Method For Algebraic Equations Research Articles

Investigating solitary wave solutions with enhanced algebraic method for new extended Sakovich equations in fluid dynamics

The present work aims to investigate solitary wave solutions for two recently developed extended equations in the context of (2+1)-dimensional and (3+1)-dimensional structures. The equations under consideration are of the Korteweg–de Vries (KdV) type, which are well-recognized as significant aspects of fluid dynamics. These equations have broad applications in physics, mathematics, and other scientific disciplines, particularly in the study of waves, soliton theory, plasma physics, biology and chemistry, and nonlinear phenomena. Its soliton solutions and integrability properties make it a fundamental model in various areas of research. This serves as the main motivation for our research work. To analyze these equations, we employ an advanced direct algebraic equation method capable of generating several sorts of solutions, including solitary and shock wave solutions, as well as their combination. In addition to these wave phenomena, singular solitons and solutions expressed in Jacobi and Weierstrass doubly periodic types have also been observed. The utilization of this outstanding technique and the subsequent acquisition of novel solutions demonstrate the originality of our study. This also allows further exploration of nonlinear models that accurately depict significant physical processes in our everyday existence.

On a mixed Galerkin method for semi-explicit index-1 integro-differential algebraic equations

On a mixed Galerkin method for semi-explicit index-1 integro-differential algebraic equations

A class of Petrov-Galerkin Krylov methods for algebraic Riccati equations

A class of Petrov-Galerkin Krylov methods for algebraic Riccati equations

A generalized ALI iteration method for nonsymmetric algebraic Riccati equations

A generalized ALI iteration method for nonsymmetric algebraic Riccati equations

Semiglobal Robust Consensus of General Linear MASs Subject to Input Saturation and Additive Perturbations.

This article considers the robust consensus problem of the general linear multiagent system (MAS) subject to both heterogeneous additive stable disturbances and input saturation. Distributed low gain feedback-based dynamic output feedback control protocols are proposed, which do not need the controller interaction. Algebraic Riccati equation and unified H∞ controller design method are employed to design the output feedback control protocols. It is established that under the assumption that each agent is asymptotically null controllable with bounded controls, semiglobal robust consensus can always be reached under the proposed controller. Furthermore, the design method specialized for leader-following consensus is addressed, under the assumption that the Laplacian matrix is diagonalizable, one can design the control protocol only with the number of follower agents. Finally, several simulations are provided to show the effectiveness of our results.

Modified Extended Lie-Group Method for Hessenberg Differential Algebraic Equations with Index-3

Hessenberg differential algebraic equations (Hessenberg-DAEs) with a high index play a critical role in the modeling of mechanical systems and multibody dynamics. Motivated by the widely used Lie-group differential algebraic equation (LGDAE) method, which handles index-2 systems, we first propose a modified extended Lie-group differential algebraic equation (MELGDAE) method for solving index-3 Hessenberg-DAEs and then provide a theoretical analysis to deepen the foundation of the MELGDAE method. Moreover, the performance of the MELGDAE method is compared with the standard methods RADAU and MEBDF on index-2 and -3 DAE systems, and it is demonstrated that the MELGDAE integrator exhibits a competitive performance in terms of high accuracy and the preservation of algebraic constraints. In particular, all differential variables in index-3 Hessenberg-DAEs achieve second-order convergence using the MELGDAE method, which suggests the potential for extension to Hessenberg-DAEs with an index of 4 or higher.

SOR-based alternately linearized implicit iteration method for nonsymmetric algebraic Riccati equations

<abstract><p>In this paper, we propose a class of successive over relaxation-based alternately linearized implicit iteration method for computing the minimal nonnegative solution of nonsymmetric algebraic Riccati equations. Under certain conditions, we prove the convergence of the iterative method. Finally, numerical examples are given to show the iterative method is efficient.</p></abstract>

Transient Analysis of the Electro-Osmotic Flow of Multilayer Immiscible Maxwell Fluids in an Annular Microchannel

This work investigates the transient multilayer electro-osmotic flow of viscoelastic fluids through an annular microchannel. The dimensionless mathematical model of multilayer flow is integrated by the linearized Poisson-Boltzmann equation, the Cauchy momentum equation, the rheological Maxwell model, initial conditions, and the electrostatic and hydrodynamic boundary conditions at liquid-liquid and solid-liquid interfaces. Although the main force that drives the movement of fluids is due to electrokinetic effects, a pressure gradient can also be added to the flow. The semi-analytical solution for the electric potential distribution and velocity profiles considers analytical techniques as the Laplace transform method, with numerical procedures using the inverse matrix method for linear algebraic equations and the concentrated matrix exponential method for the inversion of the Laplace transform. The results presented for velocity profiles and velocity tracking at the transient regime reveal an interesting oscillatory behavior that depends on elastic fluid properties via relaxation times. The time required for the flow to reach steady-state is highly dependent on the viscosity ratios and the dimensionless relaxation times. In addition, the influence of other dimensionless parameters on the flow as the electrokinetic parameters, zeta potentials at the walls, permittivity ratios, ratio of pressure forces to electro-osmotic forces, number of fluid layers, and annular thickness are investigated. The findings of this study have significant implications for the precise control of parallel fluid transport in microfluidic devices for flow-focusing applications.

The fractional analysis of fusion and fission process in plasma physics

The space-time fractional non-linear Phi-4 equation is a significant equation to describes the fission and fusion process that ensued in chemical kinematics, solid-state physics, astrophysical fusion plasma, plasma physics, and electromagnetic interactions, etc. The Phi-4 non-linear partial differential equation is reshaped utilizing the three different fractional-order derivatives and constructed transformations corresponding to every fractional-order derivative to convert the partial differential equation into an ordinary differential equation. A new extended direct algebraic equation method was successfully applied to extract the solitons solutions. The solitons solutions are developed with the exponential, trigonometric, rational, and hyperbolic functions including different unknown constant parameters. The graphical interpretations of obtained solutions are also depicted by allocating the feasible values to unknown constant parameters. The proposed scheme is an effective and functional scientific technique to investigate different fractional systems and models in engineering and physics referenced to real physical problems.

A novel linear iteration method for M-matrix algebraic Riccati equations

A novel linear iteration method for M-matrix algebraic Riccati equations

Diverse chirped optical solitons and new complex traveling waves in nonlinear optical fibers

This paper studies chirped optical solitons in nonlinear optical fibers. However, we obtain diverse soliton solutions and new chirped bright and dark solitons, trigonometric function solutions and rational solutions by adopting two formal integration methods. The obtained results take into account the different conditions set on the parameters of the nonlinear ordinary differential equation of the new extended direct algebraic equation method. These results are more general compared to Hadi et al (2018 Optik 172 545–53) and Yakada et al (2019 Optik 197 163108).

About the cover: The Fine–Petrović Polygons and the Newton–Puiseux Method for Algebraic Ordinary Differential Equations

About the cover: The Fine–Petrović Polygons and the Newton–Puiseux Method for Algebraic Ordinary Differential Equations

Inheritance properties of Krylov subspace methods for continuous-time algebraic Riccati equations

We investigate the theory behind the Krylov subspace methods for large-scale continuous-time algebraic Riccati equations. We show that the solvability of the projected algebraic Riccati equation need not be assumed but can be inherited. This study of inheritance properties is the first of its kind. We study the stabilizability and detectability of the control system, the stability of the associated Hamiltonian matrix and perturbation in terms of residuals. Special attention is paid to the stabilizing and positive semi-definite properties of approximate solutions. Illustrative numerical examples for the inheritance properties are presented.

Krylov subspace methods for discrete-time algebraic Riccati equations

We apply the Krylov subspace methods to large-scale discrete-time algebraic Riccati equations. The solvability of the projected algebraic Riccati equation is not assumed but is shown to be inherited from the original equation. Solvability in terms of stabilizability, detectability, stability radius of the associated Hamiltonian matrix and perturbation theory are considered. We pay particular attention to the stabilizing and the positive semi-definite properties of approximate solutions. Illustrative numerical examples are presented.

Comparative Performance Evaluation of an Accuracy-Enhancing Lyapunov Solver

Lyapunov equations are key mathematical objects in systems theory, analysis and design of control systems, and in many applications, including balanced realization algorithms, procedures for reduced order models, Newton methods for algebraic Riccati equations, or stabilization algorithms. A new iterative accuracy-enhancing solver for both standard and generalized continuous- and discrete-time Lyapunov equations is proposed and investigated in this paper. The underlying algorithm and some technical details are summarized. At each iteration, the computed solution of a reduced Lyapunov equation serves as a correction term to refine the current solution of the initial equation. The best available algorithms for solving Lyapunov equations with dense matrices, employing the real Schur(-triangular) form of the coefficient matrices, are used. The reduction to Schur(-triangular) form has to be done only once, before starting the iterative process. The algorithm converges in very few iterations. The results obtained by solving series of numerically difficult examples derived from the SLICOT benchmark collections for Lyapunov equations are compared to the solutions returned by the MATLAB and SLICOT solvers. The new solver can be more accurate than these state-of-the-art solvers and requires little additional computational effort.

On the numerical solution of Fisher\u2019s equation with coefficient of diffusion term much smaller than coefficient of reaction term

Li et al. (SIAM J. Sci. Comput. 20:719–738, 1998) used the moving mesh partial differential equation (MMPDE) to solve a scaled Fisher’s equation and the initial condition consisting of an exponential function. The results obtained are not accurate because MMPDE is based on a familiar arc-length or curvature monitor function. Qiu and Sloan (J. Comput. Phys. 146:726–746, 1998) constructed a suitable monitor function called modified monitor function and used it with the moving mesh differential algebraic equation (MMDAE) method to solve the same problem of scaled Fisher’s equation and obtained better results.In this work, we use the forward in time central space (FTCS) scheme and the nonstandard finite difference (NSFD) scheme, and we find that the temporal step size must be very small to obtain accurate results. This causes the computational time to be long if the domain is large. We use two techniques to modify these two schemes either by introducing artificial viscosity or using the approach of Ruxun et al. (Int. J. Numer. Methods Fluids 31:523–533, 1999). These techniques are efficient and give accurate results with a larger temporal step size. We prove that these four methods are consistent for partial differential equations, and we also obtain the region of stability.

Collocation methods for integro-differential algebraic equations with index 1

Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

Modified alternately linearized implicit iteration method for M-matrix algebraic Riccati equations

Research on the theories and efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years. In this paper, we consider numerical solution of M-matrix algebraic Riccati equation and propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. Convergence of the MALI method is proved by choosing proper parameters for the nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the MALI method is effective and efficient in some cases.

Examination of extensions to transport theory that includes rough surface scattering

Sea surface forward scattering has important effects on shallow water propagation and reverberation at mid frequencies (i.e., 1–3 kHz) under typical sea surface roughness conditions. Coupled-mode or rough surface PE modeling of these effects require averaging results over many rough surface realizations, increasing the computational effort. An alternative method is based on transport theory, where equations are developed for propagating the moments of the field, avoiding the need for utilizing rough surface realizations. Our transport theory method is based on expanding the field in unperturbed modes, and the equations of motion are for moments of the mode amplitudes. The approach has been based on keeping terms to only first-order in the surface height h(x), making the method linear in surface height. Methods for extending the approach beyond the linear model will be described, both with using realizations with coupled modes and with attempts to extend these approaches to obtain a transport theory. A key part of the approach is the use of the Differential Algebraic Equation (DAE) method in which the range-derivative of the effective boundary condition on the mean plane of the rough surface is used instead of the effective boundary condition itself. [Work supported by ONR Ocean Acoustics.]

Solvability conditions and design for H∞ & H2 almost state synchronization of homogeneous multi-agent systems

This paper studies the H∞ and H2 almost state synchronization problem for homogeneous multi-agent systems with general linear agents affected by external disturbances and with a directed communication topology. Agents are connected via diffusive full-state coupling or diffusive partial-state coupling. A necessary and sufficient condition is developed for the solvability of the H∞ and H2 almost state synchronization problem. Moreover, a family of protocols based on either an algebraic Riccati equation (ARE) method or a directed eigen structure assignment method are developed such that the impact of disturbances on the network disagreement dynamics, expressed in terms of the H∞ and H2 norm of the corresponding closed-loop transfer function, is reduced to any arbitrarily small value. The protocol for full-state coupling is static, while for partial-state coupling it is dynamic.