Results 1 to 5 of 5

Math Help - Problems on characteristics

  1. #1
    Newbie
    Joined
    May 2010
    From
    Atlanta, GA
    Posts
    9

    Smile Problems on characteristics

    Obtain a solution of

    u_t + u_x = t^a, where a = alpha, and a > 0.

    given that u = u_0(x) on t = t0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,367
    Thanks
    42
    Quote Originally Posted by Waikato View Post
    Obtain a solution of

    u_t + u_x = t^a, where a = alpha, and a > 0.

    given that u = u_0(x) on t = t0
    Characteristic equation

    \frac{dt}{1} = \frac{dx}{1} = \frac{du}{t^\alpha}

    I_1 = x- t,\;\;\;I_2 = u - \frac{t^{\alpha+1}}{\alpha+1}

    Solution

    u = \frac{t^{\alpha+1}}{\alpha+1} + f(x-t).

    Now impose your IC.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    From
    Atlanta, GA
    Posts
    9

    Initial Condition - DE

    Could you please help with imposing the IC?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,367
    Thanks
    42
    Quote Originally Posted by Waikato View Post
    Could you please help with imposing the IC?

    Thanks
    Sure

    Quote Originally Posted by Danny View Post
    Characteristic equation

    \frac{dt}{1} = \frac{dx}{1} = \frac{du}{t^\alpha}

    I_1 = x- t,\;\;\;I_2 = u - \frac{t^{\alpha+1}}{\alpha+1}

    Solution

    u = \frac{t^{\alpha+1}}{\alpha+1} + f(x-t).

    Now impose your IC.
     <br />
u(x,0) = \frac{0}{\alpha+1} + f(x) = u_0(x)<br />

    so f(x) = u_0(x) and the final solution is

    u = \frac{t^{\alpha+1}}{\alpha+1} + u_0(x-t).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2010
    From
    Atlanta, GA
    Posts
    9

    IC's and help

    Thanks for the IC's help.

    Can you check this question out? You might be able to help.

    Consider u_t + (v.grad)u = 0 , where grad = upside down triangle, and v is a vector, dotted with the grad symbol.

    where v = (-ax, ay) represents 2D stagnation point flow and u = u(x,y,t). Obtain a solution of the Cauchy problem given that u = u0(x,y) at t = 0

    (u0 is read as "u naught")
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Characteristics of y=tan(x)
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: June 13th 2011, 12:29 PM
  2. characteristics of x^3 - 6x^2 + 12x +18
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 3rd 2010, 11:20 AM
  3. How does one show this using characteristics?
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: June 11th 2010, 04:35 AM
  4. Characteristics of an IFR pdf
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: April 9th 2009, 12:01 AM
  5. Characteristics of Polynomials
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: January 5th 2009, 10:06 PM

Search Tags


/mathhelpforum @mathhelpforum