m+1)²+(V_j^m)²].

m refers to time-step number, so does not mean raised to the m-power, but m refers to the current time-step, while m+1 refers to the next time-step. j refers to distance-step number.
I hope someone can help me with these two problems. Thank you.

 

[Ishuda]

Hi everyone. I have two problems I do not manage to solve.
- Problem number 1:
I am supposed to use summation by parts to prove that

V_j^m*(V_j-1^m-2V_j+V_j+1^m)=-

(V_j+1-V_j^m)<sup>2

- Problem number 2:
In this problem, we are given the fact that a*b is less or equal to 0.5*(a^2+b^2). This might be used to prove that:</sup>

[(V_j^m+1)+0,5*([(V_j-1^m)²+(V_j+1^m)²]] is less or equal to

[(V_j^m+1)²+(V_j^m)²].

m refers to time-step number, so does not mean raised to the m-power, but m refers to the current time-step, while m+1 refers to the next time-step. j refers to distance-step number.
I hope someone can help me with these two problems. Thank you.

What is the summation over, the j or the m or both? What are the indices for the summation, i.e. j=1 to j=n_j; m=1 to n_m? What does a V_j without an m mean?

 

[Bede]

The summation is j from 1 to n. So it is only summation sign for j. The v was supposed to have an m over it.