Results 1 to 2 of 2

Math Help - Partial Differential Equation using D'Alembert's approach

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    78

    Partial Differential Equation using D'Alembert's approach

    Solve by using D'Alembert's solution with the even extensions of f(x) and g(x)

    \mu_{tt} = c^2\mu_{xx}     where 0\leq x <\infty

    \mu(x,0) = f(x), \mu_t(x,0) = g(x) where 0\leq x <\infty

    \mu_x(0,t) = 0 where t\ge 0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by flaming View Post
    Solve by using D'Alembert's solution with the even extensions of f(x) and g(x)

    \mu_{tt} = c^2\mu_{xx}     where 0\leq x <\infty

    \mu(x,0) = f(x), \mu_t(x,0) = g(x) where 0\leq x <\infty

    \mu_x(0,t) = 0 where t\ge 0
    Since we have the boundary condition u_x(0,t)=0 we would consider even extensions. Let f_1(x)\text{ and }g_1(x) be even extensions, and we will also assume that these extensions are well-behaved to satisfy the condition of D'Alembert's solution. Thus, we have that u_tt = c^2u_{xx} for (x,t) \in \mathbb{R}^2. Thus, the solution is given by:
    u(x,t) = \frac{1}{2}[f_1(x+ct) - f_1(x-ct)] + \frac{1}{2a}\int_{x-ct}^{x+ct}g_1(\xi) d\xi.
    Notice that u(-x,t) = u(x,t) therefore u_x(0,t) = 0, this is precisely what we want.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Partial derivatives (from D'Alembert)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 22nd 2011, 09:51 AM
  2. Partial Differential Equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 16th 2011, 03:48 AM
  3. partial differential equation
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: April 23rd 2011, 06:45 AM
  4. Partial differential equation-wave equation - dimensional analysis
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: August 28th 2009, 11:39 AM
  5. 3D Partial Differential Equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: March 14th 2009, 04:01 PM

Search Tags


/mathhelpforum @mathhelpforum