scholarly journals Analytic Continuation for Solutions to the System of Trinomial Algebraic Equations

2020 ◽  
pp. 114-130
Author(s):  
Irina A. Antipova ◽  
Ekaterina A. Kleshkova ◽  
Vladimir R. Kulikov
Keyword(s):  
Integral Representation ◽  
Analytic Continuation ◽  
Algebraic System ◽  
Extension Problem ◽  
Algebraic Equations ◽  
Puiseux Expansions

In the paper, we deal with the problem of getting analytic continuations for the monomial function associated with a solution to the reduced trinomial algebraic system. In particular, we develop the idea of applying the Mellin-Barnes integral representation of the monomial function for solving the extension problem and demonstrate how to achieve the same result following the fact that the solution to the universal trinomial system is polyhomogeneous. As a main result, we construct Puiseux expansions (centred at the origin) representing analytic continuations of the monomial function

scholarly journals New efficiency algorithm for solving fractional order differential-algebraic system

2016 ◽  
Vol 5 (3) ◽  
pp. 152
Author(s):  
Sameer Hasan ◽  
Eman Namah
Keyword(s):  
Exact Solution ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Algebraic System ◽  
Analytic Solution ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Differential Algebraic System ◽  
Fractional Differential

This work provided the evolution of the algorithm for analytic solution of system of fractional differential-algebraic equations (FDAEs).The algorithm referred to good effective method for combination the Laplace Iteration method with general Lagrange multiplier (LLIM). Through this method we have reached excellent results in comparison with exact solution as we illustrated in our examples.

A Numerical Algorithm for Solving Stiff ODEs and DAEs in Multibody Mechanics

1999 ◽  
Author(s):  
Hajrudin Pasic
Keyword(s):  
Algebraic System ◽  
Numerical Solutions ◽  
Solution Procedure ◽  
Differential Algebraic Equations ◽  
Good Precision ◽  
Algebraic Equations ◽  
Constant Matrix ◽  
Fully Implicit ◽  
Collocation Points ◽  
Collocation Technique

Abstract Presented is an algorithm suitable for numerical solutions of multibody mechanics problems. When s-stage fully implicit Runge-Kutta (RK) method is used to solve these problems described by a system of n ordinary differential equations (ODE), solution of the resulting algebraic system requires 2s3 n3 / 3 operations. In this paper we present an efficient algorithm, whose formulation differs from the traditional RK method. The procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. In terms of number of multiplications, the algorithm is about s2 / 2 times faster than the original, nondiagonalized system, as well as s2 times in terms of number of additions/multiplications. With s = 3 the method has the same precision and stability property as the well-known RADAU5 algorithm. However, our method is applicable with any s and not only to the explicit ODEs My′ = f(x, y), where M = constant matrix, but also to the general implicit ODEs of the form f (x, y, y′) = 0. In the solution procedure y is assumed to have a form of the algebraic polynomial whose coefficients are found by using the collocation technique. A proper choice of locations of collocation points guarantees good precision/stability properties. If constructed such as to be L-stable, the method may be used for solving differential-algebraic equations (DAEs). The application is illustrated by a constrained planar manipulator problem.

scholarly journals Seminonlinear boundary value problems for nondegenerate differential-algebraic system

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Differential Algebraic Equations ◽  
Problem Solution ◽  
Algebraic Equations ◽  
Arbitrary Continuous Function ◽  
Differential Algebraic System

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

scholarly journals On the stationary vibrations of a rectangular plate subjected to stress prescribed partially at the circumference

1991 ◽  
Vol 14 (3) ◽  
pp. 581-586
Author(s):  
M. G. El Sheikh ◽  
A. H. Khater ◽  
D. K. Callebaut
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Rectangular Plate ◽  
Singular Integral ◽  
Fourier Transformation ◽  
Algebraic System ◽  
Infinite System ◽  
Algebraic Equations ◽  
Periodical Problem

The stationary periodical problem of a vibrating rectangular plate, stressed at a segment while fixed elsewhere at one of its edges, is considered. Using the finite Fourier transformation, the problem is converted to a singular integral equation that in turn can be reduced to an infinite system of algebraic equations. The truncation of the algebraic system is justified.

scholarly journals Distribution-Based Identification of Yield Coefficients in a Baker’s Yeast Bioprocess

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Dorin Sendrescu
Keyword(s):  
Algebraic System ◽  
Linear Equations ◽  
Distribution Theory ◽  
Baker’S Yeast ◽  
Baker's Yeast ◽  
State Variables ◽  
Algebraic Equations ◽  
Test Functions ◽  
State Equations ◽  
Unknown Parameters

A distribution-based identification procedure for estimation of yield coefficients in a baker’s yeast bioprocess is proposed. This procedure transforms a system of differential equations to a system of algebraic equations with respect to unknown parameters. The relation between the state variables is represented by functionals using techniques from distribution theory. A hierarchical structure of identification is used, which allows obtaining a linear algebraic system of equations in the unknown parameters. The coefficients of this algebraic system are functionals depending on the input and state variables evaluated through some test functions from distribution theory. First, only some state equations are evaluated throughout test functions to obtain a set of linear equations in parameters. The results of this first stage of identification are used to express other parameters by linear equations. The process is repeated until all parameters are identified. The performances of the method are analyzed by numerical simulations.

A numerical approach for solving nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

2016 ◽  
Vol 14 (05) ◽  
pp. 1650036 ◽  
Author(s):  
P. K. Sahu ◽  
S. Saha Ray
Keyword(s):  
Differential Equations ◽  
Present Method ◽  
Algebraic System ◽  
Numerical Approximation ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Nonlinear Algebraic Equations

In this paper, a numerical approximation based on Legendre wavelets has been developed to solve nonlinear fractional Volterra–Fredholm integro-differential equations. Legendre wavelets are generated by dilation and translation of Legendre polynomials. The properties of the Legendre wavelets are presented in the paper. The proposed wavelet method transforms the integral equations to a system of nonlinear algebraic equations and this algebraic system has been solved numerically by Newton’s method. Convergence analysis of the proposed method has been discussed in this paper. Some examples have been illustrated to show the applicability and accuracy of the present method.

scholarly journals Exact Solution of a Linear Difference Equation in a Finite Number of Steps

2018 ◽  
Vol 14 (1) ◽  
pp. 7560-7563
Author(s):  
Sergey Mikhailovich Skovpen ◽  
Albert Saitovich Iskhakov
Keyword(s):  
Exact Solution ◽  
Difference Equation ◽  
Finite Number ◽  
Algebraic System ◽  
Linear Difference Equation ◽  
Algebraic Equations ◽  
Necessary And Sufficient Condition ◽  
Linear Algebraic Equations ◽  
Iterative Equations ◽  
Necessary And Sufficient

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

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