MHB Understanding the Heat Equation: What Does $T_j^n$ Represent?

[nacho-man]

Please refer to the attached image,
Question 1, which i have pointing an arrow at.Is this simply asking me to sub in t=0 into (5),
which would leave me with $B_{l}\sin(\pi l x)$ inside the sum?

would they expect anything further?

Thanks!

 

[MarkFL]

I recommend when you have such a wide image, to edit it to put the portion on the right underneath. As it is it goes way off the page.

 

[Dustinsfl]

After reading the question, I am not even sure what they want to achieve. Additionally, since the book says use excel, it should be burned. What mathematician, engineer, or physicists writes a book and says let's use excel?

 

[nacho-man]

The excel part is just to examine the solutions we obtain.

Question 1 is a bit dodgy, the first part with delta x and kappa i have done. But it is not related to the second part, which I Have underlined.

that is
"Given the initial condition (3)..." is an entirely unrelated question to the first sentence.
Maybe if I reword it clearer -
how do I use the intial condition (3) : $T(x,0) = \sin(\frac{\pi x}{L})$
to obtain an exact solution?

Also, the subscripts and superscripts are incredibly confusing.
For example, what is meant by $T_{j}^{0}$ or $T_{j}^{n} $

 

[HallsofIvy]

After reading the question, I am not even sure what they want to achieve. Additionally, since the book says use excel, it should be burned. What mathematician, engineer, or physicists writes a book and says let's use excel?

One who owns stock in "excel"?

 

[Opalg]

how do I use the intial condition (3) : $T(x,0) = \sin(\frac{\pi x}{L})$
to obtain an exact solution?

The initial condition (3) tells you that all the terms in the infinite series (5) vanish except for the first one. So $B_l=0$ except when $l=1$, and the series reduces to the single term $T(x,t) = B_1e^{-(\pi^2\hat{\kappa}^2t)/L^2}\sin\bigl(\frac{\pi x}L\bigr).$ You then need to take $B_1=1$ so that $T(x,0) = \sin\bigl(\frac{\pi x}L\bigr).$

 

[nacho-man]

The initial condition (3) tells you that all the terms in the infinite series (5) vanish except for the first one. So $B_l=0$ except when $l=1$, and the series reduces to the single term $T(x,t) = B_1e^{-(\pi^2\hat{\kappa}^2t)/L^2}\sin\bigl(\frac{\pi x}L\bigr).$ You then need to take $B_1=1$ so that $T(x,0) = \sin\bigl(\frac{\pi x}L\bigr).$

Okay, that's what I ended up doing. I need some help with question 3 too, (refer to image attached to this post

I am finding the notation incredibly confusing, why use both sub and superscripts, for example
$T_j^n, j=1,...,N-1$ ?What does that even mean. Actually, I've attached a graph, and want to ask if the point $(j,n)$ is what $T_j^n$ refers to. Please check that out also.

With that aside, it says i must use $(2)$ to plot the points up to $T_j^n$. I have $\alpha$ and $\Delta x$, but how do i incorporate the boundary values into equation $(2)$?