An extension of the method of Trefftz for finding local bounds on the solutions of boundary value problems, and on their derivatives
Author:
Philip Cooperman
Journal:
Quart. Appl. Math. 10 (1953), 359-373
MSC:
Primary 36.0X
DOI:
https://doi.org/10.1090/qam/52010
MathSciNet review:
52010
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Additional Information
- E. Trefftz, Konvergenz und Fehlerschätzung beim Ritzschen Verfahren, Math. Ann. 100 (1928), no. 1, 503–521 (German). MR 1512498, DOI https://doi.org/10.1007/BF01448859
O. Friedrichs, Die Randwert-und-Eigenwertprobleme aus der Theorie der elastischen Platten, Math. Ann. 98, 206, (1927-28). Ein Verfahren der Variationsrechnung..., Nach. der Ges. d. Wiss. zu Gottingen, 13, 1929.
- R. Courant and D. Hilbert, Methoden der Mathematischen Physik. Vols. I, II, Interscience Publishers, Inc., N.Y., 1943. MR 0009069
- J. L. Synge, The method of the hypercircle in function-space for boundary-value problems, Proc. Roy. Soc. London Ser. A 191 (1947), 447–467. MR 25903, DOI https://doi.org/10.1098/rspa.1947.0127
- J. L. Synge, Upper and lower bounds for the solutions of problems in elasticity, Proc. Roy. Irish Acad. Sect. A 53 (1950), 41–64. MR 0039484
- J. L. Synge, Pointwise bounds for the solutions of certain boundary-value problems, Proc. Roy. Soc. London Ser. A 208 (1951), 170–175. MR 43281, DOI https://doi.org/10.1098/rspa.1951.0151
- W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI https://doi.org/10.1090/S0033-569X-1947-25902-8
- H. J. Greenberg, The determination of upper and lower bounds for the solution of the Dirichlet problem, J. Math. Physics 27 (1948), 161–182. MR 0026171
- J. B. Diaz and H. J. Greenberg, Upper and lower bounds for the solution of the first biharmonic boundary value problem, J. Math. Physics 27 (1948), 193–201. MR 0026862
- J. B. Diaz and Alexander Weinstein, Schwarz’ inequality and the methods of Rayleigh-Ritz and Trefftz, J. Math. Phys. Mass. Inst. Tech. 26 (1947), 133–136. MR 22458, DOI https://doi.org/10.1002/sapm1947261133
- J. B. Diaz, Upper and lower bounds for quadratic functionals, Proceedings of the Symposium on Spectral Theory and Differential Problems, Oklahoma Agricultural and Mechanical College, Stillwater, Okla., 1951, pp. 279–289. MR 0043280
Cooperman, The Legendre Transformation, Master’s thesis (unpub.) New York University (1948).
- Clair G. Maple, The Dirichlet problem: Bounds at a point for the solution and its derivatives, Quart. Appl. Math. 8 (1950), 213–228. MR 40499, DOI https://doi.org/10.1090/S0033-569X-1950-40499-8
J. McConnell, The hypercircle of approximation for a system of partial differential equations of the second order, Proc. Roy. Irish Acad., 41, 53A.
Trefftz, Ein Gegenstück zum Ritzschen Verfahren, Proc. 2nd Int. Congress Appl. Mech., Zurich, 131, (1926). Konvergenz und Fehlerschätzung beim Ritzschen Verfahren, Math. Ann. 100, 503, (1928).
O. Friedrichs, Die Randwert-und-Eigenwertprobleme aus der Theorie der elastischen Platten, Math. Ann. 98, 206, (1927-28). Ein Verfahren der Variationsrechnung..., Nach. der Ges. d. Wiss. zu Gottingen, 13, 1929.
Courant and D. Hilbert, Die Methoden der Mathematischen Physik, 1, Interscience Publishers, New York, 1943, 228-231; 199-209; 2, Chap. 7.
L. Synge, The method of the hypercircle in function-space for boundary-value problems, Proc. Roy. Soc. A 191, 447, (1947).
Upper and lower bounds for the solutions of problems in elasticity, Proc. Roy. Irish Acad. 53A, 41, (1950).
Pointwise bounds for the solutions of certain boundary-value problems, Proc. Roy. Soc. A 208, 170, (1951).
Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Q. Appl. Math, 5, 241, (1947).
J. Greenberg, The determination of upper and lower bounds for the solution of the Dirichlet problem, J. Math. Phys. 27, 161, (1948).
B. Diaz and H. J. Greenberg, Upper and lower bounds for the solutions of the first biharmonic boundary value problem, J. Math Phys. 27, 193, (1948).
B. Diaz and A. Weinstein, Schwarz’s inequality and the methods of Rayleigh-Ritz and Trefftz, J. Math. Phys. 27, 133, (1948).
B. Diaz, Upper and lower bounds for quadratic functionals, Proc. Symposium on Spectral Theory, Oklahoma A. & M. College, Stillwater, 279, 1951.
Cooperman, The Legendre Transformation, Master’s thesis (unpub.) New York University (1948).
G. Maple, The Dirichlet problem: bounds at a point for the solution and its derivatives, QAM, 213, (1950).
J. McConnell, The hypercircle of approximation for a system of partial differential equations of the second order, Proc. Roy. Irish Acad., 41, 53A.
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Article copyright:
© Copyright 1953
American Mathematical Society
