Nonlinear Matrix Differential Equations Research Articles

On the Ψ - Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

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Adaptive Wavelet Precise Integration Method for Nonlinear Black-Scholes Model Based on Variational Iteration Method

An adaptive wavelet precise integration method (WPIM) based on the variational iteration method (VIM) for Black-Scholes model is proposed. Black-Scholes model is a very useful tool on pricing options. First, an adaptive wavelet interpolation operator is constructed which can transform the nonlinear partial differential equations into a matrix ordinary differential equations. Next, VIM is developed to solve the nonlinear matrix differential equation, which is a new asymptotic analytical method for the nonlinear differential equations. Third, an adaptive precise integration method (PIM) for the system of ordinary differential equations is constructed, with which the almost exact numerical solution can be obtained. At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical result shows the method's higher numerical stability and precision.

On The Ψ – Asymptotic Stability Of Nonlinear Lyapunov Matrix Differential Equations

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Developed Adomian method for quadratic Kaluza–Klein relativity

We develop and modify the Adomian decomposition method (ADecM) to work for a new type of nonlinear matrix differential equations (MDE's) which arise in general relativity (GR) and possibly in other applications. The approach consists in modifying both the ADecM linear operator with highest order derivative and ADecM polynomials. We specialize in the case of a 4 × 4 nonlinear MDE along with a scalar one describing stationary cylindrically symmetric metrics in quadratic five-dimensional GR, derive some of their properties using ADecM and construct the most general unique power series solutions. However, because of the constraint imposed on the MDE by the scalar one, the series solutions terminate in closed forms exhausting all possible solutions.

New Kamenev-type theorems for super-linear matrix differential equations

By using Riccati transformation and the integral averaging technique, some new Kamenev-type oscillation criteria are established for the super-linear matrix differential systems X ″ ( t ) + ( X n ( t ) Q ( t ) X ∗ n ( t ) ) X ( t ) = 0 and X ″ ( t ) + ( X ∗ n ( t ) Q ( t ) X n ( t ) ) X ( t ) = 0 , t ⩾ t 0 > 0 , n ≥ 1 , where Q ( t ) is an m × m continuous symmetric and positive definite matrix for t ∈ [ t 0 , ∞ ) . The results improve and complement those given by Tomastik [E.C. Tomastik, Oscillation of nonlinear matrix differential equations of second-order, Proc. Amer. Math. Soc. 19 (1968) 1427–1431], Ahlbrandt et al. [C.D. Ahlbrandt, J. Ridenhour, R.C. Thompson, Oscillation of super-linear matrix differential equation, Proc. Amer. Math. Soc. 105 (1989) 141–148] and Ou [L.M. Ou, Atkinson’s super-linear oscillation theorem for matrix dynamic equations on a time scale, J. Math. Anal. Appl. 299 (2004) 615–629], which is illustrated by an example at the end of the paper.

On Ψ-stability of nonlinear Lyapunov matrix differential equations

We prove necessary and sufficient conditions for Ψ−(uniform) stability of the trivial solution of a nonlinear Lyapunov matrix differential equation.

Adaptive interpolation wavelet and homotopy perturbation method for partial differential equations

The homotopy perturbation method proposed by Ji-Huan He has been developed to solve nonlinear matrix differential equations. This paper constructs an adaptive multilevel quasi-wavelet operator according to the interpolation wavelet theory, with which the nonlinear partial differential equations can be discretized adaptively in physical spaces as a matrix differential equation, its numerical solution can be obtained by using the homotopy perturbation method. Numerical results show that the homotopy perturbation method is not sensitive to the time step, so the arithmetic error mainly arises in the space step. Burgers equation is taken as examples to illustrate its effectiveness and convenience.

On the Error in QR Integration

An important change of variables for a linear time varying system $\dot x=A(t)x, t\ge 0$, is that induced by the QR-factorization of the underlying fundamental matrix solution: $X=QR$, with $Q$ orthogonal and $R$ upper triangular (with positive diagonal). To find this change of variable, one needs to solve a nonlinear matrix differential equation for $Q$. Practically, this means finding a numerical approximation to $Q$ by using some appropriate discretization scheme, whereby one attempts to control the local error during the integration. Our contribution in this work is to obtain global error bounds for the numerically computed $Q$. These bounds depend on the local error tolerance used to integrate for $Q$, and on structural properties of the problem itself, but not on the length of the interval over which we integrate. This is particularly important, since—in principle—$Q$ may need to be found on the half-line $t\ge 0$.

Oscillation of nonlinear second order matrix differential equations

Abstract In this paper, sufficient conditions are obtained for oscillation of all nontrivial, prepared, symmetric solutions of a class of nonlinear second order matrix differential equations of the form $$(P(t)Y')' + Q(t)F(Y) = 0, t \geqslant 0,$$ and $$Y'' + Q(t)F(Y) = 0, t \geqslant 0.$$

Coupling technique of variational iteration and homotopy perturbation methods for nonlinear matrix differential equations

The variational iteration method proposed by Ji-Huan He is a new analytical method to solve nonlinear equations. This paper applies the method to search for exact analytical solutions of linear differential equations with constant coefficients. Furthermore, based on the precise integration method, a coupling technique of the variational iteration method and homotopy perturbation method is proposed to solve nonlinear matrix differential equations. A dynamic system and Burgers equation are taken as examples to illustrate its effectiveness and convenience.

New oscillation criteria for second-order nonlinear matrix differential equations

Some new oscillation criteria are established for the second-order matrix differential system (r(t)Z′(t))′+p(t)Z′(t)+Q(t)F(Z′(t))G(Z(t))=0, t≥0>0, are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.

Reduction of dimensionality in choosing optimal control

AbstractGiven that x(t) = (x1(t),…,xn(t)) represents a trajectory of the considered dynamical system, we demonstrate that reduction of dimensionality of x(t) by using its principal components in choosing the optimal control u(t) is necessary to overcome the computational difficulties that might arise in obtaining a solution of the optimal control problem. We obtain explicit solutions for linear time‐varying systems with quadratic cost functional and show how to overcome difficulties appearing in non‐linear matrix differential equations of Riccati type. A gradient method of searching for the close‐to‐optimal trajectory and control in non‐linear optimal control problems is also considered. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Matrix approach to Lagrangian fluid dynamics

A new approach to ideal-fluid hydrodynamics based on the notion of continuous deformation of infinitesimal material elements is proposed. The matrix approach adheres to the Lagrangian (material) view of fluid motion, but instead of Lagrangian particle trajectories, it treats the Jacobi matrix of their derivatives with respect to Lagrangian variables as the fundamental quantity completely describing fluid motion.A closed set of governing matrix equations equivalent to conventional Lagrangian equations is formulated in terms of this Jacobi matrix. The equation of motion is transformed into a nonlinear matrix differential equation in time only, where derivatives with respect to the Lagrangian variables do not appear. The continuity equation that requires constancy of the Jacobi determinant in time takes the form of an algebraic constraint on the Jacobi matrix. An accompanying linear consistency condition, which is responsible for the dependence on spatial variables and does not include time derivatives, ensures completeness of the system and reconstruction of the particle trajectories by the Jacobi matrix.A class of exact solutions to the matrix equations that describes rotational non-stationary three-dimensional motions having no analogues in the conventional formulations is also found and investigated. A distinctive feature of these motions is precession of vortex lines (rectilinear or curvilinear) around a fixed axis in space. Boundary problems for the derived exact solutions including matching of rotational and potential motions across the boundary of a vortex tube are addressed. In particular, for the cylindrical vortex of elliptical cross-section involved in three-dimensional precession, the outer potential flow is constructed and shown to be a non-stationary periodic straining flow at a large distance from the vortex axis.

Reflection and Transmission Coefficients for Two- and Three-Dimensional Quantum Wires

The problem of a quantum wire connected smoothly on both sides to leads of variable cross section is studied. A method for solving this problem in terms of a set of nonlinear first order matrix differential equations for the variable reflection amplitude is discussed. The reflection coefficient obtained in this way is directly related to the conductivity, and is calculated as a function of the energy of the ballistic electrons. This formulation can be applied to three-dimensional as well as two dimensional quantum wires. For two specific cases the reflection coefficient is obtained as a function of the wave number of the incident electron, and the contribution of quantum tunneling to the transmission in each case is demonstrated. Also a model with a dissipative force is introduced and its effect on the transmission of the electrons is investigated.

Matrix Riccati equation formulation for radiative transfer in a plane-parallel geometry

In this paper, we formulate the radiative transfer problem as an initial value problem via a pair of nonlinear matrix differential equations (matrix Riccati equations or MREs) which describe the reflection (R) and transmission (T) matrices of the specific intensities in a plane-parallel geometry. One first computes R and T matrices of some small but finite layer thickness (equivalent optical thickness τ∼0.01 and then repetitively applies the doubling method to the reflection and transmission matrices R(τ)and T(τ) until reaching the desired layer thickness. The initial matrices R(τ0)and T(τ0) can be computed quite accurately by either of the following methods: multiple-order, multiple-scattering approximation, iterative method or fourth-order Runge–Kutta techniques. In addition, the reflection coefficient matrix of a semi-infinite medium satisfies an algebraic matrix equation which can be solved repetitively by a matrix method. MREs offer an alternative way of solving plane-parallel radiative transport problems. This method requires only elementary matrix operations (addition, multiplication and inversion). For vector and/or beam-wave radiative transfer problems, large matrices are required to describe the physics adequately, and the MRE method provides a significant reduction in computer memory and computation time.

On the existence of an invariant manifold for weakly nonlinear matrix differential equations

For a system of nonlinear matrix differential equations with small parameter, we establish conditions of the existence of an invariant manifold in terms of the Green function.

On similarities of class Cp and applications to matrix differential equations

This paper deals with similarities of class C p , in particular reduction of class C p to Jordan form and unitary triangularization of class C p of a matrix function. The main result is applied to establish a method to lower the order of nonlinear matrix differential equations belonging to a broad class of equations. In particular, this method permits one to transform nonlinear equations of the first order into an algebraic equation joint to a linear differential equation of the first order. The method is applied to the matrix differential equation XX′ − X′ X = X, and a particular solution is easily obtained.

Nonlinear operator equations and applications to modelling

The main purpose of this paper is to treat the problem of numerical solutions of Lyapunov matrix equations and nonlinear matrix equations such as the Riccati type which occur in many areas of applications such as in control theory, robotics, mathematical modelling and computer simulation, etc. The main approach is to obtain solutions of such type equations in a more direct manner as in contrast to completely interactive methods. Parallel processing of algorithms is made use of in the computation of solutions of such nonlinear matrix equations associated with nonlinear matrix differential equations arising in the study of nonlinear dynamical control systems.

Finite element analysis of maneuvering spacecraft truss structures

A finite element modeling and solution technique capable of determining the time response of flexible spacecraft truss structures undergoing large angle slew maneuvers has been developed. The elastic deformations of the structure are coupled with large nonsteady translational and rotational motions with respect to an inertial reference frame. The governing equations of motion of the system are derived using momentum conservation principles and the principle of virtual work. The finite element approximation is applied to the equations of motion and the resulting set of nonlinear second order matrix differential equations is solved timewise by an iterative direct numerical integration scheme based on the trapezoidal rule. The solution technique is tested on both planar and three-dimensional maneuvering spacecraft truss structures.

Joint Estimation and Control of Jump Linear Systems With Multiplicative Noises

Jump Linear Quadratic Gaussian systems are considered in the presence of state-and control-dependent noises. Assuming that the jumps of the model parameters are perfectly observed, it is possible to formulate and solve an optimal input synthesis problem. It is found that the optimal solution does not present the certainty equivalence property, so that the estimation and control synthesis must be treated simultaneously. Optimal equations for the filter and regulator gains are obtained in terms of a set of coupled nonlinear matrix differential equations.