New Development of Numerical Methods for Partial Differential Equations with Singularities

• Project/Area Number: 24540108
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Yamagata University
• Principal Investigator: FANG QING 山形大学, 理学部, 教授 (10243544)
• Research Collaborator: ZHANG Xiao-Yu
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2014: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 偏微分方程式 / 初期境界値問題 / 特異性 / 数値解法 / 高精度 / 有限差分法 / 数値解析 / 誤差評価 / 放物型偏微分方程式 / 初期値―境界値問題 / 高精度数値解法 / 有限差分スキーム / 被食者―捕食者システム / Hopf 分岐 / 数値シミュレーション / 数値数学 / 有理型関数 / 値分布 / 数値計算 / 微分方程式 / 境界値問題 / 応用数学

Outline of Final Research Achievements

Mathematical models in partial differential equations with singularities have been used frequently in studies of natural phenomena and social phenomena. They are important research themes for the development of every fields in engineering and financial engineering and so on. In this research project, the representative proposed a higher-order finite difference scheme for nonlinear two-point bpundary value problems. The representative also made progresses in the study of numerical solutions of initial-boundary value problems of parabolic equations on two dimensional spatial domains. Results of numerical analysis of the higher order are obtained.

Research Products

• [Journal Article] Convergence of the Swartztrauber-Sweet Method for Parabolic Problems on a Disk with Singularities (2014)
  • Author(s): Xiao-Yu Zhang and Qing Fang
  • Journal Title: Numerical Functional Analysis and Optimization Volume: 35 Issue: 3 Pages: 356-369
  • DOI: 10.1080/01630563.2013.811599
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Effect of Environment on Prey-Predator Systems with Numerical Simulation (2014)
  • Author(s): Qing Fang and Xiao-Yu Zhang
  • Journal Title: Communications in Computer and Information Science Volume: 461 Pages: 420-423
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Uniqueness of Meromorphic Functions and Differential Polynomials (2013)
  • Author(s): Xiao-Yu Zhang and Qing Fang
  • Journal Title: Information Volume: Vol.16, No.7(A) Pages: 4497-4508
  • Peer Reviewed
• [Journal Article] Superconvergence of FDM for IBV Problems of Parabolic Equations (2013)
  • Author(s): Xiao-Yu Zhang and Qing Fang
  • Journal Title: Information Volume: Vol.16, No.11 Pages: 7817-7826
  • Peer Reviewed
• [Journal Article] Effective condition number of finite difference method for poisson's equation involving boundary singularities (2012)
  • Author(s): Zi-Cai Li, Hung-Tsai Huang, Chien-Sen Huang, Tzon-Tzer Lu and Qing Fang
  • Journal Title: Numerical Functional Analysis and Optimization Volume: 32 Issue: 6 Pages: 659-681
  • DOI: 10.1080/01630563.2011.572267
  • Peer Reviewed
• [Journal Article] A sixth order numerical method for a class of nonlinear two-point boundary value problems (2012)
  • Author(s): Xiao-Yu Zhang and Qing Fang
  • Journal Title: Numerical Algebra, Control and Optimization Volume: 2 Issue: 1 Pages: 31-43
  • DOI: 10.3934/naco.2012.2.31
  • Peer Reviewed
• [Presentation] Superconvergence of finite difference solutions of elliptic and parabolic equations (2015)
  • Author(s): Qing Fang
  • Organizer: 日本数学会東北支部会
  • Place of Presentation: 東北大学
  • Year and Date: 2015-02-14
  • Invited
• [Presentation] Effect of Environment on Prey-Predator Systems with Numerical Simulation (2014)
  • Author(s): Qing Fang and Xiao-Yu Zhang
  • Organizer: The International Conference on Life System Modeling and Simulation 2014 \& Intelligent Computing for Sustainable Energy and Environment 2014
  • Place of Presentation: Shanghai University (China)
  • Year and Date: 2014-09-22
• [Presentation] Superconvergence of Solution Derivatives for FDM to Parabolic Problems (2014)
  • Author(s): Qing Fang
  • Organizer: Workshop on Computational Fluid Dynamics 2014
  • Place of Presentation: Hohai University (China)
  • Year and Date: 2014-09-18
  • Invited
• [Presentation] Finite Differential Methods for Elliptic Boundary Value Problems with Singularities (2013)
  • Author(s): Qing Fang
  • Organizer: Workshop on Computational Fluid Dynamics
  • Place of Presentation: Hohai University, Nanjing, China
  • Invited
• [Presentation] Effective Condition Numbers of Finite Difference Methods for Elliptic Equations with Singularities (2012)
  • Author(s): Xiao-Yu Zhang and Qing Fang
  • Organizer: 日本数学会２０１２年度秋期総合分会
  • Place of Presentation: 九州大学
• [Presentation] Superconvergence of solution derivatives for the FDM and FEM approximation to elliptic boundary value problems (2012)
  • Author(s): Qing Fang
  • Organizer: Workshop on Numerical Analysys
  • Place of Presentation: Hohai University, Nanjing, China
  • Invited
• [Presentation] Superconvergence of Solution Derivatives for FDM to Parabolic Problems
  • Author(s): Xiao-Yu Zhang and Qing Fang
  • Organizer: The Sixth International Conference on Information
  • Place of Presentation: 法政大学（アルカディア市ヶ谷 私学会館）
  • Invited
• [Presentation] Java による被食者―捕食者モデルのバースト振動への考察
  • Author(s): 川谷淳・方青
  • Organizer: 第42回数値解析シンポジウム
  • Place of Presentation: 愛媛大学（四国道後舘）
• [Presentation] On Solution Uniqueness of Elliptic Boundary Value Problem by Numerical Approach
  • Author(s): Qing Fang
  • Organizer: Workshop on Numerical Analysis and Scientific Computing
  • Place of Presentation: Beijing Forestry University (China)
  • Invited
• [Presentation] Effective Condition Numbers of Linear Systems Arising from Discretization of Elliptic Equations with Singularities
  • Author(s): Xiao-Yu Zhang and Qing Fang
  • Organizer: 日本応用数理学会２０１３年度年会
  • Place of Presentation: 九州大学（アクロス福岡）