Time discretization in the backward solution of parabolic equations. I
Author:
Lars Eldén
Journal:
Math. Comp. 39 (1982), 5368
MSC:
Primary 65M30
MathSciNet review:
658214
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Abstract: The problem of solving a parabolic partial differential equation backwards in time by a method related to the TikhonovPhillips regularization method is considered. Time discretizations based on Padé approximations of the exponential function are studied, and a priori estimates of the step length are given, which guarantee an almost optimal error bound. The computational efficiency of different discretizations is discussed. Some numerical examples are given. In Part II of this paper we study the backward beam method, and the same error estimates are obtained. A new scheme for time descretization based on Padé approximation is discussed.
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 L. Eldén, "Regularization of the backward solution of parabolic problems," Inverse and Improperly Posed Problems in Differential Equations (G. Anger, ed.), AkademieVerlag, Berlin, 1979. MR 536169 (80e:65109)
 [4]
 R. E. Ewing, "The approximation of certain parabolic equations backward in time by Sobolev equations," SIAM J. Math. Anal., v. 6, 1975, pp. 283294. MR 0361447 (50:13892)
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 J. N. Franklin, "Minimum principles for illposed problems," SIAM J. Math. Anal., v. 9, 1978, pp. 638650. MR 498340 (80g:65065)
 [6]
 A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. MR 0445088 (56:3433)
 [7]
 P. M. Hummel & C. L. Seebeck, Jr., "A generalization of Taylor's expansion," Amer. Math. Monthly, v. 56, 1949, pp. 243247. MR 0028907 (10:516i)
 [8]
 P. Manselli & K. Miller, "Dimensionality reduction methods for efficient numerical solution, backward in time, of parabolic equations with variable coefficients," SIAM J. Math. Anal., v. 11, 1980, pp. 147159. MR 556505 (81h:65097)
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 V. A. Morozov, "On the restoration of functions with guaranteed accuracy," Numerical Analysis in Fortran, Moscow Univ. Press, Moscow, 1979, pp. 4665. (Russian)
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 H. Padé, "Sur la représentation approchée d'une fonctions par des fractions rationelles," Thesis, Ann. École Norm. (3), v. 9, 1892.
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 L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, Pa., 1975. MR 0463736 (57:3678)
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 D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind," J. Assoc. Comput. Mach., v. 9, 1962, pp. 8497. MR 0134481 (24:B534)
 [13]
 V. N. Strakhov, "Solution of incorrectlyposed problems in Hilbert space," Differential Equations, v. 6, 1970, pp. 11361140. (Russian)
 [14]
 A. N. Tikhonov, "Solution of incorrectly formulated problems and the regularization method," Dokl. Akad. Nauk SSSR, v. 151, 1963, pp. 501504; English transl, in Soviet Math. Dokl., v. 4, 1963, pp. 10351038.
 [15]
 R. S. Varga, "On higher order stable implicit methods for solving parabolic partial differential equations," J. Math, and Phys., v. 40, 1961, pp. 220231. MR 0140191 (25:3613)
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DOI:
http://dx.doi.org/10.1090/S00255718198206582149
PII:
S 00255718(1982)06582149
Article copyright:
© Copyright 1982
American Mathematical Society
