Symdyn: an automated algebraic solution for high-order quantum systems

The purpose of this paper is to derive a means of scaling the dependent variables of a system of linear differential equations so that the normalized system accurately reflects the dependence of the system on initial data. The theory is developed for the special case of a system of first order hyperbolic partial differential equations in two independent variables. Several other systems of differential equations, to which the same normalization could be applied, are also indicated. The scaling is equivalent to row and column scaling commonly used in linear algebra. For the case of positive matrices (that is, for the case where the diagonalizing matrix and its inverse have no zero elements), the theory for the “1” and “$\infty $ ” norms has been thoroughly developed by Bauer [1]. The assumption of positive matrices is too stringent for some applications. Thus, the theory is generalized to show that for irreducible matrices a best scaling exists (for both the “1” and “ $\infty $” norms), while for simply reducible matrices a best scaling exists for at least one of these two norms. Several examples are given.

Applications of the Lanczos method

Lie symmetries for Lie systems: Applications to systems of ODEs and PDEs

In the previous chapters we explored varieties of periodic traits of linear difference equations, piece-wise difference equations, rational difference equations, and Max-Type difference equations. First of all, we observed how the periodicity nature varied from the existence of unique periodic cycles to every solution being periodic. Second, we discovered the necessary and sufficient conditions for the existence of periodic cycles. Furthermore, we perceived the disparity between patterns of the even ordered periodic cycles and odd ordered periodic cycles with uniqueness of solutions and alternating patterns. Our next aim is to broaden our developed knowledge to ascertain the periodicity essence of third and higher order difference equations, systems of difference equations, when the periodic sequence \(\{A_{n}\}_{n=0}^{\infty }\) is periodic with period-k, (k ≥ 3) and study of new systems of piecewise difference equations and neural networking models. How will these criteria for the existence of periodic cycles be similar or different compared to our current mastery in previous chapters? What new periodic traits can we expect that will be quite different? How will the delay m ≥ 3 affect the new periodic traits? Will even and odd values of delay m ≥ 3 influence the new periodic traits? We will pose conjectures that will lead to new theorems and results based on our previous studies and will pose new research questions.

71 - Integrable Combinations

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Pade approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations.

Construction of similarity transformations and analytic solutions for a liquid film on an unsteady stretching sheet using lie point symmetries

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

We considered a non-stationary optimal control system with delay. Non-stationary optimal control systems are described by systems of differential or differential-algebraic equations. Variable coefficients do not allow, in general, to construct a solution to such systems in quadratures. Numerical or asymptotic methods are used to solve such systems. In this paper, asymptotic methods are used, in particular, the Feshchenko-Shkil' method for integrating of singularly perturbed systems and the Wasow’s method for systems with an unstable spectrum. In this paper, we construct a transformation that reduces the optimal delay control system to a system that does not contain terms without delay. This transformation makes it possible to integrate the system by the method of steps without solving systems of differential equations at each step. The system of equations obtained as a result of the transformation of the original system is somewhat easier to study in terms of building a solution. However, the problem of optimizing the control of both systems requires both a separate mathematical study and clarification of the practical reality of spectrum instability in such systems. If the instability of the spectrum is caused by the degeneracy of the main matrix, this leads to the unboundedness of the system solution as the small parameter approaches zero. The aforementioned growth of the solution can create emergency situations in real systems. Future research concerns solving the problem of optimal control of systems with an unstable spectrum and studying the question of the reality and physical meaning of turning points in specific systems. In future studies, it is also planned to construct an asymptotic representation of the solution of a control system with delays and degeneracies. Keywords: optimal control system, systems of differential equations with delay, asymptotic solution, turning point.

A variable gravity model of the otolith membrane

Non-local symmetries and conservation laws for one-dimensional gas dynamics equations

A digital technique for manipulating symbols defining a mathematical expression is introduced and it is shown how this technique can be applied so that a digital computer can automatically compile an analog computer program directly from a mathematical description of the problem to be solved. The mathematical problem considered is that of solving a system of nonlinear first-order differential equations with initial conditions. It is assumed that these equations have been prescaled and are expressed in terms of the physical variables rather than the original mathematical variables. A resume of a method for reducing the original symbol sequence defining the differential equations to symbol subsequences from which the analog program can be formed is given. It is then shown how the analog program is to be generated from these symbol sequences. To illustrate the detailed application of the technique an analog program is compiled for a particular system of differential equations.

Using the Lie symmetry approach, we have examined herein the system of partial differential equations corresponding to the Einstein–Maxwell equations for a static axially symmetric spacetime. The method used reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations is all optimal subgroups. The system of ordinary differential equations is further solved in general to obtain exact solutions. Several new physically important families of exact solutions are derived.

This paper is concerned with estimating the internal state of a dynamic system by processing measurements taken from the system output. An exact analytic representation of the probability density functions characterizing the estimate may not be possible to obtain. Even when available, it may be too complex or not practical because, for example, recursive application is required. Hence, approximations are generally inevitable. Gaussian mixture approximations are convenient for a number of reasons. However, calculating appropriate mixture parameters that minimize a global measure of deviation from the true density is a tough optimization task. Here, we propose a approximation method that minimizes the squared integral deviation between the true density and its mixture approximation. Rather than trying to solve the original problem, it is converted into a corresponding system of explicit ordinary first-order differential equations. This system of differential equations is then solved over a finite time interval, which is an efficient way of calculating the desired optimal parameter values. For polynomial measurement nonlinearities, closed-form analytic expressions for the coefficients of the system of differential equations are derived.

The author consider initial-boundary value problem for a system of differential equations in partial derivatives of third order. We study questions of existence and uniqueness of classical solution of the problem, and ways for its evaluation. The introduction of a new desired function enables us to reduce the problem to equivalent nonlocal problem with integral condition for a system of integro-differential euqtions of hyperbolic type with a functional relation. We establish conditions for unique solvability of the nonlocal problem by means of the method of functional parameters, propose algorithms for solving of the equivalent problem and prove their convergence. As a result, there are obtained conditions for existence of unique classical solution of initial-boundary value problem for the considered system of differential equations of third order in terms of initial data.