On Solutions of Nonlinear Wave Equations
On Solutions of Nonlinear Wave Equations
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The solution formula, which has been determined by J. B. Diaz and G. S. S, Ludford, is (cf. [3], equations (23), (Ul). The sign of their integral is incorrect in (3?) and (hi). ) (29) u(x, t) = u^(x, t) + ^ j dt^ Jj V(6, t, tg)f u(x+ac[t-tJ, tQ)Jdadp. In (29) u is the solution of the initial-value problem when f 2 0, a denotes the 2 2 2 vector (a, p), 5 -a + p and V is given by (30) v(6, t, t^) - ^ ^° ^'^° — - n^, |, \, z). (l-62)l/2[(tn^)2. E2(t. T, )2]k/2 Here F is the hype rgeorae trie funct...ion and (31) z - ^ - ^^ *^*o^, 2 - o t-t^^ o Since 1 the hj^pergeometric function F in (30) is non-negative. This follows at once from the series for F"-^*^*^* U;J^ Therefore V is also non-negative and the integral operator in (29) has a non-negative kernel. Since (29) can be written in the form (8) with a non-negative integral operator L, and the integration extends over the same cone C(x, t) as in the previous case, the comparison theorem again applies.
What to read after On Solutions of Nonlinear Wave Equations?
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To quickly assess the difficulty of the text, read a short excerpt:
The solution formula, which has been determined by J. B. Diaz and G. S. S, Ludford, is (cf. [3], equations (23), (Ul). The sign of their integral is incorrect in (3?) and (hi). ) (29) u(x, t) = u^(x, t) + ^ j dt^ Jj V(6, t, tg)f u(x+ac[t-tJ, tQ)Jdadp. In (29) u is the solution of the initial-value problem when f 2 0, a denotes the 2 2 2 vector (a, p), 5 -a + p and V is given by (30) v(6, t, t^) - ^ ^° ^'^° — - n^, |, \, z). (l-62)l/2[(tn^)2. E2(t. T, )2]k/2 Here F is the hype rgeorae trie funct...ion and (31) z - ^ - ^^ *^*o^, 2 - o t-t^^ o Since 1 the hj^pergeometric function F in (30) is non-negative. This follows at once from the series for F"-^*^*^* U;J^ Therefore V is also non-negative and the integral operator in (29) has a non-negative kernel. Since (29) can be written in the form (8) with a non-negative integral operator L, and the integration extends over the same cone C(x, t) as in the previous case, the comparison theorem again applies.
What to read after On Solutions of Nonlinear Wave Equations?
You can find similar books in the "Read Also" column, or choose other free books by Joseph Bishop Keller to read onlineMoreLess
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