On solving systems of linear algebraic equations by Gibbs’s method

The paper presents an algorithm of approximate solution of a system of linear algebraic equations by the Monte Carlo method superimposed with ideas of simulating Gibbs and Metropolis fields. A solution in the form of a Neumann series is evaluated, the whole vector of solutions is obtained. The dimension of a system may be quite large. Formulas for evaluating the covariance matrix of a single simulation run are given. The method of solution is conceptually linked to the method put forward in a 2009 paper by Ermakov and Rukavishnikova. Examples of 3 × 3 and 100 × 100 systems are considered to compare the accuracy of approximation for the method proposed, for Ermakov and Rukavishnikova’s method and for the classical Monte Carlo method, which consists in consecutive estimation of the components of an unknown vector.     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.     

Solutions of Emden–Fowler equations by homotopy-perturbation method

In this paper, approximate and/or exact analytical solutions of the generalized Emden–Fowler type equations in the second-order ordinary differential equations (ODEs) are obtained by homotopy-perturbation method (HPM). The homotopy-perturbation method (HPM) is a coupling of the perturbation method and the homotopy method. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. In this work, HPM yields solutions in convergent series forms with easily computable terms, and in some cases, only one iteration leads to the high accuracy of the solutions. Comparisons with the exact solutions and the solutions obtained by the Adomian decomposition method (ADM) show the efficiency of HPM in solving equations with singularity.     

Exp-function method for solving Kuramoto–Sivashinsky and Boussinesq equations

In this paper, we use the Exp-function method to construct the generalized solitary and periodic solution of the Kuramoto–Sivashinsky and Boussinesq equations. These equations play very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The results show the reliability and efficiency of the proposed method.     

Modified projection method for solving a system of monotone equations with convex constraints

In this paper, we propose a modified projection method for solving a system of monotone equations with convex constraints. At each iteration of the method, we first solve a system of linear equations approximately, and then perform a projection of the initial point onto the intersection set of the feasible set and two half spaces containing the current iterate to obtain the next one. The iterate sequence generated by the proposed algorithm possesses an expansive property with regard to the initial point. Under mild condition, we show that the proposed algorithm is globally convergent. Preliminary numerical experiments are also reported.     

Redefined cubic B-splines collocation method for solving convection–diffusion equations

In this work, we discuss collocation method based on redefined cubic B-splines basis functions for solving convection–diffusion equation with Dirichlet’s type boundary conditions. Stability of this method has been discussed and shown that it is unconditionally stable. The developed method is tested on various problems and the numerical results are reported in tabular form. The computed results are compared wherever possible with those already available in literature. The method is shown to work for Péclet number ? 10. Easy and economical implementation process is the strength of it. This method can be easily extended to handle non-linear convection–diffusion partial differential equations.     

On solving the problems of stability by Lyapunov’s direct method

We consider problems of stability and instability of the trivial solution to nonautonomous systems of differential equations. We suggest new theorems of Lyapunov’s direct method with the use of semi-definite auxiliary functions. The idea is based on the use of the additional function that evaluates the rate of convergence of the solutions to the set, where Lyapunov’s function vanishes. We formulate theorems on the non-asymptotic stability and instability. The results are illustrated by examples, where we give a comparison with known results.     

On solving equations of algebraic sum of equal powers

It is well known that a system of equations of sum of equal powers can be converted to an algebraic equation of higher degree via Newton's identities. This is the Viete-Newton theorem. This work reports the generalizations of the Viete-Newton theorem to a system of equations of algebraic sum of equal powers. By exploiting some facts from algebra and combinatorics, it is shown that a system of equations of algebraic sum of equal powers can be converted in a closed form to two algebraic equations, whose degree sum equals the number of unknowns of the system of equations of algebraic sum of equal powers.     

Modified Newton method in circular interval arithmetic

The modified Newton method for multiple roots is organized in an interval method to include simultaneously the distinct roots of a given polynomialP in complex circular interval arithmetic. A condition on the starting disks which ensures convergence is given, and convergence is shown to be quadratic. As a consequence, a simple parallel algorithm to approach all the distinct roots ofP is derived from the modified Newton method.The research reported in this paper has been made possible through the support and the sponsorship of the Italian Government through the Ministero per l'Universitá e la Ricerca Scientifica under Contract MURST 60%, 1990 at the Universitá di L'Aquila.     

Decomposition and interval arithmetic applied to global minimization of polynomial and rational functions

A recent global optimization algorithm using decomposition (GOP), due to Floudas and Visweswaran, when specialized to the case of polynomial functions is shown to be equivalent to an interval arithmetic global optimization algorithm which applies natural extension to the cord-slope form of Taylor's expansion. Several more efficient variants using other forms of interval arithmetic are explored. Extensions to rational functions are presented. Comparative computational experiences are reported.     

The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree

One of the considerable discussions in data interpolation is to find the optimal number of data which minimizes the error of the interpolation polynomial. In this paper, first the theorems corresponding to the equidistant nodes and the roots of the Chebyshev polynomials are proved in order to estimate the accuracy of the interpolation polynomial, when the number of data increases. Based on these theorems, then we show that by using a perturbation method based on the CESTAC method, it is possible to find the optimal degree of the interpolation polynomial. The results of numerical experiments are presented.     

A new simultaneous method of fourth order for finding complex zeros in circular interval arithmetic

Starting from disjoint disks which contain polynomial complex zeros, the new iterative interval method for simultaneous finding of inclusive disks for complex zeros is formulated. The convergence theorem and the conditions for convergence are considered, and the convergence is shown to be fourth. Numerical examples are included.     

Equivalent method for accurate solution to linear interval equations

Based on linear interval equations, an accurate interval finite element method for solving structural static problems with uncertain parameters in terms of optimization is discussed. On the premise of ...     

A new Chebyshev polynomial approximation for solving delay differential equations

The purpose of this study is to give a Chebyshev polynomial approximation for the solution of mth-order linear delay differential equations with variable coefficients under the mixed conditions. For this purpose, a new Chebyshev collocation method is introduced. This method is based on taking the truncated Chebyshev expansion of the function in the delay differential equations. Hence, the resulting matrix equation can be solved, and the unknown Chebyshev coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.     

An error-free algorithm to solve a linear system of polynomial equations

The paper presents an error-free algorithm to solve a system of linear equations with polynomial coefficients. Modular arithmetic in residual polynomial class and in residual numeric class is employed. The algorithm is iterative and well suited for implementation for computers with vector operations and fast and error-free convolutors.     

The use of interval arithmetic in solving a non-linear rational expectation based multiperiod output-inflation process model: The case of the IN/GB method

The use of interval mathematics to solve non-linear problems is an attractive alternative to traditional real-number techniques. It was demonstrated in a previous paper [Stradi, B., Haven, E., 2005. Optimal investment strategy via interval arithmetic. International Journal of Theoretical and Applied Finance 8(2), 185–205] that interval arithmetic in the form of the Interval-Newton Generalized Bisection (IN/GB) method is effective in solving highly non-linear problems. In this paper we solve a rational expectations models with the help of the IN/GB method. This method is capable of (i) rapidly eliminating no solution sections of the multidimensional space and (ii) concentrate computational efforts on those areas of multidimensional space where there may be a solution.     

A strongly polynomial algorithm for solving two-sided linear systems in max-algebra

An algorithm for solving m×n systems of (max,+)-linear equations is presented. The systems have variables on both sides of the equations. After O(m⁴n⁴) iterations the algorithm either finds a solution of the system or finds out that no solution exists. Each iteration needs O(mn) operations so that the complexity of the presented algorithm is O(m⁵n⁵).     

On the computation of solutions of systems of interval polynomial equations

Systems of algebraic equations with interval coefficients are very common in several areas of engineering sciences. The computation of the solution of such systems is a central problem when the characterization of the variables related by such systems is desired.In this paper we characterize the solution of systems of algebraic equations with real interval coefficients. The characterization is obtained considering the approach introduced in J. Comput. Appl. Math. 136 (2001) 271.