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On existence, uniqueness, and numerical evaluation of solutions of ordinary and hyperbolic differential equations

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Summary

After a preliminary survey of related results, a general uniqueness theorem for the ordinary differential equation dy/dx=f(x, y) is given in section 4. The general uniqueness theorem for the hyperbolic partial differential equation uxy=f(x, y, u), proved in section 5, is an exact analogue of the general uniqueness theorem for the ordinary differential equation dy/dx=f(x, y).

Bibliography

  1. [1]

    Picard, E.,Sur les méthodes d'approximations successives dans la théorie des équations différentielles, (Note I to vol. 4 ofG. Darboux,Lecons sur la Théorie Générale des Surfaces, Paris, 1896, pp. 353–367).

  2. [2]

    Lewy, H.,Uber das Anfangswertproblem einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhängigen Veränderliche, « Math. Annalen », 98, 179–190 (1928).

  3. [3]

    Beckert, H.,Existenz- und Eindeutigkeitsbeweise für das Differenzen-verfahren zur Losung des Anafangswertproblems, des gemischten Angangs-Randwert- und des charakteristischen Problems einer hyperbolischen Differentialgleichung zweiter Ordnung mit zwei unabhangigen Variablen, « Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-naturwiss. », Kl. 97, H. 4 (1950).

  4. [4]

    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.

  5. [5]

    Zwirner, G.,Sull' approssimazione degli integrali del sistema differenziale ∂ 2 z/∂x∂y = f(x, y, z), z(x 0,y) = ϕ(y), z(x, y 0) =ϕ(x), « Atti dell'Istituto Veneto di Scienze, Lettere ed Arti, Cl. Sci. Fis. Nat. », 109, 219–231 (1959–51).

  6. [6]

    Hartman, P. andWintner, A.,On hyperbolic partial differential equations, « American Journal of Mathematics », 74, 834–864 (1952).

  7. [7]

    Conti, R.,Sul problema di Darboux per l'equazione z xy =f(x, y, z, z x ,z y ), « Annali dell'Università di Ferrara », Nuova ser., Sez. 7, Sc. Mat. 2, 129–140 (1953).

  8. [8]

    Moore, R. H.,Proof of an existence and uniqueness theorem of Picard for a non-linear hyperbolic partial differential equation, M. A. thesis, University of Maryland, June, 1955.

  9. [9]

    Aziz, A. K.,On higher order boundary value problems for hyperbolic partial differential equations in two and three variables, Ph. D. thesis, University of Maryland, June, 1958.

  10. [10]

    Conti, R. Sull'equazione integrodifferenziale di Darboux-Picard, « Le Matematiche » 13, 30–39 (1958).

  11. [11]

    Diaz, J. B.,On an analogue of the Euler-Cauchy polygon method for the numerica-solution of uxy =f(x, y, u, u x ,u y ), « Archive for Rational Mechanics and Analysis » 1, 357–390 (1958) (also issued as NAVORD Report 4451, U. S. Naval Ordnance Labo, ratory, White Oak, Maryland, January 16, 1957).

  12. [12]

    Guglielmino, Francesco,Sulla risoluzione del problema di Darboux per l'equazione s = f(x, y, z), « Boll. Unione Mat. Italiana », (3), vol. 13, 308–318 (1958).

  13. [13]

    Villari, Gaetano,Su un problema al contorno per l'equazione classe di sistemi di eguazione alle derivate parziali, « Boll. Unione Mat. Italiana » (3), vol. 13, 1–8 (1958).

  14. [14]

    Conlån, James,The Cauchy problem and the mixed boundary value problem for a non-linear hyperbolic partial differential equation in two independent variables, 3, 355–380 (1959). (Ph. D. thesis, University of Maryland, June, 1958).

  15. [15]

    Diaz, J. B., andWalter, W. L.,On uniqueness theorems for ordinary differential equations and for partial equations of hyperbolic type, Technical Note BN-177, University of Maryland. July, 1959; « Transactions of the American Mathematical Society », vol. 96, 1960, 90–100.

  16. [16]

    Moore, R. H.,On approximation of the solutions of the Goursat problem for second order quasi-linear equations, Ph. D. thesis, University of Michigan, June 1959 (to appear in the Archive for Rational Mechanics and Analysis).

  17. [17]

    Santoro, Paolo,Sul problema di Darboux per l'equazione s = f(x, y, z, p, q) e il fenomeno di Peano, « Rendiconti dell'Accademia Nazionale dei XL », Ser. IV, vol. X (82)-3–17 (1959).

  18. [18]

    Walter W.,Eindeutigkeitssatze fur die Differentialgleichung u xy =f(x, y, u, u x ,u y ), « Math. Zeit. », 71, 398–324 (1959) (see also Technical Note BN-159, University of Maryland, 1960).

  19. [19]

    Billings, J. H.,Extensions of the Laplace method, Ph. D. thesis, University of Mary, land, June 1960 (see also Technical Note BN-209, University of Maryland, 1960).

  20. [20]

    Sternberg, H. M.,The solution of the characteristic and the Cauchy problems for the Bianchi partial differential equation in n independent variables by a generalization of Riemann's method, Ph. D. thesis, University of Maryland, June 1960.

  21. [21]

    Aziz, A. K., andDiaz, J. B.,On higher order boundary value problems for a linear hyperbolic partial differential equation in three independent variables (to appear in the Indiana Journal of Mathematics and Mechanics).

  22. [22]

    Shanahan, John, P.,On uniqueness questions for hyperbolic differential equations (to appear in the Pacific Journal of Mathematics).

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Additional information

This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research and Development Command under Contract No. AF 49(638)-228.

This is a detailed account of lectures given at the Sixth Conference of Arsenal Mathematicians, held at Duke University Durham, North Carolina, June 1, 2, 1960, and at the Symposium on the Numerical Treatment of Ordinary Differential Equations, Integral and Integro-Differential Equations, Rome, 20–24 September 1960.

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Diaz, J.B. On existence, uniqueness, and numerical evaluation of solutions of ordinary and hyperbolic differential equations. Annali di Matematica 52, 163–181 (1960) doi:10.1007/BF02415674

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Numerical Evaluation
  • Related Result