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MATHEMATICAL ANALYSIS AND NUMERICAL ANALYSIS OF SEVERAL KINDS OF DIFFERENTIAL EQUATIONS.

Research Project

Project/Area Number 06640335
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionWASEDA UNIVERSITY

Principal Investigator

MUROYA Yoshiaki  WASEDA UNIVERSITY,Sch.Sci.& Eng., PROFESSOR, 理工学部, 教授 (90063718)

Co-Investigator(Kenkyū-buntansha) TANAKA Kazunaga  SCH.SCI.AND ENG., WASEDA UNIVERSITY ASSOCIATE PROFESSOR, 理工学部, 助教授 (20188288)
TSUTSUMI Masayoshi  SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (70063774)
KORI Toshiaki  SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (50063730)
OTANI Mitsuharu  SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (30119656)
YAMADA Yoshio  SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (20111825)
和田 淳蔵  早稲田大学, 教育学部, 教授 (50063342)
Project Period (FY) 1994 – 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1996: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 1995: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1994: ¥1,400,000 (Direct Cost: ¥1,400,000)
KeywordsIMPROVED SOR METHOD / MULTIPLE RELAXATION PARAMETERS / ADAPTIVE IMPROVED SOR METHOD / IMPROVED SSOR METHOD / 改良SOR法 / 反復解法 / SOR法 / -
Research Abstract

To solve non-symmetric linear systems derived from the discretization of singular pertur-bation problems, we propose a generalized SOR method with multiple relaxation parameters, that is the improved SOR method with orderings and study its theory and practical use.
In the case of tridiagonal matrices, optimal choices of the parameters are examined : It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which are computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.
We apply the above method to two-dimensional cases, and propose the "adaptive improved block SOR method with orderings" for block tridiafonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special multiple relaxation parameters are selected and used with this method for an n * n block tridiagonal matrix whose block matrices are all n * n matrices, then this iterative method converges at most n^2 iterations.
We also proposed the improved SSOR method with orderings, which converges at most only one iteration for a tridiagonal system, and n iterations for a block tridiagonal system.
The generalized convergence theorems to the improved SOR method with orderings are also considered, and we study necessary and sufficient conditions for a matrix to be a generalized diagonally dominant.
Using the notation 'basic LUL factorization' of matrices, we give some techniques to obtain special multiple relaxation parameters such that the spectral radius of the iterative matrix is zero for the Hessenberg matrices and a class of matrices.

Report

(4 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • 1995 Annual Research Report
  • 1994 Annual Research Report
  • Research Products

    (50 results)

All Other

All Publications (50 results)

  • [Publications] 石渡恵美子: "Main convergence theorems for the inproved SOR meth of with orderings." Intern,J.Conput.Math.66. 123-147 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 石渡恵美子: "New criteria for generalized diagonally dominant matrices." Intern,J.Conput.Math.予定.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 石渡 恵美子: "Improved SOR method with orderings and rest methods." Japan J.Indust,Appl.Math.16No.2(予定). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 山田 義雄: "Global solutions for quasilinear parabolic systems with cross-diffusion effects." Nonlinear Analysis,Theory,Methods & Applications. 24. 1395-1412 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 山田 義雄: "Acertain class of reaction-diffusoon systems with feed back effects." Advances in Mathemctical Saiences and Applicctions. 5. 477-485 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 大谷 光春: "Almost periodic solutions of pericdic systems governed by subdiffereutiel operstors." Procedings of the A.M.S.(1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 田中和永: "Homoclimic orbits on now compact Riemannian manifolds for second order Hamittomiar systems." Rend.Sem.Math.Unin.Padova. 93. 153-176 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 田中和永: "A remarh or positore radicl selutions of the elliptic equation Δv+K(|x|)u^<(n+2)/(n-2)>=0 in R^n." Aner Math.Soc.123. 527-531 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 田中和永: "Periodic solutions of first order singular Hamiltonian systems" Nonlinear Analysis T.M.A.26. 691-706 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 井戸川 知之: "On munorical results for the linearizable generelized wrve shortening equstion." Norlinear Times and Digert. 2. 173-178 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 石渡哲哉: "A mumerical study of blow-uf solutions to u_t=u^δ(Δu+μu)." Proceedings of the Fourth MST Inter.Res.Tnst vol T.43. 187-199 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 石渡哲哉: "A Numerical Study of Blowup Solations to u_t=u^δ(Δu+μu)(Iapauese)" Transactions of the Japan Societz for Tnd.Aml.Meth.6. 219-231 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Mitsui: "Numerecal Analysis of Ordinary Differential Eorational and its Applicctions." World Scientific, 228 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] EMIKO,ISHIWATA: "MAIN COMERGENCE THEROREMS FOR THE IMPROVED SOR METHOD WITH ORDERINGS." INTERN.J.COMP.MATH.66. 123-147 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] EMIKO,ISHIWATA: "NEW CRITERIA FOR GENERALIZED DIAGONALLY DOMINANT METRICES." INTERN.J.COMP.MATH.(TO APPEAR).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] EMIKO ISHIWATA: "IMPROVED SOR MEIHOD WITH ORDERINGS AND DIRECT METHOD." JAPAN J.INDUST.APPLMATH.16(TO APPEAR). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] YOSHIO YAMADA: "GLOBAL SOLUTIONS FOR QUASI-LINEAR PARABOLIC SYSTEMS WITH CROSS-DIFFUSION EFFECTS." NONLENEAR ANALYSIS,THEORY,METHODS.AND APPLICATIONS. 1395-1412 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] YOSHIO YAMADA: "A CERTAIN CLASS OF REACTION-DIFFUSION SYSTEMS WITH TEED BACK EFFECTS." ADVANCES IN MATHEMATICAL SCIENCES AND APPLICATIONS. 5. 477-485 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] MITSUHARU OTANI: "ALMOST PERIODIC SOLUTIONS OF PERIODIC SYSTEMS GOVERNED BY SUBDIFFERENTIAL" PROCEEDINGS OF THE A.M.S.(1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] KAZUNAGA TANAKA: "HOMOCLINIC ORBITS ON NON-COMPACT RIEMANNIAN MANIFOLDS FOR SECOND ORDER HAMTLTONIAN SYSTEMS." REND.SEM.MATH.UNIV.PADOVA. 92. 153-176 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] KAZUNAGA TANAKA: "PERIODIC SOLUTIONS OF FIRST ORDER SINGULAR HAMTLTONIAN SYSTEMS." NONLINEAR ANALYSIS T.M.A.26. 691-706 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] TOMOYUKI IDOGAWA: "ON NUMERICAL RESULTS FOR THE LINEARIZABLE GENERALIZED CURVE SHORTENING EQUATION" NONLINEAR TIMES AND DIGEST. 2. 173-178 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] TETSUYA ISHIWATA: "A NUMERICAL STUDY OF BLOW-UP SOLUTIONS TO u_t=u^<delta> (DELTAu+muu)." PROCEEDINGS OF THE FOURTH MSJ INTERNATIONAL RES.INST.wlI. 43. 187-199 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] TETSUYA ISHIWATA: "A NUMERICAL STUDY OF BLOW-UP SOLUTIONS TO u^<2t>=u^<delta> (DELTAu+muu) (JAPANESE)" TRANSACTIONS OF THE JAPAN SOCIETY FOR INDUSTRIAL AND APPLIED MATH. 6. 219-231 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] TAKETOMO MITSUI: "NUMERICAL ANALYSIS OF ORDINARY DIFFERENTIAL EQUARIONS AND ITS APPLICATIOND". WORLD SCIENTIFIC PUBLISHING, 228 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] 石原和夫: "On nonlinear SOR-like methods,II- Enovergence of the SOR-Newton method for miluly nonlinear equctions" Japan Joronal of Industrial and Applied Mathematics. 14. (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] 石渡恵美子: "Simple Disortminmant for Matrices to be Generalized Diagonally Dominant" Intern.J.Conp.Math.(予定).

    • Related Report
      1996 Annual Research Report
  • [Publications] 中島主恵: "Positive steady states for prey-predator models with cross-diffusion" Adv.Differential Equations. 1. 1099-1122 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] 中島主恵: "On positive steady-states for some reactions diffusion system" Adv.Math.Sci.Appl.6. 279-289 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] 田中和永: "Periodic selutions of firstorder singulan Hamilton systems" Nonlinear Analysio : T.M.A.26. 691-706 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] 田中和永: "Home clinic orbits on non-compact Riemannian manifolds for second order haimltonian systems" Rend.Sem.Muth.Univ.Padova. 93. 153-176 (1995)

    • Related Report
      1996 Annual Research Report
  • [Publications] 山田 義雄: "Global solntions for guasilineia parabolic systems with crossdifferstocn effects" Nonlinear Analysis, Theay, Mahoas & Applications. (予定). (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 山田 義雄: "A certain class of reaction-diffuscon systems with feed boch effects" Advances in Matlematical Sciences an a Applications. (予定). (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 大谷 光春: "Almost periodic sdutions of periodic systems gorerned by subdifferantial operatard" Proceedings of the A. M. S.(予定). (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 田中 和永: "Homachimic orlits on non-compact Rieninnien nanibdas for secand order Heniltorion systems" Rond. Sem. Path. Unir. Padova. 93. 153-176 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 田中 和永: "A sencrk on pasitive radial Solutions of the elleptic equation Δμ+K(ba)μ^<(K-12)/(R-2)>=0inR^2," Amer. Math. Soc. 123. 527-531 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 田中 和永: "Periodic solutions of first order singnler Hamiltonian systems" Nonlinear Analysis: T. M. A.26. 691-706 (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] T. Mitsni: "Numorical Analysis of Ordinary Elifferenticl Eqmetions and its Applications" World Scientibic, 228 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] 星野弘喜: "Asymptotic behavion of glabal solutions for some reaction-diffusion systems" Nonlinear Analysis,Fleory.Methoes & Applications. 23. 639-650 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 山田義雄: "Glabal solutions for quasilinear parabclic systems with cross-diffusion effects" Nonlinear Analysis,Thery,Methods & Applications. (予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 山田義雄: "A certain class of reactions-diffusion systems with feedback effects" Advances in Mathematical Sciences and Applications. (予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 井戸川知之: "Analyticity and the best possille cinstants for Sobolev-Poincere in equelities" Advabces in Mathematcal Sciences and Applications. 4. 71-78 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 井上 弘: "Strong solutions of intral boundary value problems for best convection equations in non cylindrical domains" Nonlinear Analysis,Theory,Methods & Applications. 22. (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 大谷光春: "Almost periodic solutons of periodic systems governed by subdifferential operators" Proceedings of the A.M.S.(予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 郡 敏昭: "Index of the Dirac operator on S^4 and the infinite dimevsional Grassmanian on S^3" Japanese Journal of Math.(予定).

    • Related Report
      1994 Annual Research Report
  • [Publications] 郡 敏昭: "Lie algebra of the Infinitesimal automorphison on S^3 and its centrel extension" Kyoto Jaurnal of Math.(予定).

    • Related Report
      1994 Annual Research Report
  • [Publications] 和田淳蔵: "Compact homomorphisms on algebras of continuous functiuns" Tokyo J. Math.18. (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 田中直樹: "C-semigroups and the abstract Cauchyproblem II" International Conference on Evolution Equations,Glasgow. (予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 牧野慶吾: "Sufficieutly decomposable surfaces in the 3-sphere" Yokohama Math.J.42. 95-102 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 日野原幸利: "局所コホモロジー加群のcofinitenessとそのsocleの有限性について" 早大・教育学部学術研究. 43. (1995)

    • Related Report
      1994 Annual Research Report

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Published: 1994-04-01   Modified: 2016-04-21  

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