Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    9
    ReputationRep:
    Divide [0,1] into N subintervals (x_k,x_{k+1}), k=0,...,N-1 using x_k=kh,k=0,...,N, h=\frac{1}{N}, N\geq 2.

    Let \phi_k(x)=\left(1-\left| \frac{x-x_k}{h}\right|\right)_{+}, k=1,...,N-1 and let p,q be constant functions over [0,1].

    I'm trying to find the piecewise solution to the integral \int_0^1\left[ p \phi_j' \phi_i' + q \phi_j \phi_i\right]. The solution is given in my notes without any explanation so I must be missing something because I'm not sure how to do it. Can someone help?
    Online

    17
    ReputationRep:
    What do you mean by the '+' sign in the definition of phi_k. If it means that f(x)_+ = \max(f(x), 0) (which is my best guess), then for any particular i, there are not many values of j s.t. \phi_i \phi_j \neq 0 (and same for the derivatives). It's all fairly straightforward (if a little fiddly).
    • Thread Starter
    Offline

    9
    ReputationRep:
    (Original post by DFranklin)
    What do you mean by the '+' sign in the definition of phi_k. If it means that f(x)_+ = \max(f(x), 0) (which is my best guess),
    That's correct.

    then for any particular i, there are not many values of j s.t. \phi_i \phi_j \neq 0 (and same for the derivatives).
    Can you explain a bit further how you know this? I've just started reading about a new topic today so I may be unfamiliar with this kind of work.
    Online

    17
    ReputationRep:
    (Original post by 0-))
    That's correct.


    Can you explain a bit further how you know this? I've just started reading about a new topic today so I may be unfamiliar with this kind of work.
    Draw a sketch - each function is only non-zero over a small region. For the product to be non-zero the "non-zero" bits of each function will have to overlap.
    • Thread Starter
    Offline

    9
    ReputationRep:
    (Original post by DFranklin)
    Draw a sketch - each function is only non-zero over a small region. For the product to be non-zero the "non-zero" bits of each function will have to overlap.
    OK I can see that the product will be non-zero only if |i-j|=1 or i=j. I'm not how to do the next part - the integration.

    Could you show me how to find e.g. \displaystyle \int_0^1 \phi_i \phi_{i+1} \ dx. It should all click into place after that.
    • Thread Starter
    Offline

    9
    ReputationRep:
    Can anyone help? I'm looking at this for the second time and I'm still struggling.
    Online

    17
    ReputationRep:
    Well, suppose i = j. What is \phi_i^2? There's an interval where it's \left(1 - \frac{x - i / N}{1/ N}\right)^2 (which you can make a little nicer by multiplying through by N), and another interval where it's \left(1 + \frac{x - i / N}{1/ N}\right)^2.

    Neither of these is terribly difficult to integrate, and then you just add.

    \phi_i \phi_{i+1} is somewhat similar (but I think there's only one interval to consider - I haven't drawn the sketch).
    • Thread Starter
    Offline

    9
    ReputationRep:
    (Original post by DFranklin)
    Well, suppose i = j. What is \phi_i^2? There's an interval where it's \left(1 - \frac{x - i / N}{1/ N}\right)^2 (which you can make a little nicer by multiplying through by N), and another interval where it's \left(1 + \frac{x - i / N}{1/ N}\right)^2.

    Neither of these is terribly difficult to integrate, and then you just add.

    \phi_i \phi_{i+1} is somewhat similar (but I think there's only one interval to consider - I haven't drawn the sketch).
    I have the solution to \int \phi_i \phi_j for i=j as \frac{4h}{6} but after putting the sum of the integrals into Maple I get 2+\frac{2}{3}N^2-2Ni+2i^2. Can you see what's going wrong?
    Online

    17
    ReputationRep:
    You've not exactly posted much of your working. But almost certainly, you've got the limits of your integrals wrong.
    • Thread Starter
    Offline

    9
    ReputationRep:
    (Original post by DFranklin)
    You've not exactly posted much of your working. But almost certainly, you've got the limits of your integrals wrong.
    :facepalm: Finally I realise where I've been going wrong. Sorry it took so long!

    Thank you for your help.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Has a teacher ever helped you cheat?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Write a reply...
    Reply
    Hide
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.