Laplace transform: a new approach in solving linear quaternion differential equations

Abstract

The theory of real quaternion differential equations has recently received more attention, but significant challenges remain the non‐commutativity structure. They have numerous applications throughout engineering and physics. In the present investigation, the Laplace transform approach to solve the linear quaternion differential equations is achieved. Specifically, the process of solving a quaternion different equation is transformed to an algebraic quaternion problem. The Laplace transform makes solving linear ODEs and the related initial value problems much easier. It has two major advantages over the methods discussed in literature. The corresponding initial value problems can be solved without first determining a general solution. More importantly, a particularly powerful feature of this method is the use of the Heaviside functions. It is helpful in solving problems, which is represented by complicated quaternion periodic functions. Copyright © 2017 John Wiley & Sons, Ltd.

Commutativity of quaternion‐matrix–valued functions and quaternion matrix dynamic equations on time scales

“…In this section, we will introduce the quaternion-related research developments description, which includes the qualitative analysis of quaternion dynamic equations, the solving methods of the quaternion dynamic equations (see Refs. 4,5,10,11,15,33), the quaternion matrix and the corresponding determinant algorithm (see Refs. 19,[34][35][36][37][38][39][40], the derivatives theory in quaternion space (see Refs.…”

“…1, 2) and a quaternionic dynamical structure commonly appears in many related research subjects such as differential geometry, fluid mechanics, attitude dynamics, neural networks (see Refs. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. In the literature, 11 Kou and Xia established a basic theory of linear quaternion differential equations and proposed the conjugate transposed matrix algorithms to obtain the Liouville formula for 2 × 2 quaternion dynamic equations.…”

Existence and global exponential stability of almost periodic solution for quaternion‐valued high‐order Hopfield neural networks with delays via a direct method

“…On the one hand, because quaternion‐valued functional differential equations have important applications in the fields of physics, communication, computer graphics, and neural networks, the study of various qualitative properties of quaternion‐valued functional differential equations has important theoretical and practical values . However, the quaternion multiplication does not satisfy the commutative law, which makes it difficult to study the qualitative behaviors of quaternion‐valued differential equations.…”

Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays

“…Subsequently, the author of [15,16] studied the existence of periodic solution of the quaternion Riccati equation with two-sided coefficients. For more works related the problem of the existence of periodic solutions of the quaternion differential equations, we refer to [17,18] and the references cited therein. As we know, anti-periodic functions as a special class of the quasi-periodic functions are periodic functions, but not all periodic functions are anti-periodic ones.…”