Quaternionic-valued ordinary differential equations. The Riccati equation

Wilczyński, Paweł

doi:10.1016/j.jde.2009.06.015

Cited by 51 publications

(44 citation statements)

References 16 publications

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“…But nevertheless, the solution of system (14) exists, that's why solution of the system of equations (3) should exist also (for the presented, previously chosen form of the solution (5)+ (6)). Approximate solutions of (14) are presented in the next section.…”

Section: Presentation Of the Time-dependent Part Of Solutionmentioning

confidence: 99%

A Riccati-type solution of 3D Euler equations for incompressible flow

Ershkov

Shamin

2020

Journal of King Saud University - Science

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In fluid mechanics, a lot of authors have been reporting analytical solutions of Euler and Navier-Stokes equations. But there is an essential deficiency of non-stationary solutions indeed.In our presentation, we explore the case of non-stationary flows of the Euler equations for incompressible fluids, which should conserve the Bernoulli-function to be invariant for the aforementioned system.We use previously suggested ansatz for solving of the system of Navier-Stokes equations (which is proved to have the analytical way to present its solution in case of conserving the Bernoulli-function to be invariant for such the type of the flows).Conditions for the existence of exact solution of the aforementioned type for the Euler equations are obtained. The restrictions at choosing of the form of the 3D nonstationary solution for the given constant Bernoulli-function B are considered. We should especially note that pressure field should be calculated from the given constant Bernoulli-function B, if all the components of velocity field are obtained.

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Section: Presentation Of the Time-dependent Part Of Solutionmentioning

confidence: 99%

A Riccati-type solution of 3D Euler equations for incompressible flow

Ershkov

Shamin

2020

Journal of King Saud University - Science

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“…Later, Campos and Mawhin studied the existence of periodic solutions of one‐dimensional first‐order periodic quaternion differential equation . Wilczynski continued this study and payed more attention on the existence of two periodic solutions of quaternion Riccati equations. Gasull et al.…”

Section: Introductionmentioning

confidence: 96%

Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

Kou

Xia

2018

Stud Appl Math

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Quaternion‐valued differential equations (QDEs) are a new kind of differential equations which have many applications in physics and life sciences. The largest difference between QDEs and ordinary differential equations (ODEs) is the algebraic structure. Due to the noncommutativity of the quaternion algebra, the set of all the solutions to the linear homogenous QDEs is completely different from ODEs. It is actually a right‐free module, not a linear vector space. This paper establishes a systematic frame work for the theory of linear QDEs, which can be applied to quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modeling, attitude dynamics, Kalman filter design, spatial rigid body dynamics, etc. We prove that the set of all the solutions to the linear homogenous QDEs is actually a right‐free module, not a linear vector space. On the noncommutativity of the quaternion algebra, many concepts and properties for the ODEs cannot be used. They should be redefined accordingly. A definition of Wronskian is introduced under the framework of quaternions which is different from standard one in the ODEs. Liouville formula for QDEs is given. Also, it is necessary to treat the eigenvalue problems with left and right sides, accordingly. Upon these, we studied the solutions to the linear QDEs. Furthermore, we present two algorithms to evaluate the fundamental matrix. Some concrete examples are given to show the feasibility of the obtained algorithms. Finally, a conclusion and discussion ends the paper.

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“…Zo ladek [10] given the complete description of dynamics of the Riccati equation. Wilczynski proved the existence of two periodic solutions of quaternionic Riccati equations in [11], and considered some sufficient conditions for the existence of at least one periodic solution of the quaternionic polynomial equations in [12,13]. Gasull et al [14] proved the existence of periodic orbits, homoclinic loops, invariant tori for a one-dimensional quaternionic autonomous homogeneous differential equation.…”

Section: Introductionmentioning

confidence: 99%