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Solving Quaternion Ordinary Differential Equations with Two-Sided Coefficients

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Abstract

The theory of quaternion differential equations (QDEs) has recently received a lot of attention. They have numerous applications in physics and engineering problems. In the present investigation, a new approach to solve the linear QDEs is achieved. Specifically, the solutions of QDEs with two-sided coefficients are studied via the adjoint matrix technique. That is, each quaternion can be uniquely expressed as a form of linear combinations of two complex numbers. By applying the complex adjoint representation of quaternion matrix, the connection between QDEs, with unilateral or two-sided coefficients, and a system of ordinary differential equations is achieved. By a novel specific algorithm, the solutions of QDEs with two-sided coefficients are fulfilled.

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References

  1. Adler, S.: Quaternionic quantum field theory. Commun. Math. Phys. 104(4), 611–656 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adler, S.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  3. Leo, S., Ducati, G.: Delay time in quaternionic quantum mechanics. J. Math. Phys. 53(2), 022102.1–022102.8 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Leo, S., Ducati, G., Nishi, C.: Quaternionic potentials in non-relativistic quantum mechanics. J.Phys. A Math. Gen. 35(26), 5411–5426 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Wertz, J.: Spacecraft Attitude Determination and Control. Kluwer Academic Publishers, Boston (1978)

    Book  Google Scholar 

  6. Bachmann, E., Marins, J., Zyda, M., Mcghee, R., Yun, X.: An extended kalman filter for quaternion-based orientation estimation using marg sensors. In: Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 4, pp. 2003–2011. IEEE (2001)

  7. Udwadia, F., Schttle, A.: An alternative derivation of the quaternion equations of motion for rigid-body rotational dynamics. J. Appl. Mech. 77(4), 044505.1–044505.4 (2010)

    Article  Google Scholar 

  8. Gibbon, J.: A quaternionic structure in the three-dimensional euler and ideal magneto-hydrodynamics equation. Phys. D Nonlinear Phenom. 166(1–2), 17–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gibbon, J., Holm, D., Kerr, R., Roulstone, I.: Quaternions and particle dynamics in the euler fluid equations. Nonlinearity 19(8), 1969–1983 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rubtsov, V., Roulstone, I.: Examples of quaternionic and keller structures in hamiltonian models of nearly geostrophicow. J. Phys. A Math.Gen. 30(4), L63–L68 (1997)

    Article  MATH  Google Scholar 

  11. Rubtsov, V., Roulstone, I.: Holomorphic structures in hydrodynamical models of nearly geostrophicow. Proc. Math. Phys. Eng. Sci. 457(2010), 1519–1531 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Handson, A., Ma, H.: Quaternion frame approach to streamline visualization. IEEE Trans. Vis. Comput. Gr. 1(2), 164–172 (1995)

    Article  Google Scholar 

  13. Yoshida, M., Kuroe, Y., Mori, T.: Models of hopfield-type quaternion neural networks and their energy functions. Int. J. Neural Syst. 15(01–02), 129–135 (2005)

    Article  Google Scholar 

  14. Isokawa, T., Kusakabe, T., Matsui, N., Peper, F.: Quaternion neural network and its application. In: Palade, V., Howlett, R.J., Jain, L. (eds.) Knowledge-Based Intelligent Information and Engineering Systems. KES 2003. Lecture Notes in Computer Science, vol. 2774, pp. 318–324. Springer, Berlin, Heidelberg (2003)

  15. Campos, J., Mawhin, J.: Periodic solutions of quaternionic-valued ordinary differential equations. Ann. Mat. 185(Suppl5), S109–S127 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Leo, S., Ducati, G.: Solving simple quaternionic differential equations. J. Math. Phys. 44(5), 2224–2233 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wilczynski, P.: Quaternionic valued ordinary differential equations. The Riccati equation. J. Differ. Equ. 247(7), 2163–2187 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wilczynski, P.: Quaternionic-valued ordinary differential equations II: coinciding sectors. J. Differ. Equ. 252(8), 4503–4528 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, X.: Global structure of quaternion polynomial differential equations. Commun. Math. Phys. 303(2), 301–316 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, Y., Zhang, D., Lu, J.: Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear Dyn. 87(1), 553–565 (2017)

  21. Liu, Y., Zhang, D., Lu, J., Cao, J.: Global u-stability criteria for quaternion-valued neural networks with unbounded time-varying delays. Inf. Sci. 360, 273–288 (2016)

    Article  Google Scholar 

  22. Xu, D., Xia, Y., Mandic, D.: Optimization in quaternion dynamic systems: gradient, Hessian, and learning algorithms. IEEE Trans. Neural Netw. Learn. Syst. 27(2), 249–261 (2016)

    Article  MathSciNet  Google Scholar 

  23. Kou, K., Xia, Y.: Linear quaternion differential equations: Basic theory and fundamental results (i). arXiv:1510.02224v2 (2015)

  24. Kou, K., Liu, W., Xia, Y.: Linear quaternion differential equations: basic theory and fundamental results (ii). arXiv:1602.01660 (2016)

  25. Xia, Y., Huang, H., Kou, K.: An algorithm for solving linear nonhomogeneous quaternion-valued differential equations. arXiv:1602.08713 (2016)

  26. Kravchenko, V., Kravchenko, V.: Quaternionic factorization of the schrodinger operator and its applications to some first order systems of mathematical physics. J. Phys. A 36(44), 11285–11297 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, F., Wei, Y.: Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices. Commun. Algebra. 29(6), 2363–2375 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kreyszig, E.: Advanced Engineering Mathematics, 9th edn. Wiley and Sons, Chichester (2005)

    MATH  Google Scholar 

  30. Huang, J.: A solution of two demensional linear homogeneous second order differential eguation with constant coefficients. J. Fuzhou Univ. 30(1), 20–22 (2002)

    Google Scholar 

  31. Brenner, J.: Matrices of quaternions. Pac. J. Math. 1(3), 329–335 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, W., Feng, L., Yao, B.: Zeros of two-sided quadratic quaternion polynomials. Adv. Appl. Clifford Algebras 24(3), 883–902 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shpakivskyi, V.: Linear quaternionic equations and their systems. Adv. Appl. Clifford Algebras 21(3), 637–645 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, Q., Yu, S., Xie, W.: Extreme ranks of real matrices in solution of the quaternion matrix equation axb = c with applications. Algebra Colloq. 17(2), 345–360 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Funding was provided by Universidade de Macau (Grant No. MYRG2015-00058-FST), National Natural Science Foundation of China (Grant Nos. 11401606, 11501015). Science and Technology Development Fund (Grant Nos. FDCT/099/2012/A3, FDCT/031/2016/A1).

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  1. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China

    Zhen Feng Cai & Kit Ian Kou

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  1. Zhen Feng Cai
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  2. Kit Ian Kou
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Correspondence to Kit Ian Kou.

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Cai, Z.F., Kou, K.I. Solving Quaternion Ordinary Differential Equations with Two-Sided Coefficients. Qual. Theory Dyn. Syst. 17, 441–462 (2018). https://doi.org/10.1007/s12346-017-0246-z

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  • DOI: https://doi.org/10.1007/s12346-017-0246-z

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