Solving Quaternion Ordinary Differential Equations with Two-Sided Coefficients

• Zhen Feng Cai ^[1] ; Kou, Kit Ian ^[1]
  1. [1] University of Macau

     University of Macau

     RAE de Macao (China)

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 17, Nº 2, 2018, págs. 441-462
• Idioma: inglés
• DOI: 10.1007/s12346-017-0246-z
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • 1. Adler, S.: Quaternionic quantum field theory. Commun. Math. Phys. 104(4), 611–656 (1986)
  • 2. Adler, S.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1994)
  • 3. Leo, S., Ducati, G.: Delay time in quaternionic quantum mechanics. J. Math. Phys. 53(2), 022102.1– 022102.8 (2012)
  • 4. Leo, S., Ducati, G., Nishi, C.: Quaternionic potentials in non-relativistic quantum mechanics. J.Phys. A Math. Gen. 35(26), 5411–5426 (2002)
  • 5. Wertz, J.: Spacecraft Attitude Determination and Control. Kluwer Academic Publishers, Boston (1978)
  • 6. Bachmann, E., Marins, J., Zyda, M., Mcghee, R., Yun, X.: An extended kalman filter for quaternionbased orientation estimation using marg...
  • 7. Udwadia, F., Schttle, A.: An alternative derivation of the quaternion equations of motion for rigid-body rotational dynamics. J. Appl....
  • 8. Gibbon, J.: A quaternionic structure in the three-dimensional euler and ideal magneto-hydrodynamics equation. Phys. D Nonlinear Phenom....
  • 9. Gibbon, J., Holm, D., Kerr, R., Roulstone, I.: Quaternions and particle dynamics in the euler fluid equations. Nonlinearity 19(8), 1969–1983...
  • 10. Rubtsov, V., Roulstone, I.: Examples of quaternionic and keller structures in hamiltonian models of nearly geostrophicow. J. Phys. A Math.Gen....
  • 11. Rubtsov, V., Roulstone, I.: Holomorphic structures in hydrodynamical models of nearly geostrophicow. Proc. Math. Phys. Eng. Sci. 457(2010),...
  • 12. Handson, A., Ma, H.: Quaternion frame approach to streamline visualization. IEEE Trans. Vis. Comput. Gr. 1(2), 164–172 (1995)
  • 13. Yoshida, M., Kuroe, Y., Mori, T.: Models of hopfield-type quaternion neural networks and their energy functions. Int. J. Neural Syst....
  • 14. Isokawa, T., Kusakabe, T., Matsui, N., Peper, F.: Quaternion neural network and its application. In: Palade, V., Howlett, R.J., Jain,...
  • 15. Campos, J., Mawhin, J.: Periodic solutions of quaternionic-valued ordinary differential equations. Ann. Mat. 185(Suppl5), S109–S127 (2006)
  • 16. Leo, S., Ducati, G.: Solving simple quaternionic differential equations. J. Math. Phys. 44(5), 2224–2233 (2003)
  • 17. Wilczynski, P.: Quaternionic valued ordinary differential equations. The Riccati equation. J. Differ. Equ. 247(7), 2163–2187 (2009)
  • 18. Wilczynski, P.: Quaternionic-valued ordinary differential equations II: coinciding sectors. J. Differ. Equ. 252(8), 4503–4528 (2012)
  • 19. Zhang, X.: Global structure of quaternion polynomial differential equations. Commun. Math. Phys. 303(2), 301–316 (2011)
  • 20. Liu, Y., Zhang, D., Lu, J.: Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear...
  • 21. Liu, Y., Zhang, D., Lu, J., Cao, J.: Global u-stability criteria for quaternion-valued neural networks with unbounded time-varying delays....
  • 22. Xu, D., Xia, Y., Mandic, D.: Optimization in quaternion dynamic systems: gradient, Hessian, and learning algorithms. IEEE Trans. Neural...
  • 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...
  • 26. Kravchenko, V., Kravchenko, V.: Quaternionic factorization of the schrodinger operator and its applications to some first order systems...
  • 27. Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997) 462 Z
  • 28. Zhang, F., Wei, Y.: Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices. Commun. Algebra....
  • 29. Kreyszig, E.: Advanced Engineering Mathematics, 9th edn. Wiley and Sons, Chichester (2005)
  • 30. Huang, J.: A solution of two demensional linear homogeneous second order differential eguation with constant coefficients. J. Fuzhou Univ....
  • 31. Brenner, J.: Matrices of quaternions. Pac. J. Math. 1(3), 329–335 (1951)
  • 32. Xu, W., Feng, L., Yao, B.: Zeros of two-sided quadratic quaternion polynomials. Adv. Appl. Clifford Algebras 24(3), 883–902 (2014)
  • 33. Shpakivskyi, V.: Linear quaternionic equations and their systems. Adv. Appl. Clifford Algebras 21(3), 637–645 (2011)
  • 34. Wang, Q., Yu, S., Xie, W.: Extreme ranks of real matrices in solution of the quaternion matrix equation axb = c with applications....