Application of a Modified Newton’s Iteration Method to Construct Solutions of Eigenvalue Problems of Nonlinear Partial Differential Operators

Field: Mathematics (Physical Sciences)

Previous article Next article Application of a Modified Newton’s Iteration Method to Construct Solutions of Eigenvalue Problems of Nonlinear Partial Differential OperatorsY. M. Chen and P. L. ChristiansenY. M. Chen and P. L. Christiansenhttps://doi.org/10.1137/0118028PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Joseph B. Keller and , Lu Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math., 19 (1966), 371–420 MR0205520 (34:5347) 0284.35004 CrossrefISIGoogle Scholar[2] Martin H. Millman and , Joseph B. Keller, Perturbation theory of nonlinear boundary-value problems, J. Mathematical Phys., 10 (1969), 342–361 10.1063/1.1664849 MR0237867 (38:6146) 0169.12702 CrossrefISIGoogle Scholar[3] Felix E. Browder, Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 176–183 MR0179459 (31:3707) 0135.15802 CrossrefISIGoogle Scholar[4] Felix E. Browder, Infinite dimensional manifolds and non-linear elliptic eigenvalue problems, Ann. of Math. (2), 82 (1965), 459–477 MR0203249 (34:3102) 0136.12002 CrossrefISIGoogle Scholar[5] Felix E. Browder, Nonlinear eigenvalue problems and Galerkin approximations, Bull. Amer. Math. Soc., 74 (1968), 651–656 MR0226453 (37:2043) 0162.20302 CrossrefISIGoogle Scholar[6] Herbert B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems, Bull. Amer. Math. Soc., 74 (1968), 887–891 MR0229985 (37:5551) 0162.16501 CrossrefISIGoogle Scholar[7] Herbert B. Keller, Nonlinear bifurcation, J. Differential Equations, 7 (1970), 417–434 10.1016/0022-0396(70)90090-2 MR0264255 (41:8851) 0208.36802 CrossrefISIGoogle Scholar[8] Herbert B. Keller, Positive solutions of some nonlinear eigenvalue problems, J. Math. Mech., 19 (1969/1970), 279–295 MR0251389 (40:4619) 0188.17103 ISIGoogle Scholar[9] Herbert B. Keller, Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal., 35 (1969), 363–381 10.1007/BF00247683 MR0255979 (41:639) 0188.17102 CrossrefISIGoogle Scholar[10] Herbert B. Keller and , Donald S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16 (1967), 1361–1376 MR0213694 (35:4552) 0152.10401 ISIGoogle Scholar[11] Donald S. Cohen, Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech., 17 (1967), 209–215 MR0213695 (35:4553) 0154.12701 ISIGoogle Scholar[12] Melvyn S. Berger, An eigenvalue problem for nonlinear elliptic partial differential equations, Trans. Amer. Math. Soc., 120 (1965), 145–184 MR0181821 (31:6047) 0142.08402 CrossrefISIGoogle Scholar[13] Melvyn S. Berger, A Sturm-Liouville theorem for nonlinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 543–582 MR0211299 (35:2181) 0147.09503 Google Scholar[14] M. Berger, J. B. Kelley and , S. Antman, A bifurcation theory for nonlinear elliptic partial differential equations and related systemsBifurcation Theory and Nonlinear Eigenvalue Problems, W. A. Benjamin, New York, 1969, 113–216 0181.11603 Google Scholar[15] Helmut H. Schaefer, R. E. Langer, Some nonlinear eigenvalue problems, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962), Univ. of Wisconsin Press, Madison, Wis., 1963, 117–137 MR0147917 (26:5429) 0118.32303 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails An Iteration Method for Solving Nonlinear Eigenvalue ProblemsY. -M. J. Demoulin and Y. M. ChenSIAM Journal on Applied Mathematics, Vol. 28, No. 3 | 12 July 2006AbstractPDF (676 KB)An interation method for studying the bifurcation of solutions of the nonlinear equations,L(?)u+?R(?,u)=0Numerische Mathematik, Vol. 23, No. 1 | 1 Feb 1974 Cross Ref Interaction of longitudinal waves with transverse waves in dispersive nonlinear elastic media. IQuarterly of Applied Mathematics, Vol. 29, No. 1 | 1 January 1971 Cross Ref Volume 18, Issue 2| 1970SIAM Journal on Applied Mathematics259-538 History Submitted:18 March 1969Published online:01 August 2006 InformationCopyright © 1970 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0118028Article page range:pp. 335-345ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics