Study of non-local differential equations and convolution equations in the complex domain

• Project/Area Number: 23540186
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Chiba University
• Principal Investigator: ISHIMURA Ryuichi 千葉大学, 理学(系)研究科(研究院), 教授 (10127970)
• Co-Investigator(Renkei-kenkyūsha): OKADA Yasunori 千葉大学, 大学院理学研究科, 教授 (60224028)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 畳込み方程式 / 非局所微分方程式 / 包合的微分方程式系 / 非局所擬微分方程式 / 演算子法 / 特性集合 / 微分・差分方程式 / 代数解析学 / 包合系

Outline of Final Research Achievements

First we considered convolution equations with analytic-functional kernel in the space A-∞(D) of holomorphic functions with polynomial growth near the boundary of a convex domain D and we determined the directions to which any solutions to the corresponding homogeneous equation is prolongated. Secondely, we established the invertibility Theorem for a non-local differential equation and using this, we proved the existence and analytic continuation for the equation. As an application, we obtained the operational calculus for a differential-difference equation with constant coefficients.

Research Products

• [Int'l Joint Research] ロストフ南方数学研究所(ロシア連邦)
• [Int'l Joint Research] Nanyang 工科大学(シンガポール)
• [Journal Article] Extension of solutions of convolution equations in spaces of holomorphic functions with polynomial growth in convex domains (2012)
  • Author(s): A.V. Abanin, R. Ishimura and Le Hai Khoi
  • Journal Title: Bulletin des Sciences Mathematiques Volume: 136 Issue: 1 Pages: 96-110
  • DOI: 10.1016/j.bulsci.2011.06.002
  • Peer Reviewed
• [Journal Article] Existence and prolongation of analytic solutions of non-local differential equations (2012)
  • Author(s): R. Ishimura
  • Journal Title: Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie Volume: 55(103),No. 2 Pages: 179-197
  • Peer Reviewed
• [Journal Article] Convolution operators in A-∞ for convex domains (2012)
  • Author(s): A.V.Abanin, R, Ishimura and Le Hai Khoi
  • Journal Title: Arkiv f"or Mathematik Volume: 50-1 Issue: 1 Pages: 1-22
  • DOI: 10.1007/s11512-011-0146-4
  • Peer Reviewed
• [Journal Article] Exponential-polynomial bases for null spaces (2011)
  • Author(s): A.V.Abanin, R, Ishimura and Le Hai Khoi
  • Journal Title: Contemporary Mathematics, AMS Volume: 547 Pages: 1-15
  • Peer Reviewed