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

Abstract

<p style='text-indent:20px;'>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.</p>

Hyers-Ulam stability of linear quaternion-valued differential equations

“…This part introduces some basic symbols, definitions, and concepts of quaternion algebra [15,24,25]. Let H stand for the quaternion set, R represent the field of real numbers and C represent the field of complex numbers.…”

“…Zhang [28] studied the global structure of the quaternion Bernoulli equation. Xia et al [24,25] gave the stability results of quaternion periodic systems and the variation of constants formula in the sense of quaternion, and presented an algorithm for solving linear non-homogeneous QDEs. Chen et al [4] derived an explicit quaternion norm estimation in the sense of quaternion, proved that the first-order linear QDEs is asymptotically stable and Hyers-Ulam stable and the nth-order linear QDEs is generalized Hyers-Ulam stable.…”

Stability analysis of linear quaternion-valued differential equation using integral transform

“…A method for figuring out the basic matrix of linear systems with many eigenvalues was presented by Kou et al [13]. In the meantime, Xia et al [22] established the variation of constants formula in the quaternion sense, gave stability results for quaternion periodic systems, and presented an algorithm for solving linear nonhomogeneous QDEs. Suo et al [19] simultaneously presented solutions in both the complex numbers and quaternion settings for linear quaternion-valued impulsive differential equations.…”