On the General Theory of Mixed Problems

  • R. Hersh


We start with a simple example. Consider the wave equation in a half-plane,
$$\begin{gathered} u_{tt} = u_{xx} + u_{yy} \,\,\,in\,\,\,t > 0,\,x > 0, \hfill \\ u = u_t = 0\,\,\,\,on\,\,\,\,t = 0,\,x > 0, \hfill \\ B(D_t ,D_x ,D_y )u = f(t,y)\,\,\,\,on\,\,\,\,t > 0,\,x = 0. \hfill \\ \end{gathered}$$




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agmon, S.: Problèmes mixtes pour les équations hyperboliques d’ordre supérieur, Colloques sur les équations aux dérivées partielles. C. N. R. S 13–18 (1962).Google Scholar
  2. 2.
    Birkhoff, Garret, and Robert E. Lynch: Numerical Solution of The Telegraph and Related Equations, Numerical Solution of Partial Differential Equations. 373 pp. Proceed. Symp. Univ. Maryland 1966.Google Scholar
  3. 3.
    Duff, G. F. D.: Mixed Problems for linear systems of first order equations. Canada Jour, of Math. 10, 127–160 (1958).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Friedrichs, K. O.: Symmetric Positive Linear Differential Equations. Comm. Pure and Appl. Math., Vol. XI, No. 3, 1958.Google Scholar
  5. 5.
    Hersh, Reuben: Mixed Problems in Several Variables, Jour. Math, and Mech., Vol. 12, No. 3, 317–334 (1963).MathSciNetMATHGoogle Scholar
  6. 6.
    Hersh, Reuben: Boundary Conditions for Equations of Evolution. Arch. Rational Mech. and Anal., Vol. 16, No. 4, 243–263 (1964).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hersh, Reuben: On Surface Waves with Finite and Infinite Speed of Propagation. Arch. Rational Mech. and Anal., Vol. 19, No. 4, 308–316 (1965).MathSciNetMATHGoogle Scholar
  8. 8.
    Hersh, Reuben: On Vibration, Diffusion, Equilibrium Across a Plane Interface. Arch. Rational Mech. and Anal., Vol. 21, No. 5, 368–390 (1966).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hersh, Reuben: On the Theory of Difference Schemes for Mixed Initial Boundary Value Problems. SIAM Jour. Numer. Anal., Vol. 5, No. 2 (1968).Google Scholar
  10. 10.
    Hersh, Reuben: The Method of Reflection for Two-sided Problems of General Type. Denver Univ.-NSF Conference on Stability Theory of Initial-Boundary Value Problems (1968).Google Scholar
  11. 11.
    Lax, P. D., and K. O. Friedrichs: Boundary Value Problems for First Order Operators. Comm. on Pure and Appl. Math., Vol. XVIII, No. 1/2 (1965).MathSciNetGoogle Scholar
  12. 12.
    Lax, P. D., and R. S. Phillips: Local Boundary Conditions for Dissipative Symmetric Linear Differential Operators. Comm. on Pure and Appl. Math., Vol. XIII, No. 3 (1960).MathSciNetGoogle Scholar
  13. 13.
    Mizohata, S.: Quelques Problèmes Au Bord, Du Type Mixte Pour Des Équations Hyperboliques. College de France 1966–1967.Google Scholar
  14. 14.
    Osher, Stanley: Systems of Difference Equations with General Homogeneous Boundary Conditions. Brookhaven National Laboratory, Report No. BNL 11857, 1/16/68.Google Scholar
  15. 15.
    Sarason, Leonard: On Hyperbolic Mixed Problems. Arch, for Rational Mech. and Anal., Vol. 18, No. 4, 310–334 (1965).MathSciNetMATHGoogle Scholar
  16. 16.
    Sobolev, S. L.: On Mixed Problems for Partial Differential Equations with Two Independent Variables, Doklady. Akad. Nauk USSR 1958.Google Scholar
  17. 17.
    Strang, G.: Wiener-Hopf Difference Equations. J. Math. Mech. 13, 85–96 (1964).MathSciNetMATHGoogle Scholar
  18. 18.
    Strang, G.: Hyperbolic Initial-Boundary Value Problems in Two Unknowns J. Diff. Eq. 1969.Google Scholar
  19. 19.
    Kreiss, H. O.: Initial boundary value problems for hyperbolic systems, Uppsala University, Department of Computer Sciences, May 1969.Google Scholar
  20. 20.
    Leray, J., and Y. Ohya: Systèmes Linéaries, Hyperboliques non Stricts. Colloque de Liège, C. N. R. B. 1964.Google Scholar
  21. 21.
    Da Prato, Giuseppe: Problèmes Au Bord de Type Mixte Pour Des Équations Paraboliques ou Hyperboliques, Collége de France (1967—1968).Google Scholar
  22. 22.
    Ikawa, Mitsuru: On the Mixed Problem for the Wave Equation with an Oblique Derivative Boundary Condition, Proceed. Japan Acad., Vol. 44, No. 10, 1033—1037 (1968)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1970

Authors and Affiliations

  • R. Hersh

There are no affiliations available

Personalised recommendations