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On approximate solutions of non-linear hyperbolic partial differential equations. (English) Zbl 0099.33604


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[1] Arzelà, C.: Sull’esistenza degli integrali delle equazioni differenziali ordinarie. Mem. Reale Accad. Sci. Inst. Bologna, Ser. V, 6, 131–140 (1896).
[2] Conlan, J.: The Cauchy problem and the mixed boundary value problem for a non-linear hyperbolic partial differential equation in two independent variables. Arch. rational Mech. Anal. 3, 355–380 (1959). · Zbl 0093.31203 · doi:10.1007/BF00284187
[3] Diaz, J. B.: On an analogue of the Euler-Cauchy polygon method for the numerical solution of u xy=f(x, y, u, ux, uy). Arch. rational Mech. Anal. 1, 357–390 (1958). · Zbl 0084.11501 · doi:10.1007/BF00298015
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[7] Leehey, P.: On the existence of not necessarily unique solutions of classical hyperbolic boundary value problems for non-linear second order partial differential equations in two independent variables. Ph. D. Thesis, Brown University, June 1950.
[8] Mie, G.: Beweis der Integrierbarkeit gewöhnlicher Differentialgleichungssysteme nach Peano. Math. Ann. 43, 553–586 (1893). (This is a summary in German of Peano’s paper [9], much of which is written in logical notation.) · doi:10.1007/BF01446453
[9] Peano, G.: Démonstration de l’intégrabilité des équations différentielles ordinaires. Math. Ann. 37, 182–228 (1890). (Cf. Mie [8] above.) · doi:10.1007/BF01200235
[10] Zwirner, G.: Sull’approssimazione degli integrali del sistema differenziale \(\frac{{\partial ^2 z}}{{\partial x\partial y}} = f(x, y, z ), z(x_0 , y) = \psi (y), z(x, y_0 ) = \varphi (x)\) . Atti dell’ Istituto Veneto di Scienze, Lettere ed Arti, Cl. Sci. Fis. Mat. Nat. 109, 219–231 (1950, 51).
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