General theory of the higher-order linear quaternion q-difference equations through the quaternion determinant algorithm and the characteristic polynomial

The general theory of quaternion q-difference equations is completely different from traditional q-difference equations in the complex space for the special algebraic structure of the quaternion space, such as the non-commutativity of multiplication among elements. In this paper, some properties of basic quaternion q-discrete functions, such as the quaternion q-exponential function, are obtained. The existence, uniqueness, and extension theorems of the solution for higher-order linear quaternion q-difference equations (QQDCEs) are established through constructing quaternion q-stepwise approximation sequences, and the equivalent variable transform between higher-order linear QQDCEs and quaternion linear q-difference equations is given. Moreover, some basic results, such as the Wronskian formula, Liouville formula, and general solution structure theorems of higher-order linear QQDCEs with constant and variable coefficients, are established by applying the quaternion characteristic polynomial and the quaternion determinant algorithm. In addition, some particular and general solution formulas of non-homogeneous QQDCEs are obtained under the non-commutative condition of quaternion elements. Finally, several examples are provided to illustrate the feasibility of our obtained results.

[1]  Chao Wang,et al.  General theory of the higher-order quaternion linear difference equations via the complex adjoint matrix and the quaternion characteristic polynomial , 2021 .

[2]  D. O’Regan,et al.  Commutativity of quaternion‐matrix–valued functions and quaternion matrix dynamic equations on time scales , 2020, Studies in Applied Mathematics.

[3]  Chao Wang,et al.  Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales , 2020 .

[4]  K. Kou,et al.  Laplace transform: a new approach in solving linear quaternion differential equations , 2018 .

[5]  K. Kou,et al.  Solving Quaternion Ordinary Differential Equations with Two-Sided Coefficients , 2018 .

[6]  Yonghui Xia,et al.  A Unified Analysis of Linear Quaternion Dynamic Equations on Time Scales , 2016, 1607.00105.

[7]  Kit Ian Kou,et al.  Linear Quaternion Differential Equations: Basic Theory and Fundamental Results , 2015, 1510.02224.

[8]  Xiang Zhang,et al.  Global Structure of Quaternion Polynomial Differential Equations , 2014, 1407.7941.

[9]  Bahman Kalantari,et al.  Algorithms for quaternion polynomial root-finding , 2013, J. Complex..

[10]  Mahmoud H. Annaby,et al.  q-Fractional Calculus and Equations , 2012 .

[11]  Paweł Wilczyński,et al.  Quaternionic-valued ordinary differential equations. The Riccati equation , 2009 .

[12]  J. Llibre,et al.  One-dimensional quaternion homogeneous polynomial differential equations , 2009 .

[13]  Mahmoud H. Annaby,et al.  On the zeros of the second and third Jackson q-Bessel functions and their associated q-Hankel transforms , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Darryl D. Holm,et al.  Quaternions and particle dynamics in the Euler fluid equations , 2005, nlin/0512034.

[15]  J. Gibbon.  A quaternionic structure in the three-dimensional Euler and ideal magneto-hydrodynamics equations , 2002 .

[16]  D. O’Regan,et al.  Positive Solutions of Differential, Difference and Integral Equations , 1998 .

[17]  Fuzhen Zhang.  Quaternions and matrices of quaternions , 1997 .

[18]  Helmer Aslaksen,et al.  Quaternionic determinants , 1996 .

[19]  L. Vinet,et al.  Quantum symmetries of q‐difference equations , 1995 .

[20]  L. Vinet,et al.  Symmetries of theq-difference heat equation , 1994 .

[21]  P. Freund,et al.  The spectral problem for theq-Knizhnik-Zamolodchikov equation and continuousq-Jacobi polynomials , 1993, hep-th/9309144.

[22]  T. E. Mason.  On Properties of the Solutions of Linear q-Difference Equations with Entire Function Coefficients , 1915 .

[23]  R. D. Carmichael,et al.  The General Theory of Linear q-Difference Equations , 1912 .

[24]  F. H. Jackson.  q-Difference Equations , 1910 .

[25]  R. Agarwal,et al.  Theory of Translation Closedness for Time Scales , 2020 .

[26]  J. Morais,et al.  Real quaternionic calculus handbook , 2014 .

[27]  Jean Mawhin,et al.  Periodic solutions of quaternionic-valued ordinary differential equations , 2006 .