一类分数阶系统的稳定性和Laplace变换

摘要/Abstract

摘要：

该文研究了一类分数阶微分系统的Hyers-Ulam-Rassias稳定性.主要应用Laplace变换方法证明了这类分数阶系统是Hyers-Ulam-Rassias稳定的.通过具体的例子说明了所得理论结果的有效性.

关键词: Hyers-Ulam-Rassias稳定性, 分数阶微分系统, 分数阶微分方程, Laplace变换

Abstract:

This paper investigates the Hyers-Ulam-Rassias stability of a kind of fractional differential systems, and proves that this kind of fractional differential systems are Hyers-UlamRassias stable by the Laplace transform method. Two examples are given to illustrate the theoretical results.

Key words: Hyers-Ulam-Rassias stability, Fractional differential system, Fractional differential equation, Laplace transform

中图分类号:

O177.9

王春. 一类分数阶系统的稳定性和Laplace变换[J]. 数学物理学报, 2019, 39(1): 49-58.

Chun Wang. Stability of Some Fractional Systems and Laplace Transform[J]. Acta mathematica scientia,Series A, 2019, 39(1): 49-58.

参考文献 27

1	András S , Mészáros A R . Ulam-Hyers stability of dynamic equations on time scales via Picard operators. Appl Math Comput, 2013, 219: 4853- 4864
2	Gejji V D , Babakhani A . Analysis of a system of fractional differential equations. J Math Anal Appl, 2004, 293: 511- 522 doi: 10.1016/j.jmaa.2004.01.013
3	Gordji M E , Cho Y , Ghaemi M , Alizadeh B . Stability of the second order partial differential equations. J Inequal Appl, 2011, 81: 1- 10
4	Hegyi B , Jung S M . On the stability of Laplace's equation. Appl Math Lett, 2013, 26: 549- 552 doi: 10.1016/j.aml.2012.12.014
5	Jung S M . Hyers-Ulam stability of linear differential equations of first order. Appl Math Lett, 2004, 17: 1135- 1140 doi: 10.1016/j.aml.2003.11.004
6	Jung S M . Hyers-Ulam stability of linear differential equations of first order, Ⅲ. J Math Anal Appl, 2005, 311: 139- 146 doi: 10.1016/j.jmaa.2005.02.025
7	Jung S M . Hyers-Ulam stability of linear differential equations of first order, Ⅱ. Appl Math Lett, 2006, 19: 854- 858 doi: 10.1016/j.aml.2005.11.004
8	Jung S M . Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients. J Math Anal Appl, 2006, 320: 549- 561 doi: 10.1016/j.jmaa.2005.07.032
9	Kilbas A A , Srivastava H M , Trujillo J J . Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006
10	Li K , Peng J . Laplace transform and fractional differential equations. Appl Math Lett, 2011, 24: 2019- 2023 doi: 10.1016/j.aml.2011.05.035
11	Lungu N , Popa D . Hyers-Ulam stability of a first order partial differential equation. J Math Anal Appl, 2012, 385: 86- 91 doi: 10.1016/j.jmaa.2011.06.025
12	Obloza M . Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt Prace Mat, 1993, 13: 259- 270
13	Obloza M . Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk-Dydakt Prace Mat, 1997, 14: 141- 146
14	Podlubny I . Fractional Differential Equations. San Diego: Academic Press, 1999
15	Popa D , Raşa I . On the Hyers-Ulam stability of the linear differential equation. J Math Anal Appl, 2011, 381: 530- 537 doi: 10.1016/j.jmaa.2011.02.051
16	Rezaei H , Jung S M , Rassias T M . Laplace transform and Hyers-Ulam stability of linear differential equations. J Math Anal Appl, 2013, 403: 244- 251 doi: 10.1016/j.jmaa.2013.02.034
17	Shen Y , Chen W . Laplace transform method for the Ulam stability of linear fractional differential equations with constant coefficients. Mediterr J Math, 2017, 14: 1- 17 doi: 10.1007/s00009-016-0833-2
18	Wang J , Zhou Y . Mittag-Leffler-Ulam stabilities of fractional evolution equations. Appl Math Lett, 2012, 25: 723- 728 doi: 10.1016/j.aml.2011.10.009
19	Wang J , Li X . A uniform method to Ulam-Hyers stability for some linear fractional equations. Mediterr J Math, 2016, 13: 625- 635 doi: 10.1007/s00009-015-0523-5
20	Wang C, Xu T Z. Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions. J Funct Spaces, 2014, 2014: Article ID: 398673
21	Wang C , Xu T Z . Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives. Appl Math, 2015, 60: 383- 393 doi: 10.1007/s10492-015-0102-x
22	Wang C , Xu T Z . Hyers-Ulam stability of differential operators on reproducing kernel function spaces. Complex Anal Oper Theory, 2016, 10: 795- 813 doi: 10.1007/s11785-015-0486-3
23	Wang C , Xu T Z . Hyers-Ulam stability of a class of fractional linear differential equations. Kodai Math J, 2015, 38: 510- 520 doi: 10.2996/kmj/1446210592
24	Wang C , Xu T Z . Stability of the nonlinear fractional differential equations with the right-sided RiemannLiouville fractional derivative. Discrete Contin Dyn Syst Ser S, 2017, 10: 505- 521
25	Xu T Z , Wang C , Rassias T M . On the stability of multi-additive mappings in non-Archimedean normed spaces. J Comput Anal Appl, 2015, 18: 1102- 1110
26	Xu T Z . On the stability of multi-Jensen mappings in β-normed spaces. Appl Math Lett, 2012, 25: 1866- 1870 doi: 10.1016/j.aml.2012.02.049
27	王春, 许天周. 拟Banach空间上含参数的二次-可加混合型函数方程的解和Hyers-Ulam-Rassias稳定性. 数学物理学报, 2017, 37A (5): 846- 859 doi: 10.3969/j.issn.1003-3998.2017.05.006
	Wang C , Xu T Z . Solution and Hyers-Ulam-Rassias stability of a mixed type quadratic-additive functional equation with a parameter in quasi-Banach spaces. Acta Math Sci, 2017, 37A (5): 846- 859 doi: 10.3969/j.issn.1003-3998.2017.05.006

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