NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS

doi:10.1016/S0252-9602(18)30781-1

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (3): 756-768.doi: 10.1016/S0252-9602(18)30781-1

NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS

丁小丽¹, 蒋耀林²

1. Department of Mathematics, Xi'an Polytechnic University, Shaanxi 710048, China;
2. Department of Mathematics, Xi'an Jiaotong University, Shaanxi 710049, China

收稿日期:2016-12-09 修回日期:2017-03-11 出版日期:2018-06-25 发布日期:2018-06-25
通讯作者: Xiaoli DING E-mail:dingding0605@126.com
基金资助:
The work of the first author was supported by the Natural Science Foundation of China (NSFC) under grant 11501436 and Young Talent fund of University Association for Science and Technology in Shaanxi, China (20170701).

NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS

Xiaoli DING¹, Yaolin JIANG²

1. Department of Mathematics, Xi'an Polytechnic University, Shaanxi 710048, China;
2. Department of Mathematics, Xi'an Jiaotong University, Shaanxi 710049, China

Received:2016-12-09 Revised:2017-03-11 Online:2018-06-25 Published:2018-06-25
Contact: Xiaoli DING E-mail:dingding0605@126.com
Supported by:
The work of the first author was supported by the Natural Science Foundation of China (NSFC) under grant 11501436 and Young Talent fund of University Association for Science and Technology in Shaanxi, China (20170701).

摘要/Abstract

摘要：

Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.

关键词: Fractional differential-algebraic equations, nonnegativity of solutions, waveform relaxation, monotone convergence

Abstract:

Key words: Fractional differential-algebraic equations, nonnegativity of solutions, waveform relaxation, monotone convergence

丁小丽, 蒋耀林. NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS[J]. 数学物理学报(英文版), 2018, 38(3): 756-768.

Xiaoli DING, Yaolin JIANG. NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS[J]. Acta mathematica scientia,Series B, 2018, 38(3): 756-768.

参考文献

[1] Mainardi F. Fractals and Fractional Calculus Continuum Mechanics. Springer Verlag, 1997
[2] Malinowska A B, Torres D F M. Towards a combined fractional mechanics and quantization. Fract Calculus Appl Anal, 2012, 15:407-417
[3] Liu F, Anh V, Turner I. Numerical solution of the space fractional Fokker-Planck equation. J Comput Appl Math, 2004, 166:209-219
[4] Al-Sulami H, El-Shahed M, Nieto J J, Shammakh W. On fractional order dengue epidemic model. Mathematical Problems in Engineering, 2014, 2014:456537, 6 pages
[5] Area I, Losada J, Ndaïrou F, Nieto J J, Tcheutia D D. Mathematical modeling of 2014 Ebola outbreak (In Press). Mathematical Methods in the Applied Sciences. DOI:10.1002/mma.3794
[6] Dzielinski A, Sierociuk D, Sarwas G. Ultracapacitor parameters identification based on fractional order model. Budapest:Proc ECC'09, 2009
[7] Eslahchi M R, Mehdi Dehghan, Parvizi M. Application of the collocation method for solving nonlinear fractional integro-differential equations. J Comput Appl Math, 2014, 257:105-128
[8] Zhao J J, Xiao J Y, Ford Neville J. Collocation methods for fractional integro-differential equations with weakly singular kernels. Numer Algorithms, 2014, 65:723-743
[9] Gong C Y, Bao W M, Tang G J, Yang B, Liu J. An efficient parallel solution for Caputo fractional reaction-diffusion equation. J Supercomput, 2014, 68:1521-1537
[10] Stokes P W, Philippa B, Read W, White R D. Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart. J Comput Phys, 2015, 1:334-344
[11] Xu Q W, Hesthaven Jan S, Chen F. A parareal method for time-fractional differential equations. J Comput Phys, 2014, 293:173-183
[12] Mohammed Al-Refai, Yuri Luchko. Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives. Appl Math Comput, 2015, 257:40-51
[13] Zhang K Y, Xu J F, Donal O'Regan. Positive solutions for a coupled system of nonlinear fractional differential equations. Math Methods Appl Sci, 2015, 38:1662-1672
[14] Agarwal Ravi P, Donal O'Regan, Svatoslav Staněk. Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J Math Anal Appl, 2010, 1:57-68
[15] Mustafa G, Ismail Y. Positive solutions of higher-order nonlinear mulit-point fractional equations with integral boundary conditions. Frac Calc Appl Anal, 2016, 19:989-1009
[16] Li B X, Sun S R, Han Z L. Positive solutions for singular fractional differential equations with three-point boundary conditions. J Appl Math Comput, 2016, 52:477-488
[17] Li Y. Nonexistence of positive solutions for a semi-linear equation involing the fractional Laplcian in R^{N^*}. Acta Mathematica Scientia, 2016, 36:666-682
[18] Wang Guotao, Agarwal Ravi P, Cabada Alberto. Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl Math Lett, 2012, 6:1019-1024
[19] Tadeusz Kaczorek. Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans on Circuits and Systems-I, 2011, 6:1203-1210
[20] Lelarasmee E, Ruehli A, Sangiovanni-Vincentelli A. The waveform relaxation method for time domain analysis of large scale integrated circuits. IEEE Trans Computer-Aided Design, 1982, 1:131-145
[21] Leimkuhler B, Ruehli A E. Rapid convergence of waveform relaxation. Appl Numer Math, 1993, 11:211-224
[22] Jiang Y L. A general approach to waveform relaxation solutions of nonlinear differential-algebraic equations:The continuous-time and discrete-time cases. IEEE Trans Circuits and Systems-I, 2004, 51:1770-1780
[23] Zubik-kowal B, Vandewalle S. Waveform relaxation for functional differential equation. SIAM J Sci Comput, 1999, 21:207-226
[24] Ding X L, Jiang Y L. Waveform relaxation methods for fractional differential-algebraic equations. Fract Calculus Appl Anal, 2014, 17:585-604
[25] Ding X L, Jiang Y L. Semilinear fractional differential equations based on a new integral operator approach. Commun Nonlinear Sci Numer Simulat, 2012, 17:5143-5150
[26] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Amsterdam:Elsevier Science B V, 2006
[27] Sandberg I W. A nonnegativity-preservation property associated with certain systems of nonlinear differential equations//Proceedings of the 1974 IEEE International Conference on Systems, Man and Cybernetics. Los Alamitas, CA:IEEE Computer Society, 1974:230-233
[28] Nestruev J. Smooth manifolds and Observables. Berlin:Springer, 2003

NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS

NONNEGATIVITY OF SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL-ALGEBRAIC EQUATIONS

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