Amarnath An Elementary Course in Partial Di64256erential Equa tions Part A Uniqueness of solution for one dimensional wave equation with 64257nite length Theorem The solution of the following problem if it exists is unique tt xx xt 0 1 x 0 l x 0 l ID: 27962 Download Pdf

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Amarnath An Elementary Course in Partial Di64256erential Equa tions Part A Uniqueness of solution for one dimensional wave equation with 64257nite length Theorem The solution of the following problem if it exists is unique tt xx xt 0 1 x 0 l x 0 l

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Uniqueness Results for Solutions of (1) Wave equation and (2) Heat equation (Reference - T. Amarnath. An Elementary Course in Partial Diﬀerential Equa- tions.) Part A: Uniqueness of solution for one dimensional wave equation with ﬁnite length Theorem: The solution of the following problem, if it exists, is unique. tt xx x,t 0 (1) x, 0) = l, x, 0) = l, (0 ,t ) = l,t ) = 0 , t> Proof The above uniqueness result for IBP of wave equation is equivalent to showing that the following IBP has only trivial solution, tt xx 0 (2) x, 0) = 0 l, x, 0) = 0 l, (0 ,t ) = l,t ) =

0 , t> Let x,t ) be a solution of problem (2). Now consider, ) = dx. Observe that ) is a diﬀerentiable function of , since x,t ) is twice diﬀeren- tiable. Therefore dE dt xt tt dx, tt dx xx dx. (0 ,t ) = 0 (0 ,t ) = 0 and l,t ) = 0 l,t ) = 0 Therefore dE dt tt xx dx = 0 = constant Since x, 0) = 0 implies x, 0) = 0 and given that x, 0) = 0, therefore (0) = 0

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Hence ,v t> This is possible only of x,t ) =constant, since x, 0) = 0 , v Hence the theorem. Part B: Uniqueness of solution for one dimensional heat equation with ﬁnite length Theorem: The solution of

the following problem, if it exists, is unique. κu xx x,t 0 (3) x, 0) = l, (0 ,t ) = l,t ) = 0 , t Proof The above uniqueness result for IBP of heat equation is equivalent to showing that the following IBP has only trivial solution, κv xx 0 (4) x, 0) = 0 l, (0 ,t ) = l,t ) = 0 , t Let x,t ) be a solution of problem (4). Now consider, ) = x,t dx. Observe that ) is a diﬀerentiable function of , since x,t ) is twice diﬀeren- tiable. Therefore dE dt vv dx, vv xx dx vv dx Since (0 ,t ) = l,t ) = 0 we have dE dt dx i.e. is a decreasing function of . Also, by deﬁnition,

) is nonnegative and from the condition (( x, 0) = 0 we have (0) = 0 Therefore t> x,t 0 in 0 l, t Hence the theorem.

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