Results 1 to 9 of 9

Math Help - Using Row-Echelon Reduction With Differential Operators As Elements

  1. #1
    Newbie pi_cubed's Avatar
    Joined
    Dec 2008
    From
    Redlands, California
    Posts
    8

    Using Row-Echelon Reduction With Differential Operators As Elements

    Hello. If this question belongs in another forum, I apologize.

    When solving 'n-1' homogeneous equations in 'n' unknowns, is it permissible to use
    differential operators in the matrix elements?

    For example, when trying to solve :

    i = i1 + i2
    i = Cv' - Cx'
    i1 = x/R
    i2 = i3 + i4
    i2 = Cx' - Cy'
    i3 = y/R
    i4 = Cy' - Cu'
    i4 = u/R

    The variables i, i1, i2, i3 and i4 are instantaneous currents {i.e., i = i(t)}.
    The variables x, y, u and v are instantaneous voltages.
    The prime marks indicate first derivatives.

    I was thinking about putting these equations in row-echelon form, and then
    performing reductions. For the primed variables, I was thinking about using
    the operator D = d/dt.

    Can I do this, being aware that I treat the operator algebraically in the columns,
    but use it operationally row-wise?

    Thanks!!
    Last edited by pi_cubed; July 13th 2010 at 09:21 AM. Reason: Removed extraneous text from end of post
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Sorry, no. That's not allowed. If your matrix starts having operators in it, then finding the eigenvalues (which is ultimately what you're trying to do) becomes very much more complicated. Ph.D. theses have been written on that sort of thing (mine was!).

    Your problem looks like a circuit problem. So you have a system of ODE's, correct? How are you expecting to solve it without having n equations? Perhaps you could post the original problem here, or is that the original problem?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie pi_cubed's Avatar
    Joined
    Dec 2008
    From
    Redlands, California
    Posts
    8

    Problem Definition for Example Given

    This problem comes from Paul J. Nahin's book
    "An Imaginary Tale (The Story of {radical -1})", Princeton University Press, 1998,
    on page 139. This example is a circuit solution for a phase-shift oscillator.

    The author says :
    "With a total of eight equations in nine variables we can solve for the ratio of any
    two, and the particular ration of interest for our circuit is u/v. One way to do this
    is to first manipulate the above equations to eliminate all the variables except
    for u and v.
    This is not difficult to do, but it is rather detailed, and so I will just give you the
    answer and encourage you to verify it :
    v''' = u''' + [6/RC]u'' + [5/{(RC)^2}]u' + [1/{(RC)^3}]u ."

    I was hoping in this instance I could use row-echelon form, reducing to the two
    variables u and v. In this situation, will this method work?

    If not, would you please suggest a path to follow.

    Thanks!
    Last edited by pi_cubed; July 13th 2010 at 09:20 AM. Reason: Fixed typo
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Given the nature of what the author is asking you to do, I would plug away with substitutions and differentiation. You're going to have to differentiate in several places, or I'm mistaken. Your initial equations are not so complicated that substitution wouldn't work pretty well, I think.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie pi_cubed's Avatar
    Joined
    Dec 2008
    From
    Redlands, California
    Posts
    8
    I have solved the problem using the row-echelon method :
    1) rewrite all equations so that they are equal to zero;
    2) construct the augmented matrix - since the "augmentation column" is all zeroes it can be ignored;
    3) perform the normal row operations until reaching reduced row-echelon form, leaving the columns for
    u and v with the only non-one entries.

    Since the only operations involved in row reduction involving the single differential operator D=d/dt were addition or multiplication, the process went smoothly.
    Last edited by pi_cubed; July 13th 2010 at 09:21 AM. Reason: Fixed typo
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    You say, "...the only operations involved in row reduction involving the single differential operator D=d/dt were addition or multiplication..."

    What kind of multiplications did you come up with? Multiplication can be tricky with differential operators. Operators are slippery, and can give you wrong results unless you use a test function. Can you show me a calculation or two involving multiplication?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie pi_cubed's Avatar
    Joined
    Dec 2008
    From
    Redlands, California
    Posts
    8

    Clearing up semantic ambiguity

    Hi Adrian,

    I probably should have written more than that single sentence, which is semantically very vague - sorry about that!

    When I was multiplying by D across rows, the elements in each column were simple polynomials in D.

    For example : suppose a (zero-augmented) 2x3 echelon matrix,
    with left-to-right column headings of 'y', 'v', and 'u', containing two rows
    {1, -[(RC)^2](D^2), [4 + 6RCD + [(RC)^2](D^2)]} and
    {CD, 0, [-1/R - CD]} .

    [D is defined to be "d/dt"; R and C are constants.]

    {The two rows represent the equations :
    0 = y - [(RC)^2]v'' + 4u + 6RCu' + [(RC)^2]u'' and
    0 = Cy' - u/R - Cu' }

    Algebraically multiplying the first row by (-CD) and
    adding the result to the second row yields :

    {0, (R^2)(C^3)(D^3), [-1/R - 5CD - 6R(C^2)(D^2) - (R^2)(C^3)(D^3)]}

    which represents :
    0 = (R^2)(C^3)v''' - u/R -5Cu' - 6R(C^2)u'' - (R^2)(C^3)u'''
    (which gives the answer the author mentioned above wanted).

    This is probably so elementary a thing that I overcomplicated it with my lousy ability in English composition.
    Last edited by pi_cubed; July 14th 2010 at 04:19 AM. Reason: Making format readable.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Ah, I see what you're about. I'm going to have to take back some of my words in Post #2. You're not trying to find eigenvalues this way. You're just using these ideas to organize your elimination of variables. Your operator multiplication, I should warn you, while it does work in this case, would not work the same way if anything in your original DE's was nonlinear or if you had non-constant coefficients that have to be differentiated. Operators always look to the right, and operate on whatever is to the right. Because your coefficients are all constants, the derivative operator passes through and ends up on the right of every expression, which is where you want it. If you have non-constant coefficients, or nonlinearities, you'd have to be more careful.

    Interesting idea, though. Thanks for posting!
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie pi_cubed's Avatar
    Joined
    Dec 2008
    From
    Redlands, California
    Posts
    8
    Thanks Adrian. You said exactly what I was thinking - you are very eloquent of speech!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. tangent vector fields as linear partial differential operators
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 30th 2011, 12:35 AM
  2. Proof with polynomial differential operators
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: April 20th 2010, 05:10 AM
  3. Linear operators and Differential Equations
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 25th 2009, 12:21 PM
  4. Differential vector operators
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: October 7th 2008, 01:37 AM
  5. Replies: 3
    Last Post: November 4th 2007, 08:04 PM

Search Tags


/mathhelpforum @mathhelpforum