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

  1. #1
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    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
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  2. #2
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    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.
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