Stability of bounded solutions of linear functional equations
HTML articles powered by AMS MathViewer
- by Joel N. Franklin PDF
- Math. Comp. 25 (1971), 413-424 Request permission
Abstract:
The weak sequential compactness of reflexive Banach spaces is used to explain the fact that certain ill-posed, linear problems become well-posed if the solutions are required to satisfy a prescribed bound. Applications are made to the computability of solutions of ill-posed problems associated with elliptic and parabolic partial differential equations.References
-
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, Conn., 1923.
- Fritz John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. (4) 40 (1955), 129–142. MR 87224, DOI 10.1007/BF02416528
- Fritz John, Numerical solution of problems which are not well posed in the sense of Hadamard, Symposium on the numerical treatment of partial differential equations with real characteristics: Proceedings of the Rome Symposium (28-29-30 January 1959) organized by the Provisional International Computation Centre Libreria Eredi Virgilio Veschi, Rome, 1959, pp. 103–116. MR 0107983
- Fritz John, Continuous dependence on data for solutions of partial differential equations with a presribed bound, Comm. Pure Appl. Math. 13 (1960), 551–585. MR 130456, DOI 10.1002/cpa.3160130402
- L. E. Payne, On some non well posed problems for partial differential equations, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, N.Y., 1966, pp. 239–263. MR 0213749
- Robert D. Richtmyer, Difference methods for initial-value problems, Interscience Tracts in Pure and Applied Mathematics, Tract 4, Interscience Publishers, Inc., New York, 1957. MR 0093918
- Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Akadémiai Kiadó, Budapest, 1953 (French). 2ème éd. MR 0056821
- P. R. Garabedian, Partial differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0162045
- J. R. Cannon, Some numerical results for the solution of the heat equation backwards in time, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, N.Y., 1966, pp. 21–54. MR 0207221
- Jim Douglas Jr., Approximate continuation of harmonic and parabolic functions, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 353–364. MR 0202333
- Richard Saylor, Numerical elliptic continuation, SIAM J. Numer. Anal. 4 (1967), 575–581. MR 222461, DOI 10.1137/0704052
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Math. Comp. 25 (1971), 413-424
- MSC: Primary 47A50; Secondary 35R25
- DOI: https://doi.org/10.1090/S0025-5718-1971-0380461-7
- MathSciNet review: 0380461