American Institute of Mathematical Sciences

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July  2022, 15(7): 1685-1697. doi: 10.3934/dcdss.2021162

An algorithm for solving linear nonhomogeneous quaternion-valued differential equations and some open problems

Yonghui Xia 1,, , Hai Huang 2, and  Kit Ian Kou 3,

1. 

College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China
2. 

School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
3. 

Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China

*Corresponding author: Yonghui Xia

Received  August 2021 Revised  October 2021 Published  July 2022 Early access  December 2021

Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.

Keywords: Differential equations, quaternion, solution, nonhomogeneous, algorithm.
Mathematics Subject Classification: 34K23, 34D30, 37C60, 37C55, 39A12.
Citation: Yonghui Xia, Hai Huang, Kit Ian Kou. An algorithm for solving linear nonhomogeneous quaternion-valued differential equations and some open problems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1685-1697. doi: 10.3934/dcdss.2021162
References:
[1] S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York, 1995. 
[2]

S. L. Adler, Quaternionic quantum field theory, Commun. Math. Phys., 104 (1986), 611-656.  doi: 10.1007/BF01211069.

[3]

J. J. Buckley and T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999), 241-248.  doi: 10.1016/S0165-0114(98)00323-6.

[4]

J. Campos and J. Mawhin, Periodic solutions of quaternionic-values ordinary differential equations, Ann. Mat. Pura Appl., 185 (2006), S109–S127. doi: 10.1007/s10231-004-0139-z.

[5]

D. Chen, M. Feckan and J. Wang, On the stability of linear quaternion-valued differential equations, Qual. Theor. Dyn. Syst., 2021.

[6]

L. Chen, Definition of determinant and Cramer solution over the quaternion field, Acta Math. Sinica (N.S.), 7 (1991), 171-180.  doi: 10.1007/BF02633946.

[7]

L. Chen, Inverse matrix and properties of double determinant over quaternion field, Sci. China Ser. A, 34 (1991), 528-540. 

[8]

D. Cheng, K. Kou and Y. Xia, Floquet theory for quaternion-valued differential equations, Qual. Theor. Dyn. Syst., 19 (2020), 1-23.  doi: 10.1007/s12346-020-00355-8.

[9]
[10]

A. Gasull, J. Llibre and X. Zhang, One dimensional quaternion homogeneous polynomial differential equations, J. Math. Phys., 50 (2009), 082705.  doi: 10.1063/1.3139115.

[11]

J. D. Gibbon, A quaternionic structure in the three-dimensional Euler and ideal magnetohydrodynamics equation, Physica D, 166 (2002), 17-28.  doi: 10.1016/S0167-2789(02)00434-7.

[12]

J. D. Gibbon, D. D. Holm, R. M. Kerr and I. Roulstone, Quaternions and particle dynamics in the Euler fluid equations, Nonlinearity, 19 (2006), 1969-1983.  doi: 10.1088/0951-7715/19/8/011.

[13]

K. Kou, W. Liu and Y. Xia, Solve the linear quaternion-valued differential equations having multiple eigenvalues, J. Math. Phys., 60 (2019), 023510. 

[14]

K. Kou, Y. Lou and Y. Xia, Zeros of a class of transcendental equation with application to bifurcation of DDE, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650062.  doi: 10.1142/S0218127416500620.

[15]

K. Kou and Y. Xia, Linear quaternion differential equations: Basic theory and fundamental results, Stud. Appl. Math., 141 (2018), 3-45.  doi: 10.1111/sapm.12211.

[16]

S. Leo and G. Ducati, Delay time in quaternionic quantum mechanics, J. Math. Phys., 53 (2012), 022102.  doi: 10.1063/1.3684747.

[17]

S. Leo and G. Ducati, Solving simple quaternionic differential equations, J. Math. Phys., 44 (2003), 2224-2233.  doi: 10.1063/1.1563735.

[18]

S. Leo, G. Ducati and C. Nishi, Quaternionic potentials in non-relativistic quantum mechanics, J. Phys. A., 35 (2002), 5411-5426.  doi: 10.1088/0305-4470/35/26/305.

[19] J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. 
[20]

X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Computat., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[21]

Z. Li and C. Wang, Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales, Open Math., 18 (2020), 353-377.  doi: 10.1515/math-2020-0021.

[22]

Z. Li, C. Wang, R. P. Agarwal and D. O'Regan, Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales, Stud Appl Math., 146 (2021), 139-210.  doi: 10.1111/sapm.12344.

[23]

Y. Liu, Y. Zheng, J. Lu, J. Cao and L. Rutkowski, Constrained quaternion-variable convex optimization: A quaternion-valued recurrent neural network approach, IEEE Trans. Neu. Netw. Learning syst., 31 (2020), 1022-1035.  doi: 10.1109/TNNLS.2019.2916597.

[24]

J. Marins, X. Yun, E. Bachmann, R. McGhee and M. Zyda, An extended Kalman filter for quaternion-based orientation estimation using MARG sensors, IEEE/RSJ International Conference on Intelligent Robots and Systems Maui, 2001. doi: 10.1109/IROS.2001.976367.

[25]

D. Peng, X. Li, R. Rakkiyappan and Y. Ding, Stabilization of stochastic delayed systems: Event-triggered impulsive control, Appl. Math. Comput., 401 (2021), 126054.  doi: 10.1016/j.amc.2021.126054.

[26]

V. N. Roubtsov and I. Roulstone, Holomorphic structures in hydrodynamical models of nearly geostrophic flow, Proc. R. Soc. London. Ser. A., 457 (2001), 1519-1531.  doi: 10.1098/rspa.2001.0779.

[27]

V. N. Rubtsov and I. Roulstone, Examples of quaternionic and Keller structures in Hamiltonian models of nearly geostrophic flow, J. Phys. A., 30 (1997), 3739. 

[28]

Y. Song and X. Tang, Stability, steady-state bifurcations and Turing patterns in a predator-prey model with herd behavior and prey-taxis, Stud. Appl. Math., 139 (2017), 371-404.  doi: 10.1111/sapm.12165.

[29]

Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301 (2005), 1-21.  doi: 10.1016/j.jmaa.2004.06.056.

[30]

L. Suo, M. Feckan and J. Wang, Quaternion-valued linear impulsive differential equations, Qual. Theor. Dyn. Syst., 20 (2021), 33.  doi: 10.1007/s12346-021-00467-9.

[31]

F. Udwadia and A. Schttle, An alternative derivation of the quaternion equations of motion for rigid-body rotational dynamics, J. Appl. Mech., 77 (2010), 044505.  doi: 10.1115/1.4000917.

[32]

J. R. Wertz, Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, The Netherlands, 1978.

[33]

P. Wilczynski, Quaternionic-valued ordinary differential equations. The Riccati equation, J. Differential Equations, 247 (2009), 2163-2187.  doi: 10.1016/j.jde.2009.06.015.

[34]

P. Wilczynski, Quaternionic-valued ordinary differential equations II. Coinciding sectors, J. Differential Equations, 252 (2012), 4503-4528.  doi: 10.1016/j.jde.2012.01.005.

[35] Y. Xia, K. Kou and and Y. Liu, Theoy and Applications of Quaternion-Valued Differential Equations, Science Press, Beijing, 2021. 
[36]

B. Zhang, W. Zhu, Y. Xia and Y. Bai, A unified analysis of exact traveling wave solutions for the fractional-order and integer-order Biswas-Milovic equation: Via bifurcation theory of dynamical system, Qual. Theor. Dyn. Syst., 19 (2020), 11.  doi: 10.1007/s12346-020-00352-x.

[37]

X. Zhang, Global structure of quaternion polynomial differential equations, Commun. Math. Phys., 303 (2011), 301-316.  doi: 10.1007/s00220-011-1196-y.

[38]

Y. Zhang and Y. Xia, Traveling wave solutions of generalized Dullin-Gottwald-Holm equation with parabolic law nonlinearity, Qual. Theor. Dyn. Syst., 20 (2021), 67.  doi: 10.1007/s12346-021-00503-8.

[39]

Y. Zhao, X. Li and J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467.  doi: 10.1016/j.amc.2020.125467.

[40]

J. Zhu and J. Sun, Existence and uniqueness results for quaternion-valued nonlinear impulsive differential systems, J. Syst. Sci. Complex., 31 (2018), 596-607.  doi: 10.1007/s11424-017-6158-9.

[41]

W. Zhu, Y. Xia, B. Zhang and Y. Bai, Exact traveling wave solutions and bifurcations of the time fractional differential equations with applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950041.  doi: 10.1142/S021812741950041X.

show all references

References:
[1] S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York, 1995. 
[2]

S. L. Adler, Quaternionic quantum field theory, Commun. Math. Phys., 104 (1986), 611-656.  doi: 10.1007/BF01211069.

[3]

J. J. Buckley and T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999), 241-248.  doi: 10.1016/S0165-0114(98)00323-6.

[4]

J. Campos and J. Mawhin, Periodic solutions of quaternionic-values ordinary differential equations, Ann. Mat. Pura Appl., 185 (2006), S109–S127. doi: 10.1007/s10231-004-0139-z.

[5]

D. Chen, M. Feckan and J. Wang, On the stability of linear quaternion-valued differential equations, Qual. Theor. Dyn. Syst., 2021.

[6]

L. Chen, Definition of determinant and Cramer solution over the quaternion field, Acta Math. Sinica (N.S.), 7 (1991), 171-180.  doi: 10.1007/BF02633946.

[7]

L. Chen, Inverse matrix and properties of double determinant over quaternion field, Sci. China Ser. A, 34 (1991), 528-540. 

[8]

D. Cheng, K. Kou and Y. Xia, Floquet theory for quaternion-valued differential equations, Qual. Theor. Dyn. Syst., 19 (2020), 1-23.  doi: 10.1007/s12346-020-00355-8.

[9]
[10]

A. Gasull, J. Llibre and X. Zhang, One dimensional quaternion homogeneous polynomial differential equations, J. Math. Phys., 50 (2009), 082705.  doi: 10.1063/1.3139115.

[11]

J. D. Gibbon, A quaternionic structure in the three-dimensional Euler and ideal magnetohydrodynamics equation, Physica D, 166 (2002), 17-28.  doi: 10.1016/S0167-2789(02)00434-7.

[12]

J. D. Gibbon, D. D. Holm, R. M. Kerr and I. Roulstone, Quaternions and particle dynamics in the Euler fluid equations, Nonlinearity, 19 (2006), 1969-1983.  doi: 10.1088/0951-7715/19/8/011.

[13]

K. Kou, W. Liu and Y. Xia, Solve the linear quaternion-valued differential equations having multiple eigenvalues, J. Math. Phys., 60 (2019), 023510. 

[14]

K. Kou, Y. Lou and Y. Xia, Zeros of a class of transcendental equation with application to bifurcation of DDE, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650062.  doi: 10.1142/S0218127416500620.

[15]

K. Kou and Y. Xia, Linear quaternion differential equations: Basic theory and fundamental results, Stud. Appl. Math., 141 (2018), 3-45.  doi: 10.1111/sapm.12211.

[16]

S. Leo and G. Ducati, Delay time in quaternionic quantum mechanics, J. Math. Phys., 53 (2012), 022102.  doi: 10.1063/1.3684747.

[17]

S. Leo and G. Ducati, Solving simple quaternionic differential equations, J. Math. Phys., 44 (2003), 2224-2233.  doi: 10.1063/1.1563735.

[18]

S. Leo, G. Ducati and C. Nishi, Quaternionic potentials in non-relativistic quantum mechanics, J. Phys. A., 35 (2002), 5411-5426.  doi: 10.1088/0305-4470/35/26/305.

[19] J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. 
[20]

X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Computat., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[21]

Z. Li and C. Wang, Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales, Open Math., 18 (2020), 353-377.  doi: 10.1515/math-2020-0021.

[22]

Z. Li, C. Wang, R. P. Agarwal and D. O'Regan, Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales, Stud Appl Math., 146 (2021), 139-210.  doi: 10.1111/sapm.12344.

[23]

Y. Liu, Y. Zheng, J. Lu, J. Cao and L. Rutkowski, Constrained quaternion-variable convex optimization: A quaternion-valued recurrent neural network approach, IEEE Trans. Neu. Netw. Learning syst., 31 (2020), 1022-1035.  doi: 10.1109/TNNLS.2019.2916597.

[24]

J. Marins, X. Yun, E. Bachmann, R. McGhee and M. Zyda, An extended Kalman filter for quaternion-based orientation estimation using MARG sensors, IEEE/RSJ International Conference on Intelligent Robots and Systems Maui, 2001. doi: 10.1109/IROS.2001.976367.

[25]

D. Peng, X. Li, R. Rakkiyappan and Y. Ding, Stabilization of stochastic delayed systems: Event-triggered impulsive control, Appl. Math. Comput., 401 (2021), 126054.  doi: 10.1016/j.amc.2021.126054.

[26]

V. N. Roubtsov and I. Roulstone, Holomorphic structures in hydrodynamical models of nearly geostrophic flow, Proc. R. Soc. London. Ser. A., 457 (2001), 1519-1531.  doi: 10.1098/rspa.2001.0779.

[27]

V. N. Rubtsov and I. Roulstone, Examples of quaternionic and Keller structures in Hamiltonian models of nearly geostrophic flow, J. Phys. A., 30 (1997), 3739. 

[28]

Y. Song and X. Tang, Stability, steady-state bifurcations and Turing patterns in a predator-prey model with herd behavior and prey-taxis, Stud. Appl. Math., 139 (2017), 371-404.  doi: 10.1111/sapm.12165.

[29]

Y. Song and J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301 (2005), 1-21.  doi: 10.1016/j.jmaa.2004.06.056.

[30]

L. Suo, M. Feckan and J. Wang, Quaternion-valued linear impulsive differential equations, Qual. Theor. Dyn. Syst., 20 (2021), 33.  doi: 10.1007/s12346-021-00467-9.

[31]

F. Udwadia and A. Schttle, An alternative derivation of the quaternion equations of motion for rigid-body rotational dynamics, J. Appl. Mech., 77 (2010), 044505.  doi: 10.1115/1.4000917.

[32]

J. R. Wertz, Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, The Netherlands, 1978.

[33]

P. Wilczynski, Quaternionic-valued ordinary differential equations. The Riccati equation, J. Differential Equations, 247 (2009), 2163-2187.  doi: 10.1016/j.jde.2009.06.015.

[34]

P. Wilczynski, Quaternionic-valued ordinary differential equations II. Coinciding sectors, J. Differential Equations, 252 (2012), 4503-4528.  doi: 10.1016/j.jde.2012.01.005.

[35] Y. Xia, K. Kou and and Y. Liu, Theoy and Applications of Quaternion-Valued Differential Equations, Science Press, Beijing, 2021. 
[36]

B. Zhang, W. Zhu, Y. Xia and Y. Bai, A unified analysis of exact traveling wave solutions for the fractional-order and integer-order Biswas-Milovic equation: Via bifurcation theory of dynamical system, Qual. Theor. Dyn. Syst., 19 (2020), 11.  doi: 10.1007/s12346-020-00352-x.

[37]

X. Zhang, Global structure of quaternion polynomial differential equations, Commun. Math. Phys., 303 (2011), 301-316.  doi: 10.1007/s00220-011-1196-y.

[38]

Y. Zhang and Y. Xia, Traveling wave solutions of generalized Dullin-Gottwald-Holm equation with parabolic law nonlinearity, Qual. Theor. Dyn. Syst., 20 (2021), 67.  doi: 10.1007/s12346-021-00503-8.

[39]

Y. Zhao, X. Li and J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467.  doi: 10.1016/j.amc.2020.125467.

[40]

J. Zhu and J. Sun, Existence and uniqueness results for quaternion-valued nonlinear impulsive differential systems, J. Syst. Sci. Complex., 31 (2018), 596-607.  doi: 10.1007/s11424-017-6158-9.

[41]

W. Zhu, Y. Xia, B. Zhang and Y. Bai, Exact traveling wave solutions and bifurcations of the time fractional differential equations with applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950041.  doi: 10.1142/S021812741950041X.

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