On testing the existence of universal denominators for partial differential and difference equations

We consider the problem of testing the existence of a universal denominator for partial differential or difference equations with polynomial coefficients and prove its algorithmic undecidability. This problem is closely related to finding rational function solutions in that the construction of a universal denominator is a part of the algorithms for finding solutions of such form for ordinary differential and difference equations.     

Computational Problems of Multivariate Hypergeometric Theory

We consider computational problems of the theory of hypergeometric functions in several complex variables: computation of the holonomic rank of a hypergeometric system of partial differential equations, computing the defining polynomial of the singular hypersurface of such a system and finding its monomial solutions. The presented algorithms have been implemented in the computer algebra system MATHEMATICA.     

Improved algorithms for solving difference andq-difference equations

Improved algorithms for finding denominators of rational solutions of linear difference andq-difference equations with polynomial coefficients are proposed. The improved efficiency of these algorithms is achieved as a result of a more efficient implementation of the Abramov algorithm (due to the use of the Man and Wright algorithm for calculating the dispersion, which is extended for the case ofq-dispersion) and of the improvement of this algorithm by using an additional procedure for minimizing the degree of the denominator (similar to the Migushov algorithm). The case of difference equations is analyzed in detail, whereasq-difference equations are considered by analogy with the first case. The algorithms described were implemented in Maple V.     

Checking existence of solutions of partial differential equations in the fields of Laurent series

It is proved that the problem of checking the existence of solutions of linear partial differential equations with polynomial coefficients is algorithmically undecidable. Decidability of the problem of checking the existence of monomial solutions with real and complex exponents is established.     

Denominators of rational solutions of linear difference systems of an arbitrary order

An algorithm for finding a universal denominator of rational solutions of a system of linear difference equations with polynomial coefficients is proposed. The equations may have arbitrary orders.     

Rational solutions of linear difference equations: Universal denominators and denominator bounds

Complexities of some well-known algorithms for finding rational solutions of linear difference equations with polynomial coefficients are studied.     

A Real Polynomial Decision Algorithm Using Arbitrary-Precision Floating Point Arithmetic

We study the problem of deciding whether a system of real polynomial equations and inequalities has solutions, and if yes finding a sample solution. For polynomials with exact rational number coefficients the problem can be solved using a variant of the cylindrical algebraic decomposition (CAD) algorithm. We investigate how the CAD algorithm can be adapted to the situation when the coefficients are inexact, or, more precisely, Mathematica arbitrary-precision floating point numbers. We investigate what changes need to be made in algorithms used by CAD, and how reliable are the results we get.     

Search for Polynomial Solutions of Linear Functional Systems by Means of Induced Recurrences

The problem of searching for polynomial solutions of linear functional (differential, difference, and q-difference) equations is considered. The problem is solved by means of the construction of an induced recurrent system in the coefficients of the expansion of the desired solution in a certain basis. In the paper, practical aspects of the construction of such induced recurrences are discussed, and a new algorithm for the construction of polynomial solutions based on them is suggested.     

Evaluating the rational generating function for the solution of the Cauchy problem for a two-dimensional difference equation with constant coefficients

We propose an algorithm for evaluation of rational generating functions for solutions of the Cauchy problems for two-dimensional difference equations with constant coefficients. The coefficients of onedimensional difference equations and the initial data are used to solve the corresponding Cauchy problems. The algorithm is implemented in the Maple computer algebra system.     

Analysis of nonlinear electrical circuits using bernstein polynomials

In electrical circuit analysis, it is often necessary to find the set of all direct current (d.c.) operating points (either voltages or currents) of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.     

Value monoids of zero-dimensional valuations of rank 1

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid-1980s and developed a framework for performing such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x,y) that are suitable for this framework. We then perform such computations for ideals in the polynomial ring K[x,y]. Interestingly, for these valuations, some ideals have finite Gröbner bases with respect to a valuation that are not Gröbner bases with respect to any monomial order, whereas other ideals only have Gröbner bases that are infinite.     

Elimination of lower order harmonics in Voltage Source Inverter feeding an induction motor drive using Evolutionary Algorithms

Selective Harmonic Elimination technique is one of the control methods applied in Voltage Source Inverters to eliminate the harmonics. However, finding the solutions for the harmonic reduction is a difficult problem to be solved. This paper presents an efficient and reliable Evolutionary Algorithms based solution for Selective Harmonic Elimination (SHE) switching pattern to eliminate the lower order harmonics in Pulse Width Modulation (PWM) inverter. Determination of pulse pattern for the elimination of lower order harmonics of a PWM inverter necessitates solving a system of nonlinear transcendental equations. Evolutionary Algorithms are used to solve nonlinear transcendental equations for PWM–SHE. In this proposed method, harmonics up to 19th are eliminated using Evolutionary Algorithms without using dual transformer. The experimental results are obtained and are validated with simulations using PSIM 6.1 and MATLAB 7.0.     

Optimal Processes for Controllable Oscillators

We consider the problem of optimal parametric control for a single oscillator or an ensemble of oscillators due to a change in one of the coefficients of the system of equations characterizing them. We obtain solutions for the problem of finding the maximal change in the energy of oscillations for a given time.     

Reduced Gröbner Bases,Free Difference–Differential Modules and Difference–Differential Dimension Polynomials

We define a special type of reduction in a free left module over a ring of difference–differential operators and use the idea of the Gröbner basis method to develop a technique that allows us to determine the Hilbert function of a finitely generated difference–differential module equipped with the natural double filtration. The results obtained are applied to the study of difference–differential field extensions and systems of difference–differential equations. We prove a theorem on difference–differential dimension polynomial that generalizes both the classical Kolchin’s theorem on dimension polynomial of a differential field extension and the corresponding author’s result for difference fields. We also determine invariants of a difference–differential dimension polynomial and consider a method of computation of the dimension polynomial associated with a system of linear difference–differential equations.     

Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

We consider linear difference equations with polynomial coefficients over C and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the C-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and that the dimension of this space is equal to the order of the original equation.We also consider d-dimensional (d≥1) hypergeometric sequences, i.e., sequential and subanalytic solutions of consistent systems of first-order difference equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive).Subanalytic solutions have applications in computer algebra. We show that some implementations of certain well-known summation algorithms in existing computer algebra systems work correctly when the input sequence is a subanalytic solution of an equation or a system, but can give incorrect results for some sequential solutions.     

An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations

The tanh-function method for finding explicit travelling solitary wave solutions to non-linear evolution equations is described. The method is usually extremely tedious to use by hand. We present a Mathematica package ATFM that deals with the tedious algebra and outputs directly the required solutions. The use of the package is illustrated by applying it to a variety of equations; not only are previously known solutions recovered but in some cases more general forms of solution are obtained.     

Exact solution of a KdV equation with variable coefficients

A simple direct method is developed for finding exact solutions of nonlinear equations with variable coefficients. The (1 + 1)-dimensional KdV equation is used as an example to elucidate the solution procedure, and its exact solution is obtained.     

The use of a pocket computer in gas dynamics

A pocket computer (Casio model FX-702P) has been utilized to solve some problems arising in gas dynamics. The programs reported cover algebraic calculations, solution of transcendental equations and finite difference solutions of coupled, nonlinear ordinary differential equations. In gas dynamics terms, the algebraic computations pertain to properties variation in isentropic flow, Rayleigh flow and Fanno flow. The transcendental equations solved are those arising in Prandtl-Meyer flow and oblique shock analysis. Problems of pipe flow with simultaneous friction and heat transfer, unsteady discharge of a variable-volume tank through a convergent nozzle, Taylor-Maccoll flow over a cone and flow in a rocket nozzle with secondary injection of oxidant in the divergent section, are used as illustrative examples of coupled, nonlinear ordinary differential equations.     

Approximation of experimental data by solving linear difference equations with constant coefficients (in particular,by exponentials and exponential cosines)

This paper proposes a method for approximating experimental data points by the curves representing the solutions of linear difference equations with constant coefficients, in particular, by the curves of the expcos class. An algorithm for finding the coefficients and initial conditions of this approximation is described. The proposed approach minimizes the root mean square (RMS) deviation. The method is tested on some model examples, including the refinement of the beginning of QRS complexes on a three-dimensional ECG loop (in the form of Frank leads).     

New methods of secure outsourcing of scientific computations

In this paper, we present several methods of secure outsourcing of numerical and scientific computations. Current outsourcing techniques are inspired by the numerous problems in computational mathematics, where a solution is obtained in the form of an approximation. Examples of such problems can be found in the fields of economics, military, petroleum industry, and in other areas. Many of today’s scientific and numerical problems require large computational resources; therefore, they can only be solved on supercomputers or by using the capabilities of the largest computing systems, such as grid technology, cloud, etc. We believe that it is imperative to improve the mathematical framework to enable secure outsourcing. Therefore, the main goal of this paper is to present different methods of finding approximate solutions to some equations solved by an external computer. To accomplish this, we chose certain classes of algebraic and differential equations because, in most cases, modern computing problems are reduced to solving such systems of equations (differential equations, linear programming, etc.). As an important application example we are presenting a specific applied problem related to geological exploration.