Solving a boundary value problem for partial differential equations (heat equation)

[vague olive]

Hihi.

Crisis 1
I'm reallly struggling to reconcile my working for this problem with that of the memo's.
I've appended an image of my own working out to this post, as well as that of the memo. So this discrepancy in answer is the first part to my question.

Crisis 2
Secondly, I am struggling to understand how to solve these types of problems where they as (like in this case): By using the alternating series test one can show that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(2 n-1)^3}$ is convergent. Determine the sum of this series by utilizing the solution to the heat equation given above.

I tried to figure out an approach to this (the third attached image) based on working from a previous tutorial, but I still don't get it. I have attached the memo to that part of the question as the 4th image.

Can someone please help guide me in what approach I use to solve this part of the question as well?

In short, the attached images are:

1. Crisis 1, own working
2. crisis 1, memo
3. crisis 2, own working
4. crisis 2, memo

I would so appreciate someone's help. For some reason my university course does not have a forum for this course, and my classmates are not responding either.

[neon panther]

sarathecrewe

[silent fog]

you went wrong at this step

really annoying

sorry you had to suffer so much

substitutet x=pi where you shouldve substituted x=0

@vague olive

[vague olive]

wait where?

[vague olive]

aaaaaaaaa

noooooo

brooo

HOURS of my life

thank you!

[silent fog]

np!

[vague olive]

could do you by any chance .... help me with crisis 2 as well 😅 ? It might take a bit longer so I understand if you'd rather not at this stage

[silent fog]

you know how to do d'alembert test?

or wait

[vague olive]

I have never in my life heard of that man

[silent fog]

excuse me

owh

it just mean

you know alternating series test

?

[vague olive]

oh I googled. It's just the ratio test. Yeah, I've covered the various tests for series convergence

[silent fog]

well

do that

but here

[vague olive]

ok... but why do they tend to sub x=pi/2? Is it just a useful test value or what?

[silent fog]

mostly

we do that

for the sine

because we want an alternating series you know

[vague olive]

because the memos seem to like to use this substitution. As well as then changing n to 2k-1, so that it only aassoiciates with odd numbers to the sine cancels out or something

I'm just confused by what general approach they want me to utilise here

[silent fog]

owh wow

my bad for confusing

it also asks to determine the value

well apperantly considering this solution you should know where the alternating harmonic series or at least the variant of it converges to

[vague olive]

i understand finding the final value of a series like this, but what's confusing to me is utilising the pde equation to do so. The transformation thereof into the form of the equation we want to solve, that process is confusing to me

[silent fog]

so basically when you put in x= pi/2 and t= 0 you can use the boundary condition to solve your problem can you understand that?

since u(x,0) = x^2 nad

such that u(pi/2,0)= (pi/2)^2

[vague olive]

ok yes that makes sense, with you so far

[silent fog]

since the series you are looking for is in the expansion coefficient

you need to find the value on the series on the left

but that you apparently know from i guess a previous problem

or a given

you set your knowns then to one side

and the series you want to find its convergence value on the other side

[vague olive]

ok, so let me check this approach then:
the question wants me to evaulate a series. I cant do this yet, but I know the solution equation u(x,y) can be utilised to get there somehow

• so I inspect the u(x,y) equation, and recognise that manipulating it such that n=odd numbers will simplify the analysis, as it accounts for the non-zero values of the sine-function, when x is taken as pi/2
• n therefore I make the n substitution, as well as the x=pi/2 substitution
• I equate the LHS to the f(x) value at x=pi/2
• I simplify until my equation is usable in the form of what they asked me at the beginning
• then I find the final answer

is this the correct approach to this kind of problem then, @silent fog ?

[silent fog]

yes

[vague olive]

woohooo

[silent fog]

basically

dont forget t=0 for the x^2 part but you get that

[vague olive]

Oh yeah, I would have missed that, thanks

what a legend, thanks bro.

.close