Effective multigrid algorithms for algebraic system arising from static peridynamic systems

2021 ◽  
Author(s):  
Gwanghyun Jo ◽  
Youn Doh Ha
Keyword(s):  
Algebraic System

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Extensions of Elementary Cause-Effect Structures

2021 ◽  
Vol 179 (2) ◽  
pp. 93-111
Author(s):  
Ludwik Czaja
Keyword(s):  
Characteristic Feature ◽  
Petri Nets ◽  
Algebraic System ◽  
Formal System ◽  
Parallel Execution

Cause-effect structures are objects of a formal system devised for modeling, testing and verifying properties of tasks, where parallel execution of actions is the most characteristic feature. This is an algebraic system called a quasi-semiring. In this paper elementary cause-effect structures, a system behaviourally equivalent to 1-safe Petri nets, are extended by the following features: weighted edges, multi-valued nodes having capacities (counterpart of place/transition Petri nets), inhibitors and a model of time. The extensions are accomplished by modifying the notion of state and semantics, but leaving unchanged structure of the quasi-semiring expressions.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

2016 ◽  
Vol 6 (2) ◽  
pp. 211-221 ◽  
Author(s):  
Lei Zhu ◽  
Wei-Wei Xu ◽  
Xing-Dong Yang
Keyword(s):  
Algebraic System ◽  
Condition Numbers ◽  
Linear Algebraic System ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Interest In Science ◽  
Perturbation Bounds

AbstractWe consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
A. M. Nagy ◽  
N. H. Sweilam ◽  
Adel A. El-Sayed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Algebraic System ◽  
Order Differential Equation ◽  
Fundamental Problem ◽  
Operational Matrix ◽  
Variable Order ◽  
Engineering Problems ◽  
Fractional Differential ◽  
Fourth Kind

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

scholarly journals Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

2021 ◽  
Vol 5 (3) ◽  
pp. 70
Author(s):  
Esmail Bargamadi ◽  
Leila Torkzadeh ◽  
Kazem Nouri ◽  
Amin Jajarmi
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Algebraic System ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Chebyshev Wavelets ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

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.

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