HURWITZ MATRIX, LYAPUNOV AND DIRICHLET ON THE SUSTAINABILITY OF LYAPUNOV’S

2018 ◽  
pp. 431-436
Author(s):  
Irina Dmitrievna Kostrub
Keyword(s):  
Linear Systems ◽  
Variable Coefficients ◽  
Hurwitz Matrix ◽  
Stable Case ◽  
New Material ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Stability ◽  
Monodromy Matrices ◽  
Similar Classification

The concepts of Hurwitz, Lyapunov and Dirichlet matrices are introduced for the convenience of the stability of linear systems with constant coefficients. They allow us to describe all the cases of interest in the stability theory of linear systems with constant coefficients. A similar classification is proposed for systems of linear differential equations with periodic coefficients. Monodromy matrices of such systems can be either Hurwitz matrices or Lyapunov matrices or Dirichlet matrices (in the discrete sense) in a stable case. The new material relates to systems with variable coefficients.

On the Stability of Accelerated Motion: Some Thoughts on Linear Differential Equations with Variable Coefficients

1957 ◽  
Vol 8 (4) ◽  
pp. 309-330 ◽  
Author(s):  
A. R. Collar
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Variable Coefficients ◽  
Order Equation ◽  
Linear Differential Equations ◽  
Constant Speed ◽  
Accelerated Motion ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Stability

Summary:In studies of the stability of aeronautical systems, equations of motion are derived which have coefficients dependent on flight speed. Conventional practice treats the speed as constant, when a set of linear differential equations with constant coefficients results. Actually, since the speed varies during flight, it may be regarded as a prescribed function of time; the set of linear differential equations then has variable coefficients.The treatment of the problem of stability then becomes much more complex in this case. A simple example is given to show that a system which is stable at any constant speed can become unstable during deceleration; the ordinary constant-speed criteria are, strictly, therefore inadequate. Some approaches to the discussion of stability during acceleration are suggested; a solution is given of the single second-order equation which enables the amplitude of oscillation of the solution to be studied. Inverse methods of approach are suggested, both for single and sets of equations, in which particular forms of acceleration corresponding to prescribed solutions are derived; and some tentative conclusions are drawn. As would be expected, the effects of acceleration depend on a dimensionless “acceleration number.”

scholarly journals Monte-Carlo for solving large linear systems of ordinary differential equations

2021 ◽  
Vol 8 (1) ◽  
pp. 37-48
Author(s):  
Sergey M. Ermakov ◽  
◽  
Maxim G. Smilovitskiy ◽  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Differential Equations ◽  
Integral Equations ◽  
Virtual Machines ◽  
Variable Coefficients ◽  
Fredholm Type ◽  
Type Integral ◽  
Constant Coefficients ◽  
Linear Differential

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov

1962 ◽  
Vol 58 (4) ◽  
pp. 694-702 ◽  
Author(s):  
P. C. Parks
Keyword(s):  
Differential Equations ◽  
Direct Proof ◽  
Real Constant ◽  
Integral Calculus ◽  
Linear Ordinary Differential Equations ◽  
Hurwitz Stability ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Stability ◽  
Complex Integral

ABSTRACTThe second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.

scholarly journals V. On the calculus of symbols, with applications to the theory of differential equations

1861 ◽  
Vol 151 ◽  
pp. 69-82 ◽  
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
New Methods ◽  
Methods Of Calculation ◽  
Great Utility ◽  
Long Time ◽  
Constant Coefficients ◽  
Linear Differential

The calculus of generating functions, discovered by Laplace, was, as is well known, highly instrumental in calling the attention of mathematicians to the analogy which exists between differentials and powers. This analogy was perceived at length to involve an essential identity, and several analysts devoted themselves to the improvement of the new methods of calculation which were thus called into existence. For a long time the modes of combination assumed to exist between different classes of symbols were those of ordinary algebra; and this sufficed for investigations respecting functions of differential coefficients and constants, and consequently for the integration of linear differential equations, with constant coefficients. The laws of combination of ordinary algebraical symbols may be divided into the commutative and distributive laws; and the number of symbols in the higher branches of mathematics, which are commutative with respect to one another, is very small. It became then necessary to invent an algebra of non-commutative symbols. This important step was effected by Professor Boole, for certain classes of symbols, in his well-known and beautiful memoir published in the Transactions of this Society for the year 1844, and the object of the paper which I have now the honour to lay before the Society is to perfect and develope the methods there employed. For this purpose I have constructed systems of multiplication and division for functions of non-commutative symbols, subject to the same laws of combination as those assumed in Professor Boole’s memoir, and I thus arrive at equations of great utility in the integration of linear differential equations with variable coefficients.

Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 1

1985 ◽  
Vol 100 (3-4) ◽  
pp. 301-326 ◽  
Author(s):  
M. H. Lantsman
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Algebraic Equations ◽  
Order Of Growth ◽  
Constant Coefficients ◽  
Independent Variable ◽  
Linear Differential ◽  
Image Position ◽  
Asymptotic Representations

SynopsisWe consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equationcan be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

Paper 27. The Stability of a Ship in a Simple Sinusoidal Wave

1972 ◽  
Vol 14 (7) ◽  
pp. 194-206 ◽  
Author(s):  
W. G. Price
Keyword(s):  
Differential Equations ◽  
Rigid Body Dynamics ◽  
Linear Differential Equations ◽  
Stability Criteria ◽  
Sinusoidal Wave ◽  
Body Dynamics ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Stability ◽  
Dynamics Stability

The linear differential equations governing the motion of a ship in a sinusoidal wave disturbance are derived from the principles of rigid body dynamics. Stability criteria are obtained when the linear differential equations have ( a) constant coefficients and ( b) periodic time-dependent coefficients that are needed to describe the ship in a following sinusoidal wave. The frequency of encounter between ship and wave is not necessarily zero.

scholarly journals Fourier Operational Matrices of Differentiation and Transmission: Introduction and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
F. Toutounian ◽  
Emran Tohidi ◽  
A. Kilicman
Keyword(s):  
Exact Solution ◽  
Numerical Solution ◽  
Variable Coefficients ◽  
Operational Matrices ◽  
Residual Function ◽  
Numerical Examples ◽  
Pantograph Equation ◽  
Constant Coefficients ◽  
Collocation Scheme ◽  
Linear Differential

This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized Pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solution of Pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods.

scholarly journals On Charwat's theory of motion of tracers in planar vortex flows

1983 ◽  
Vol 25 (2) ◽  
pp. 175-189
Author(s):  
R. B. Leipnik
Keyword(s):  
Solution Method ◽  
Vortex Motion ◽  
Variable Coefficients ◽  
Vortex Flows ◽  
Flow Problems ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Neutrally Buoyant ◽  
Fowler Equation ◽  
Solid Type

AbstractThe motion of small, near neutrally buoyant tracers in vortex flows of several types is obtained on the basis of Charwat's mathematical model, which is highly non-linear.The solution method in the non-degenerate case expresses the squared orbital radius r2 as a product AA*, where the complex number A satisfies a second-order linear differential ‘factor equation’, generally with variable coefficients. The angular coordinate is expressed in terms of log(A*/A). Solid-type rotation and sinusoidally perturbed solid-type rotation correspond respectively to constant coefficients and sinusoidal coefficients. The former exactly yields a scalloped spiral tracer motion; the latter yields unstable tracer motion as t → ∞ except when the perturbing frequency and amplitude are rather specially related to the flow and tracer parameters. Free vortex motion is somewhat degenerate for this solution method but can be partially analyzed in terms of solutions of a generalized Emden–Fowler equation. The method can be used for other planar flow problems with a symmetry axis.

scholarly journals REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS AND NONPERMUTABLE VARIABLE COEFFICIENTS

2020 ◽  
Vol 25 (2) ◽  
pp. 303-322
Author(s):  
Michal Pospíšil
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Multiple Delays ◽  
Matrix Coefficients ◽  
Representation Of Solutions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Delay Differential

Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications.Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.

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