11.5.1.3 D'Alembert Method

Condition: let \ (u (x, t)- \) solution \ (\ quad \) in \ (\ quad [0.1] \ times \ bar {r} _ {+} \) mixed task \ [\ begin {arry} {l} u_ {t t} (x, t) = u_ {x x} (x, t), \ quad U (0, t) = u (1, t) = 0, \\ u (x, 0) = 0, u_ {t} (x, 0) = x^{2} (1-x). \\ \ text {Find} \ LIM _ {T \ RIGHTARROW+\ InFty} \ int_ {0} {1} \ left [u_ {t} {2} (x, t)+u_ {x} {2} (x, t) \ right] d x \\ \ Text {and} \ lim _ {t \ rightarrow 0} \ int_ {0}^{1} \ left [u_ {t} {2} (x, t)+u_ {x} {2} (x, t) \ right] d x. \ end {Array} \]