基于变分法的回火分数阶脉冲微分系统分析

数学物理学报 ›› 2021, Vol. 41 ›› Issue (2): 415-426.

基于变分法的回火分数阶脉冲微分系统分析

任晶,翟成波*()

^{山西大学数学科学学院太原 030006}

收稿日期:2020-04-29 出版日期:2021-04-26 发布日期:2021-04-29
通讯作者: 翟成波 E-mail:cbzhai@sxu.edu.cn
基金资助:
山西省自然科学基金(201901D111020);山西省研究生创新项目(2019BY014)

Analysis of Impulsive Tempered Fractional Differential System via Variational Approach

Jing Ren,Chengbo Zhai*()

^{School of Mathematical Sciences, Shanxi University, Taiyuan 030006}

Received:2020-04-29 Online:2021-04-26 Published:2021-04-29
Contact: Chengbo Zhai E-mail:cbzhai@sxu.edu.cn
Supported by:
the NSF of Shanxi Province(201901D111020);the Graduate Student Innovation Project of Shanxi Province(2019BY014)

摘要/Abstract

摘要：

该文在恰当的容许函数空间中讨论了具有瞬时和非瞬时脉冲的回火分数阶微分方程耦合系统的Dirichlet边值问题.利用变分法得到弱解存在性和唯一性的充分条件，进一步证明了每个弱解都是古典解.最后给出一个实例证明了理论结果的可行性.

关键词: 回火分数阶微分方程, Lax-Milgram定理, 变分法

Abstract:

The aim of this paper is to discuss a Dirichlet boundary value problem for tempered fractional differential system with instantaneous and non-instantaneous impulses in an appropriate admissible function space. By using variational method, we obtain the sufficient condition for the existence and uniqueness of weak solution, moreover, we show that every weak solution is a classical solution. In the end, an example is presented to highlight the feasibility of the theoretical results.

Key words: Tempered fractional differential equation, Lax-Milgram theorem, Variational method

中图分类号:

O175.8

任晶,翟成波. 基于变分法的回火分数阶脉冲微分系统分析[J]. 数学物理学报, 2021, 41(2): 415-426.

Jing Ren,Chengbo Zhai. Analysis of Impulsive Tempered Fractional Differential System via Variational Approach[J]. Acta mathematica scientia,Series A, 2021, 41(2): 415-426.

参考文献 15

1	Almeida R , Morgado M L . Analysis and numerical approximation of tempered fractional calculus of variations problems. J Comput Appl Math, 2019, 361, 1- 12
2	Agarwal R , Hristova S , O'Regan D . Non-Instantaneous Impulses in Differential Equations. Springer, Cham, 2017
3	Chipot M . Elements of Nonlinear Analysis. Boston: Birkhauser Basel, 2000
4	Heidarkhani S , Cabada A , Afrouzi G A , Moradi S , Caristi G . A variational approach to perturbed impulsive fractional differential equations. J Comput Appl Math, 2018, 341, 42- 60 doi: 10.1016/j.cam.2018.02.033
5	Jiao F , Zhou Y . Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput Math Appl, 2011, 62, 1181- 1199 doi: 10.1016/j.camwa.2011.03.086
6	Khaliq A , ur Rehman M . On variational methods to non-instantaneous impulsive fractional differential equation. Appl Math Lett, 2018, 83, 95- 102 doi: 10.1016/j.aml.2018.03.014
7	Kilbas A , Srivastava H , Trujillo J . Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier Science, 2006
8	Li D P , Chen F Q , An Y K . The existence of solutions for an impulsive fractional coupled system of (p, q)-Laplacian type without the Ambrosetti-Rabinowitz condition. Math Methods Appl Sci, 2019, 42, 1449- 1464 doi: 10.1002/mma.5435
9	Riewe F . Nonconservative Lagrangian and Hamiltonian mechanics. Phys Rev E, 1996, 53, 1890 doi: 10.1103/PhysRevE.53.1890
10	Sabzikar F , Meerschaert M M , Chen J H . Tempered fractional calculus. J Comput Phys, 2015, 293, 14- 28 doi: 10.1016/j.jcp.2014.04.024
11	Tian Y , Zhang M . Variational method to differential equations with instantaneous and non-instantaneous impulses. Appl Math Lett, 2019, 94, 160- 165 doi: 10.1016/j.aml.2019.02.034
12	Yang D , Wang J R , O'Regan D . A class of nonlinear non-instantaneous impulsive differential equations involving parameters and fractional order. Appl Math Comput, 2018, 321, 654- 671
13	姚旺进. 基于变分方法的脉冲微分方程耦合系统解的存在性和多重性. 数学物理学报, 2020, 40A (3): 705- 716 doi: 10.3969/j.issn.1003-3998.2020.03.017
	Yao W J . Existence and multiplicity of solutions for a coupled system of impulsive differential equations via variational method. Acta Math Sci, 2020, 40A (3): 705- 716 doi: 10.3969/j.issn.1003-3998.2020.03.017
14	Zhang W , Liu W B . Variational approach to fractional Dirichlet problem with instantaneous and non-instantaneous impulses. Appl Math Lett, 2020, 99, 105993 doi: 10.1016/j.aml.2019.07.024
15	Zhao Y L , Chen H B , Xu C J . Nontrivial solutions for impulsive fractional differential equations via Morse theory. Appl Math Comput, 2017, 307, 170- 179

[1]	姚旺进. 基于变分方法的脉冲微分方程耦合系统解的存在性和多重性[J]. 数学物理学报, 2020, 40(3): 705-716.
[2]	李容星,王文清,曾小雨. 带椭球势阱的Kirchhoff型方程的变分问题[J]. 数学物理学报, 2019, 39(6): 1323-1333.
[3]	吉蕾,廖家锋. 一类带临界指数的非齐次Kirchhoff型问题第二个正解的存在性[J]. 数学物理学报, 2019, 39(5): 1094-1101.
[4]	张晶. 一类次临界Bose-Einstein凝聚型方程组的渐近收敛行为和相位分离[J]. 数学物理学报, 2019, 39(3): 441-450.
[5]	康东升, 熊萍. 一类带有不同Rellich项的临界双调和方程组的非平凡解[J]. 数学物理学报, 2017, 37(6): 1105-1118.
[6]	黄德成, 陈守信. Ginzburg-Landau方程极小能量解存在性的新证明[J]. 数学物理学报, 2017, 37(2): 299-306.
[7]	徐嘉, 王燕霞, 代国伟. 与Fisher-Kolmogorov's方程相关的一个差分方程的最快异宿解[J]. 数学物理学报, 2016, 36(6): 1145-1156.
[8]	张琦. 带有位势的Schrödinger-Poisson系统解的存在性与多解性[J]. 数学物理学报, 2016, 36(1): 49-64.
[9]	宋洪雪, 闫庆伦. 一类带有Neumann边值条件的拟线性椭圆外部问题的多解性[J]. 数学物理学报, 2015, 35(5): 845-854.
[10]	张亚静, 郝江浩. 一个非齐次临界Neumann问题的多正解[J]. 数学物理学报, 2013, 33(4): 661-672.
[11]	熊明; 刘嘉荃; 曾平安. 一维 p-Laplacian方程的两点奇异边值问题正解的存在性[J]. 数学物理学报, 2007, 27(3): 549-558.
[12]	甘在会;张健. 一类耦合非线性Klein-Gordon方程组的驻波[J]. 数学物理学报, 2006, 26(4): 559-569.
[13]	康东升, 邓引斌. SobolevHardy不等式与临界双重调和问题[J]. 数学物理学报, 2003, 23(1): 106-114.