An Asymptotic Solution of Linear Second Order Hyperbolic Differential Equations
An Asymptotic Solution of Linear Second Order Hyperbolic Differential Equations
Morris Kline
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In performing the differentiation with respect to x. First and then with respect to t one does end up with the term U. (x, 0) f(t)in addition to the other terms in (3. 6). However since 'J(x, t) = for t = except on S, we conclude that Uj(x, 0) = . - 11 - tt a' a ; at' (3. -?) J u(x. T-r)f(V)d'r'= a** I J U^^(x, t-y)f('r)dY+ [U(x. T^)]f' (t-t^) * [nt(x, ti)]f(t-t^) r . To perform the differentia tiocs called for in the first-order terms in (3.^) we may use the results already ohtained. For k = l..., 2, ... , n-l, (3. 8) h^^ jTiu, t-r)f(Y)d'T' = •b'^l J u^Cx. T-iOf (Y)d'y - [u(^. Til]f (t-t^vj i . For k = n v/e have (3. 9) b*^ f n(x, t-V)f(Y)dy= h* J U^U, t-'Y)f(V)dV+ [U(x, t^i]f(t-t^) I . I ° J Tor the remsininf term in (3.^) we merely restate that (3. 10) c J u(x, t-r)f(r)dr = c I" uCx. T-T^f (r)dr . To verify the correctness of the Duhamel principle, that is, that (3.^^) is correct, we must now pdd up the terms in equations (3. 5) to (3. 10). We find on the right side th8. T the terms involving integrals amount to J L(u(x, t-'0) f(r)dV, Since U(j', t) is required to satisfy L(u) = for t-Y t^, these terms add up to zero.
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