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Application of Fractional Calculus to Degenerate Hyperbolic Equations

Research Project

Project/Area Number 03640194
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 解析学
Research InstitutionFukuoka University

Principal Investigator

SAIGO Megumi  Fukuoka University, Department of Applied Mathematics, Professor, 理学部, 教授 (10040403)

Co-Investigator(Kenkyū-buntansha) SENBA Takashi  Fukuoka University, Department of Applied Mathematics, Lecturer, 理学部, 講師 (30196985)
FUKUSHIMA Yukio  Fukuoka University, Department of Applied Mathematics, Associate Professor, 理学部, 助教授 (40099007)
WATANABE Masafumi  Fukuoka University, Department of Applied Mathematics, Professor, 理学部, 教授 (70078559)
SUYAMA Yoshihiko  Fukuoka University, Department of Applied Mathematics, Professor, 理学部, 教授 (70028223)
EBIHARA Yukiyoshi  Fukuoka University, Department of Applied Mathematics, Professor, 理学部, 教授 (00078601)
Project Period (FY) 1991 – 1992
Project Status Completed (Fiscal Year 1992)
Budget Amount *help
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1992: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1991: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsFractional calculus / Degenerate hyperbolic equations / Hypergeometric functions / Laplace and Stieltjes transforms / Axisymmetric differential operator / Theory of univalent functions / Mellin transform / Dual integral equations / 超幾何級数 / 積分変換
Research Abstract

The research on definitions and properties of generalized fractional calculus, and the investigation on solutions of the Euler-Darboux equation as an application of the fractional calculus were already published. Further, various properties of hypergeometric functions, especially behavior of the functions near the boundaries of convergence regions, were reported. In 1991 and 1992, we have the following results : On relations of products of generalized fractional integrals with the Laplace and Stieltjes transforms in the generalized function space ; On the composition of the fractional integrals and axisymmetric differential operator in potential theory ; On the problem of mapping property by the fractional integrals of the convex and starlike functions in the theory of univalent functions ; On representability of the fractional integrals by products of the Mellin and the Laplace transforms ; On applications of the fractional calculus to the H-function ; On the relation of the fractional calculus with the Laplace transform ; On solutions of dual integral equations involving the fractional calculus ; On the definition and properties of the multidimensional modified fractional calculus ; On the establishment of new integral representation formulas for multidimensional hypergeometric series ; On new integral transforms with Meijer's G-function as kernels and their convolutions ; On the application of the fractional calculus to certain probabilistic density function with multidimensional hypergeometric functions ; On the behavior near boundaries of convergence regions of the Lauricella series F_S, F_T, the Srivastava series H_C, and the Horn series H_3, G_1, G_2 ; On estimations and mapping properties of operators with power-logarithmic kernel in the generalized Holder space ; On properties in the space L_<nu,p> of integral transforms with the H-function as kernel.

Report

(3 results)
  • 1992 Annual Research Report   Final Research Report Summary
  • 1991 Annual Research Report
  • Research Products

    (39 results)

All Other

All Publications (39 results)

  • [Publications] Megumi SAIGO,Vu Kim Tuan: "Some integral representatious of multivariable hypergeometic functions" Rendicouti del Circolo Matematica di Palerwo. 41. 69-80 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi SAIGO,H.M.Srivastava: "The asymptotic behavior of Srivastava hypergeometic series Hc near the loundary of its convergence region" Kyungpook Mathematical Journal. 32. (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Vu Kim Tuan,Megumi SAIGO: "Multidimensional modified fractional calculus operators" Mathematische Nachrichter. 161. (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi SAIGO,H.M.Srivastava: "Some asymptotic formulas exhibiting the behaviors of the triple hypergeonetic series Fs and Ft near the loundaries of convergence regions" Funkcialay Ekvacioj. 34. 423-448 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] H.-J.Glaeshe,Megumi SAIGO: "Products of Laplace transform and fractional integrals on spaces of generalized functions" Mathematica Japonica. 37. 373-382 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] H.M.Srivastava,Megumi SAIGO,R.K.Raina: "Same existence and connectrion theorems associated with the Laplace transform and a certain class of integral operators" Journal of Mathematical Analysis and Applications. 172. 1-10 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] 西郷 恵 他6名: "改訂新版 線形代数学" 学術図書出版社, 239 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo and H.M.: "Srivastava : Some asymtotic formulas exhibiting the behaviors of the triple hypergeometric series F_S and F_T near the boundaries of their convergence regions" Funkcialaj Ekvacioj. Vol.34, No.3. 423-448 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo and R.K.Raina: "On the fractional calculus operator involving Gauss's series and its application to certain statistical distributions, Universidad del Zulia, Facultad de Ingenieria" Revista Technica. Vol.14, No.1, Edicion Especial. 53-62 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] A.A.Kilbas, Megumi Saigo and V.A.Zhuk: "On the composition of operators of generalized fractional integration with a differential operator in axisymmetric potential theory, (in Russian) Differential'nye Uravnenija" Vol.27, No.9. 1640-1642 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Shigeyoshi Owa, Megumi Saigo and David M.Furuto: "On a certain fractional operator, Proceedings of the Japan Academy, Series A" Vol.67, No.5. 151-153 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo and Nguyen Thanh Hai: "The asymptotic representations of Appell series F_1 and Horn series H_3 near boundaries of convergence regions" Fukuoka University Science Reports. Vol.21, No.1. 17-23 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo and S.B.Yakubovich: "On the of convolution integrals for G-transforms" Fukuoka University Science Reports. Vol.21, No.2. 181-193 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] H.-J.Glaeske and Megumi Saigo: "Products of Laplace transform and fractional integrals on spaces of generalized functions" Mathematica Japonica. Vol.37, No.2. 373-382 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo and Vu Kim Tuan: "Some integral representations of multivariable hypergeometric functions" Rendiconti del Circolo Matematico di Palermo. Serie II, Tomo 41, Fasc.1. 69-80 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo, R.K.Saxena and Jeta Ram: "Certain properties of operators of fractional integrations associated with Mellin and Laplace transformations" Current Topics in Analytic Function Theory (H.M.Srivastava and S.Owa, Editors). World Scientific. 291-304 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo and Nguyen Thanh Hai: "Asymptotic representations of the Horn series G_1 and G_2 near boundaries of their convergence regions" Fukuoka University Science Reports. Vol.22, No.1. 1-9 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo, R.K.Saxena and Jeta Ram: "On the fractional calculus operator associated with the H-function" Ganita Sandesh. Vol.6, No.1. 36-47 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] H.M.Srivastava, Megumi Saigo and R.K.Raina: "Some existence and connection theorems associated with the Laplace transform and a certain class of integral operators" Journal of Mathematical Analysis and Applications. Vol.172, No.1. 1-10 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] A.A.Kilbas, Megumi Saigo and S.A.Shlapakov: "Integral transforms with Fox's H-function in Lnu,gamma spaces" Fukuoka University Science Reports. Vol.23, No.1. 9-31 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo, R.K.Saxena and Jeta Ram: "Application of generalized fractional calculus operators in the solution of certain dual integral equations" On New Developments in the Theory of Univalent Functions, Surikaisekikenkyusyo Kokyuroku, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan. 821. 56-69 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] A.A.Kilbas, Megumi Saigo and S.A.Shlapakov: "Integral transforms with Fox's H-function in spaces of summable functions" On New Developments in the Theory of Univalent Functions, Surikaisekikenkyusyo Kokyuroku, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan. 821. 70-86 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] A.A.Kilbas, Megumi Saigo and Silla Bubakar: "Zygmund type estimates and mapping properties of operators with power-logarithmic kernels in generalized Holder spaces" On New Developments in the Theory of Univalent Functions, Surikaisekikenkyusyo Kokyuroku, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan. 821. 87-100 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Megumi Saigo and H.M.Srivastava: "The asymptotic behavior of the Srivastava hypergeometric series Hc near the boundary of its convergence region" kyungpook Mathematical Journal. Vol.32, No.2. 5-10 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Vu Kim Tuan and Megumi Saigo: "Multidimensional modified fractional calculus operators" Mathematische Nachrichten. Band 161. (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] H.-J.Glaeske and Megumi Saigo: "Stieltjes transforms and fractional integrals on spaces of generalized functions" Mathematica Japonica.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] H.-J.Glaeske: "Products of Laplace transform and fractional integrals on spaces of generalized functions" Mathematica Japonica. 37. 373-382 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Megumi Saigo: "Certain properties of operators of fractional integrations associated with Mellin and Laplace transformations" Current Topics in Analytic Function Theory (H.M.Srivastava and S.Owa,Editors). 291-304 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Megumi Saigo: "On the fractional calculus operators associated with the H-function" Ganita Sandesh. 6. 36-47 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Vu Kim Tuan: "Multidimeusional modified factional calculus operators" Mathematische Nachrichten. 161. (1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] A.A.Kilbas: "Integral transforms with Fox's H-function in Lv,r spaces" Fukuoka University Scienu Reports. 23. 9-31 (1993)

    • Related Report
      1992 Annual Research Report
  • [Publications] Yukiyoshi Ebihara: "Global Estimates for Solutions of initial-boumnday value problem for a semilinear wave eguation with a singulan damping" Advanus in Mathematisal Scienus and Applications. 1. 99-111 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Takasi Senba: "On level sets of the solutins for Ut=Uxy+F(u) with F(u)=u^α or u(log(1/u))^β_+" Funkcialaj Ekvacioj. 35. 117-135 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Megumi Saigo: "Some integral Representations of multivariable hypergeometric functions" Rendiconti del Circolo Matematico di Palermo,Serie II. 41. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] Megumi Saigo: "Some asymototic formulas exhibiting the behaviors of the triple hypergeometric series F_s and F_T near the boundaries of their convergence regions" Funkcialaj Ekvacioj. 34. 423-448 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Megumi Saigo: "On the fractional calculus operator involving Gauss's series and its qpplication to certain statistical distributions" Universidad del Zulia,Facultad de Ingenieria,Revista Technica. 14. 53-62 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] А.А.Килбас,[A.A.Kilbas]: "О композиции операторов обобщенного дробного интегрирования с дифференциальным опеартором осесимметрической теории потенциала[On Compositions of Generalized Factional Calculus Operators with the Axisymmetric Differential Operator of the Potential Theory]" Дцфференчаные Уравненця[Differentsal'nye Uravneniya]. 27. 1640-1642 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Yukiyoshi Ebihara: "Global estimates for solutions of initial-boundary value problem for a semilinear wave equation with a singular daming" Advances in Mathematical Sciences. 1. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] Masafumi Watanabe: "Strong convergence of an unbounded stochastic approximation algorithm under general conditions" Bulletin of Information and Cybernetics. 25. (1992)

    • Related Report
      1991 Annual Research Report

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Published: 1991-04-01   Modified: 2016-04-21  

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