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Application of Fractional Calculus to Degenerate Hyperbolic Equations

分数阶微积分在简并双曲方程中的应用

基本信息

批准号：
03640194
负责人：
SAIGO Megumi
金额：
$ 1.02万
依托单位：
Fukuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1992
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03640194/
关键词：
Fractional 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

项目摘要

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.

关于广义分数阶微积分的定义和性质的研究，以及对作为分数阶微积分应用的欧拉-达布方程的解的研究已经发表。此外，还报告了超几何函数的各种性质，特别是收敛区域边界附近函数的行为。 1991年和1992年，我们得到了以下成果：关于广义分数积分的乘积与广义函数空间中的Laplace和Stieltjes变换的关系；论位论中分数阶积分与轴对称微分算子的复合；关于单价函数理论中凸函数和星函数的分数阶积分的映射性质问题；关于梅林变换和拉普拉斯变换乘积的分数积分的可表示性；分数阶微积分在 H 函数中的应用；论分数阶微积分与拉普拉斯变换的关系；涉及分数阶微积分的对偶积分方程的解；论多维修正分数阶微积分的定义和性质；关于多维超几何级数新的积分表示公式的建立；以 Meijer 的 G 函数为核的新积分变换及其卷积；分数阶微积分在某些具有多维超几何函数的概率密度函数中的应用；关于 Lauricella 系列 F_S、F_T、Srivastava 系列 H_C 和 Horn 系列 H_3、G_1、G_2 收敛区域边界附近的行为；广义Holder空间中幂对数核算子的估计和映射性质；以H函数为核的积分变换空间L_<nu,p>的性质。