Discontinuous Initial Value Problems And Asymptotic Expansion of Steady State So
Discontinuous Initial Value Problems And Asymptotic Expansion of Steady State So
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To quickly assess the difficulty of the text, read a short excerpt:
In Section 6 we will show that these hypersurfaces can never have a tangent plane parallel to the t-axis, so they can be defined by equations of the form t « T^°'(x), With this information, the asymptotic expansion of the integral (122) is obtained simply by repeated integrations by parts. At each step we obtain contributions from the lower end point of the integral and from the dis- continuities of u and its derivatives. We shall assume that the discontinuities - 68 - of u and its derivatives ...with respect to t all occur on a finite number of hyper- surfaces t - f (x)j a - l, . , . , p. We write ^(1) ^(2) 00 (125) v(x) - [' * [ * ••• + u(t, x)e^*dt .
Integrating by parts (126) v(x) . - ^%S) - 2: ?-, _ \_. T-\ \ *■■■*[ • r-0 (lo)) a r"0 l_t_l « If g(x) is sufficiently smooth, we know from Part I that u will be smooth and the terms involving [u ]*^ will vanish. Now compare (130) with (10), We see that V « y is indeed an asymptotic expansion to M terms of v. In the general case, where g and its derivatives are discontinuous across p, we must determine the functions (-1) ) [u 1'^ which appear in (130), We shall identify them with the terms z^°'^ "^ in the formal expansion (see (3I4) and (13)).
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To quickly assess the difficulty of the text, read a short excerpt:
In Section 6 we will show that these hypersurfaces can never have a tangent plane parallel to the t-axis, so they can be defined by equations of the form t « T^°'(x), With this information, the asymptotic expansion of the integral (122) is obtained simply by repeated integrations by parts. At each step we obtain contributions from the lower end point of the integral and from the dis- continuities of u and its derivatives. We shall assume that the discontinuities - 68 - of u and its derivatives ...with respect to t all occur on a finite number of hyper- surfaces t - f (x)j a - l, . , . , p. We write ^(1) ^(2) 00 (125) v(x) - [' * [ * ••• + u(t, x)e^*dt .
Integrating by parts (126) v(x) . - ^%S) - 2: ?-, _ \_. T-\ \ *■■■*[ • r-0 (lo)) a r"0 l_t_l « If g(x) is sufficiently smooth, we know from Part I that u will be smooth and the terms involving [u ]*^ will vanish. Now compare (130) with (10), We see that V « y is indeed an asymptotic expansion to M terms of v. In the general case, where g and its derivatives are discontinuous across p, we must determine the functions (-1) ) [u 1'^ which appear in (130), We shall identify them with the terms z^°'^ "^ in the formal expansion (see (3I4) and (13)).
What to read after Discontinuous Initial Value Problems And Asymptotic Expansion of Steady State So?
You can find similar books in the "Read Also" column, or choose other free books by Robert M Lewis to read onlineMoreLess
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