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A novel bond stress-slip model for 3-D printed concretes
An algorithm for solving linear nonhomogeneous quaternion-valued differential equations and some open problems
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 |
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.
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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