Solution Of Algebraic Equations Research Articles

Numerical solution of time-varying delay systems by Chebyshev wavelets

The solution of time-varying delay systems is obtained by using Chebyshev wavelets. The properties of the Chebyshev wavelets consisting of wavelets and Chebyshev polynomials are presented. The method is based upon expanding various time functions in the system as their truncated Chebyshev wavelets. The operational matrix of delay is introduced. The operational matrices of integration and delay are utilized to reduce the solution of time-varying delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Numerical solution ofnth-order integro-differential equations using trigonometric wavelets

The main aim of this paper is to apply the trigonometric wavelets for the solution of the Fredholm integro-differential equations of nth-order. The operational matrices of derivative for trigonometric scaling functions and wavelets are presented and are utilized to reduce the solution of the Fredholm integro-differential equations to the solution of algebraic equations. Furthermore, we get an estimation of error bound for this method. Copyright © 2011 John Wiley & Sons, Ltd.

Analysis of time-varying delay systems via triangular functions

This paper presents a method for finding the analysis of time-varying delay systems using triangular functions. We present the properties of the triangular functions. The operational matrices of integration, delay and product are utilized to reduce the solution of delay systems to the solution of algebraic equations. Illustrative examples is included to demonstrate the validity and applicability of the technique.

Collective excitation frequencies and stationary states of trapped dipolar Bose-Einstein condensates in the Thomas-Fermi regime

We present a general method for obtaining the exact static solutions and collective excitation frequencies of a trapped Bose-Einstein condensate (BEC) with dipolar atomic interactions in the Thomas-Fermi regime. The method incorporates analytic expressions for the dipolar potential of an arbitrary polynomial density profile, thereby reducing the problem of handling non-local dipolar interactions to the solution of algebraic equations. We comprehensively map out the static solutions and excitation modes, including non-cylindrically symmetric traps, and also the case of negative scattering length where dipolar interactions stabilize an otherwise unstable condensate. The dynamical stability of the excitation modes gives insight into the onset of collapse of a dipolar BEC. We find that global collapse is consistently mediated by an anisotropic quadrupolar collective mode, although there are two trapping regimes in which the BEC is stable against quadrupole fluctuations even as the ratio of the dipolar to s-wave interactions becomes infinite. Motivated by the possibility of fragmented BEC in a dipolar Bose gas due to the partially attractive interactions, we pay special attention to the scissors modes, which can provide a signature of superfluidity, and identify a long-range restoring force which is peculiar to dipolar systems. As part of the supporting material for this paper we provide the computer program used to make the calculations, including a graphical user interface.

Oscillation processes as a tool of physics cognition

Based on research of resonant and transitional processes in electric LC-, RL-, RC-, and LCRcircuits, the article offers theoretical and experimental proof of physical reality of complex frequency and other concrete complex numbers which are a function of complex frequency. Analysis of resonant processes in electric circuits has shown inconsistency in their conventional interpretation at real frequencies. On the contrary, interpretation of resonance at complex frequencies allowed explaining all the inconsistencies of its interpretation at real frequencies. We provide the description of experiments confirming physical reality of resonance at complex frequencies. As for transitional processes in electric circuits, their analysis by means of differential equations requires the solution of the respective characteristic algebraic equations. In this respect we demonstrate that the solution of algebraic equations with the use of real numbers rules out the existence of oscillation transitional process, and, therefore, does not conform to physical reality.

Bernstein Ritz‐Galerkin method for solving an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

AbstractIn this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Time and amplitude reconstruction from sampling measurements of the PHOS signal profile in the ALICE project

This paper presents the analysis of procedures dedicated for time and amplitude reconstruction from sampling measurements of the ALICE electromagnetic spectrometer PHOS signal profile. The proposed approach is based on the least squares method to fit the profile measured in discrete number of points to the known impulse response function of electronic filter. Assuming semi-Gaussian shape of this function one can reduce the problem of simultaneous determination of signal time mark t0 and amplitude A to the solution of algebraic equation of the nth power, where n is the order of used electronic filter. The proper analytical formulae to define parameters A and t0 as well as their uncertainties σA and σt are proposed. This approach allows one to reduced substantially the processing time for effective estimation of A and t0 with simplification of the used software. The application of obtained formulae to the analysis of beam test data from the prototype of electromagnetic spectrometer PHOS equipped with the ALTRO electronics are considered together with corresponding Monte-Carlo simulation.

Predictive control for a class of distributed delay systems using Chebyshev polynomials

In this paper, the model predictive control (MPC) is developed for linear time-varying systems with distributed time delay in state. The Chebyshev operational matrices of product, integration and delay are utilized to transform the solution of distributed delay differential equation to the solution of algebraic equations. The Chebyshev functions are also applied to derive approximate solution of finite horizon optimal control problem involved in MPC. The proposed method is simple and computationally advantageous. Illustrative example demonstrates the validity and applicability of the technique.

Numerical solution of the linear two-dimensional Fredholm integral equations of the second kind via two-dimensional triangular orthogonal functions

In this paper, we will give some results for developing the two-dimensional triangular orthogonal functions (2D-TFs) for numerical solution of the linear two-dimensional Fredholm integral equations of the second kind. The product of 2D-TFs and some formulas for calculating definite integral of them are derived and utilized to reduce the solution of two-dimensional Fredholm integral equation to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Numerical examples are presented and results are compared with analytical solution to demonstrate the validity and applicability the method.

Collocation and finite difference-collocation methods for the solution of nonlinear Klein–Gordon equation

Two numerical techniques based on the finite difference and collocation methods are presented for the solution of nonlinear Klein–Gordon equation. The operational matrix of derivative for the cubic B-spline scaling functions is presented and is utilized to reduce the solution of nonlinear Klein–Gordon equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new techniques.

B-polynomial multiwavelets approach for the solution of Abel's integral equation

A numerical method for solving Abel's integral equation as singular Volterra integral equations is presented. The method is based upon Bernstein polynomial (B-polynomial) multiwavelet basis approximations. The properties of B-polynomial multiwavelets are first presented. These properties are then utilized to reduce the singular Volterra integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A Social History of the “Galois Affair” at the Paris Academy of Sciences (1831)

ArgumentThis article offers a social history of the “Galois Affair,” which arose in 1831 when the French Academy of Sciences decided to reject a paper presented by an aspiring mathematician, Évariste Galois. In order to historicize the meaning of Galois's work at the time he tried to earn recognition for his research on the algebraic solution of equations, this paper explores two interrelated questions. First, it analyzes scholarly algebraic practices and the way mathematicians were trained in the nineteenth century to reconstruct the scientific meaning of Galois's research. By emphasizing the historically contingent nature of scientific debates, the paper argues that to historicize the work of a mathematician, we must historicize the scientific world as well. The second part of this article goes further to seek a fresh perspective on Galois's short career by analyzing the cultural, social, and institutional dynamics that governed how the mathematical community came to recognize an aspiring mathematician as one of its own.

Dynamic fracture analysis of adhesive bonded material under normal loading

Dynamic fracture behavior of a Griffith crack along the interface of an adhesive bonded material under normal loading is studied. The singular integral equations are obtained by employing integral transformation and introducing dislocation density functions. By adopting Gauss–Jacobi integration formula, the problem is reduced to the solution of algebraic equations, and their solutions can be obtained by collocation dots method. Based on the parametric discussions presented in the paper, the following conclusions can be drawn: (1) Mode I dynamic stress intensity factor (DSIF) increases with increasing initial crack length and decreasing visco-elastic layer thickness, revealing distinct size effect; (2) The influence of the visco-elastic adhesive relaxation time on the DSIF should not be ignored.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups Bm(𝕊2\{x1,…,xn}) of the n-punctured sphere includes the usual Artin braid groups Bm(for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type [Formula: see text] (for n = 3).We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of Bmis completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients.When n ≥ 2, we prove that the lower central series (respectively, derived series) of Bm(𝕊2\{x1,…,xn}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B2(𝕊2\{x1,x2}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B2(𝕊2\{x1,x2}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ2* ℤ. Further, its lower central series quotients Γi/Γi + 1are direct sums of copies of ℤ2, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γi/Γi + 1are 2-elementary finitely-generated groups.

Optimization of time delay systems by hybrid functions

A method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed. The properties of the hybrid functions which consists of block-pulse functions plus Taylor series are presented. The method is based upon expanding various time functions in the system as their truncated hybrid functions. Operational matrices of integration and delay are presented and are utilized to reduce the solution of an optimazation problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation

AbstractRecently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences. In this article a numerical method for solving the one‐dimensional hyperbolic telegraph equation is presented. The method is based upon Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with Galerkin method are then utilized to reduce the telegraph equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Legendre multiscaling functions for solving the one‐dimensional parabolic inverse problem

AbstractAn inverse problem concerning diffusion equation with a source control parameter is investigated. The approximation of the problem is based on the Legendre multiscaling basis. The properties of Legendre multiscaling functions are first presented. These properties together with Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method

An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Bernstein polynomial basis. The properties of Bernstein polynomials are first presented. The Bernstein polynomial basis vanishes except the first polynomial at x = 0, which is equal to 1 and the last polynomial at x = R, which is also equal to 1 over the interval [0, R]. This provides greater flexibility in which the boundary conditions are imposed at the end points of the interval. These properties together with the Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A numerical analysis of sheet metal formability for automotive stamping applications

A theoretical failure model is presented for the numerical prediction of the forming limit strains of automotive sheets. The model uses the Swift’s diffuse necking and Hill’s localized necking concepts in describing tearing-type sheet metal failures and a computational scheme is proposed in which the failure conditions are expressed in incremental forms. The Bauschinger effect is included properly in the deformation modeling using an additive backstress form of the nonlinear-kinematic hardening rule. The necking conditions and plasticity model are transformed into a set of algebraic equations that may be applied both for proportional and non-proportional strain-controlled loadings. An iterative approach is employed in the incremental solution of algebraic equations. The formability analyses are conducted using the proposed theoretical model and the forming limit strains of two new generation auto sheets (Trip600 1.4mm, DP980 1.15mm) are estimated. The numerically generated FLC are compared with the experimental data and the FLC calculated with the Keeler–Brazier equation. For both steels, the model produced conservative plain–strain intercept values, FLC0, when compared with the predictions of Keeler–Brazier equation. Also the negative minor strain part of the experimental FLD’s is estimated with sufficient accuracy. For the positive minor strain side, however, the predictions are lower than both the experimental fit and the standard curve.

Using triangular orthogonal functions for solving Fredholm integral equations of the second kind

The present work proposes a method for solving Fredholm integral equations. This is demonstrated by using a complementary pair of orthogonal triangular functions set derived from the well-known block pulse functions set. The operational matrices for integration, product of two triangular functions and some formulas for calculating definite integral of them are derived and utilized to reduce the solution of Fredholm integral equation to the solution of algebraic equations. Illustrative examples are included to show the high accuracy of the estimation, and to demonstrate validity and applicability of the method.