Infinite String: consider the initial/boundary value problem Pu x E R, t > 0, dx2' u(х, 0) %3D f(x), x E R ди (x, 0) = g(x), dt x E R where f and g are two given twice differentiable functions. We...

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Extracted text: Infinite String: consider the initial/boundary value problem Pu x E R, t > 0, dx2' u(х, 0) %3D f(x), x E R ди (x, 0) = g(x), dt x E R where f and g are two given twice differentiable functions. We showed in class that this problem is solved by he d'Alembert's solution 1 u(x, t) = [f(x+ ct) + f(x – ct)] + IG(x + ct) – G(x – c | 2c where G is an antiderivative of g. Sketch the solution for t = 0, 1, 2, 3, where f(x) and g(x) are given below. a) x2 f(x) = { |x| < 1="" i지=""> 1’ g(x) = b) sin TX f(x) = 0 g(x) = {" |x| < 1="" |지=""> 1