Number Of Particular Solutions Research Articles

Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control

A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL(n,R), which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given.

Solution of Boolean equations via atomic decomposition into independent switching equations

ABSTRACTThis paper considers the problem of solving a system of Boolean equations over a finite (atomic) Boolean algebra other than the two-valued one, referred to herein as a ‘big’ Boolean algebra. The paper suggests the replacement of this system of equations by a single Boolean equation, and then proposes a novel method for solving this equation by using its atomic decomposition into several independent switching equations. This method has many advantages including the efficient derivation of a complete compact listing of all particular solutions in a form similar to the recently developed permutative additive form, but obtained in a more direct fashion without using parameters. Many detailed examples are used to illustrate the proposed new method. The examples demonstrate how the consistency condition might force a collapse of the underlying Boolean algebra into a subalgebra, and also how to list a huge number of particular solutions in a very small space.

Exact Solution of the Problem of Elastic Bending of a Multilayer Beam under the Action of a Normal Uniform Load

An exact solution of the theory of elasticity is presented for the problem of a narrow multilayer bar section transverse bending under the action of a normal uniform load on longitudinal faces. The solution is built using the principle of superposition, by imposing common solutions to the problems of bending a multilayer cantilever with uniform loads on the longitudinal faces and an arbitrary load on the free end, and allows to take into account the orthotropy of the materials of the layers, as well as transverse shear deformation and compression. On the basis of a built-in general solution, a number of particular solutions are obtained for multi-layer beams with various ways of the ends fixing.

Interior field methods for Neumann problems of Laplace's equation in elliptic domains, comparisons with degenerate scales

The interior field method (IFM) is applied to Neumann problems for Laplace's equation in elliptic domains. The polynomial convergence rates are derived, and small condition number as O(N) can be obtained, where N is the number of particular solutions used. Moreover, the effective condition number as O(1) is explored, to display excellent stability (Li et al., 2015 [21]). Numerical experiments are carried out, to support the analysis made. The error analysis of the IFM for Dirichlet problems in circular domains is reported in Li et al. (2016) [19]. The error and stability analysis of the IFM for Neumann problems in elliptic domains is more advanced and challenging; this is the first goal of this paper. The second goal is to compare the Neumann problems with the degenerate scales of Dirichlet problems; some useful guidances are found for application. From the comparisons, the conservation law is essential to guarantee the unique solutions. The adaptive processes are also proposed to deal with the algorithm singularity of Dirichlet problems; they may be applied to the boundary element method (BEM), the original NFM, and the indirect BIEM for the arbitrary smooth boundary or the convex polygonal boundary.

Convex optimization under inequality constraints in rank-deficient systems

Many geodetic applications require the minimization of a convex objective function subject to some linear equality and/or inequality constraints. If a system is singular (e.g., a geodetic network without a defined datum) this results in a manifold of solutions. Most state-of-the-art algorithms for inequality constrained optimization (e.g., the Active-Set-Method or primal-dual Interior-Point-Methods) are either not able to deal with a rank-deficient objective function or yield only one of an infinite number of particular solutions. In this contribution, we develop a framework for the rigorous computation of a general solution of a rank-deficient problem with inequality constraints. We aim for the computation of a unique particular solution which fulfills predefined optimality criteria as well as for an adequate representation of the homogeneous solution including the constraints. Our theoretical findings are applied in a case study to determine optimal repetition numbers for a geodetic network to demonstrate the potential of the proposed framework.

Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

The variable-entered Karnaugh map (VEKM) is shown to be the natural map for representing and manipulating general ‘big’ Boolean functions that are not restricted to the switching or two-valued case. The VEKM is utilized herein in producing a compact general solution of a system of Boolean equations. It serves as a powerful manual tool for function inversion, implementation of the solution procedure, handling don't-care conditions and minimization of the final expressions. The rules of using the VEKM are semi-algebraic and collective in nature, and hence are much easier to state, remember and implement than are the tabular and per-cell rules of classical maps. As a result, the maps used are significantly smaller than those required by classical methods. As an offshoot, the paper contributes some pictorial insight into the representation of ‘big’ Boolean algebras and functions. It also predicts the correct number of particular solutions of a Boolean equation, and produces an exhaustive list of particular solutions. Details of the method are carefully explained and further demonstrated via an illustrative example.

Effective condition number for simplified hybrid Trefftz methods

The simplified hybrid Trefftz method was first proposed in Trefftz [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7] in 1926 for solving Laplace's equation, where the harmonic functions are chosen as admissible functions, and their linear combination is sought to satisfy the boundary conditions. The error analysis of the hybrid TM is provided in [Li ZC, Chen YL, Georgiou GG, Xenohontos C. Special boundary approximation methods for Laplace equation problems with boundary singularities—applications to the Motz problem. Int Comput Math Appl 2006;51:115–42; Li ZC, Lu TT, Hu HY, Cheng AH-D. Trefftz and collocation methods. Southampton: WIT Publishers; 2007], but no stability analysis exists so far. Also the simplified hybrid techniques have been applied for the TM to couple with the finite element method (FEM) in our previous study and only the error analysis has been made. Hence, the stability analysis is important for the simplified hybrid TM. In this paper, we will apply the effective condition number Cond _ eff . For the original hybrid TM [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7], uniform particular solutions satisfying the governed equation (e.g., the harmonic functions satisfying Laplace's equation) were chosen. In general, piecewise particular solutions can be used for wide application of the hybrid TM, and the interior continuity conditions may be dealt with by hybrid techniques. Their algorithms and error analysis are provided in [Huang HT, Li ZC, Herrera I. Coupling techniques of Trefftz methods. Technical Report, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan; 2006] without stability analysis. In this paper, two cases of the simplified hybrid TM are considered: Case I: uniform particular solutions used, and Case II: piecewise particular solutions used. It is proved that both Cond _ eff and Cond grow exponentially, with respect to the number of particular solutions used. In Case I, Cond is huge and Cond _ eff is significantly smaller than Cond; but in Case II, Cond is moderately large, and Cond _ eff is significantly smaller than Cond. Hence the ill-conditioning of the simplified hybrid TM for Case II is not severe. Such theoretical results have been validated by the numerical experiments. The study of Cond _ eff in this paper provides a complete and comprehensive knowledge of the simplified hybrid TM.

On application of classical Eshelby approach to calculating effective elastic moduli of dispersed composites

The problem of finding effective elastic moduli of media with spheroid inclusions in case of small concentration of these inclusions is addressed. A number of particular solutions, both known and new, were obtained as limit transitions and asymptotical expansion of the general solution, based on Eshelby’s approach. A special attention was paid to determining the ranges of applicability of the obtained asymptotical solutions. It was shown that for spheroid inclusions the areas of applicability of the asymptotic solutions are determined by two parameters: the ratio of elastic moduli of the inclusion and the matrix and aspect ratio of the inclusions.

Inequalities between distance-based graph polynomials

In a recent paper [ I. Gutman, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur) 131 (2005) 1-7], the Hosoya polynomial H = H(G,?) of a graph G, and two related distance-based polynomials H1 = H1(G, ?) and H2 = H2(G, ?) were examined. We now show that max-?H1 - ?2H, ?H1 - ?2H} ? H2 ? ?H1 - ??H holds for all graphs G and for all ? ? 0, where ? and ? are the smallest and greatest vertex degree in G. The answer to the question which of the terms ?H1 -?2H and ?H1 -?2H is greater, depends on the graph G and on the value of the variable ?. We find a number of particular solutions of this problem. AMS Mathematics Subject Classification (2000): 05C12, 05C05.

Generalized cofactors and nonlinear superposition principles

It is known from Lie's works that the only ordinary differential equation of first order in which the knowledge of a certain number of particular solutions allows the construction of a fundamental set of solutions is, excepting changes of variables, the Riccati equation. For planar complex polynomial differential systems, the classical Darboux integrability theory exists based on the fact that a sufficient number of invariant algebraic curves permits the construction of a first integral or an inverse integrating factor. In this paper, we present a generalization of the Darboux integrability theory based on the definition of generalized cofactors.

Nonlinear superposition formulae based on the Lie groupSO(n+ 1,n)

Systems of nonlinear ordinary differential equations are constructed, for which the general solution is algebraically expressed in terms of a finite number of particular solutions. Expressions of that type are called nonlinear superposition formulae. These systems are connected with local Lie group transformations on their homogeneous spaces. In the work presented here, the nonlinear superposition formulae are constructed for the case of the SO(3,2) group and some aspects of the general case of SO(n + 1, n) are studied.

Nonlinear superposition formulas based on imprimitive group action

Systems of nonlinear ordinary differential equations are constructed for which the general solution is expressed algebraically in terms of a finite number of particular solutions. The equations and the corresponding nonlinear superposition formula are based on a nonlinear action of the Lie group SL(N,C) on a homogeneous space M. The isotropy group of the origin of this space is a nonmaximal parabolic subgroup of SL(N,C). Such equations can occur as Bäcklund transformations for soliton equations on flag manifolds.

The solution of diagonal decoupling problem by dynamic output feedback and constant precompensator: the general case

In this paper, the solution to diagonal decoupling problem via dynamic output feedback and constant precompensator is presented in the general case. The literature contains a number of particular solutions for this problem. These solutions are valid, for instance, for square transfer matrices or they employ restricted dynamic output feedback schemes. Here, we start with a nonsquare transfer matrix and consider a dynamic output feedback scheme where there is a proper compensator in the feedback loop accompanied by a constant gain (precompensator) in the feedforward path. In this respect, the problem being considered is the dynamic output feedback version of the well-known Morgan's problem. The approach taken involves the characterization of open-loop diagonalizers which admit a dynamic output feedback realization. The key concepts involved in the solution are the generalized version of diagonal causality degree dominance and the set of attainable infinite zero orders. An example is also included to illustrate the main result. >

Analytical solutions to classical non-linear wave equations and their quantum meaning

A number of particular solutions of a class of non-linear Klein-Gordon equations are analysed from the viewpoint of their usefulness as quantum bound states in many-body systems. Similarities and differences between quantum and classical solutions are discussed.

Penalty‐combined approaches to the Ritz‐Galerkin and finite element methods for singularity problems of elliptic equations

AbstractPenlty coupling techniques on an interface boundary, artificial or material, are first presented for combining the Ritz–Galerkin and finite element methods. An optimal convergence rate first is proved in the Sobolev norms. Moreover, a significant coupling strategy, L + 1 = O(|ln h|), between these two methods are derived for the Laplace equation with singularities, where L + 1 is the total number of particular solutions used in the Ritz–Galerkin method, and h is the maximal boundary length of quasiuniform elements used in the linear finite element method. Numreical experiments have been carried out for solving the benchmark model: Motz's problem. Both theoretical analysis and numreical experiments clearly display the importance of penalty‐combined methods is solving elliptic equations with singularities.

Superposition formulas for nonlinear superequations

Nonlinear superequations, for which the general solution can be expressed algebraically in terms of a finite number of particular solutions, are obtained. They are based on the orthosymplectic supergroup OSP(m,2n) and its action on a homogeneous superspace. Superposition formulas are discussed for the cases m=1, n arbitrary, and m=2, n=1. For OSP(2,2) the number of particular solutions needed to reconstruct the general solution depends on the dimension of the underlying Grassmann algebra, whereas for OSP(1,2n) it does not.

A nonconforming combination for solving elliptic problems with interfaces

Nonconforming combinations are provided for solving interface problems of elliptic equations. In these approaches, the Ritz-Galerkin method with particular solutions is used for the part of a solution domain where there are interface singular points; and the conventional finite element method is used for the rest of the solution domain. In addition, admissible functions chosen are constrained to be continuous only at the element nodes on the common boundary of the subdomains. Error bounds are derived in the Sobolev norms, and numerical experiments are given for solving a model interface problem of the equation, − Δu + u = 0. Moreover, a significant coupling relation, L + 1 = O(|ln h|), is found for interface problems by using the nonconforming combinations, where ( L + 1) is the total number of particular solutions used in the Ritz-Galerkin method, and h is the maximal boundary length of triangular elements in the finite element method.

Particular solutions of the equation for the quark propagator in the infra-red region of QCD

A number of particular solutions of the Schwinger-Dyson equation for the quark propagator in the infra-red region of QCD is obtained taking account of the gauge identities. The solutions are of a nonperturbative character, some of them break the chiral invariance. The obtained solutions are analysed with the help of the effective potential method.

Superposition rules for nonlinear coupled first-order differential equations

Pairs of coupled nonlinear differential equations of the polynomial type have been studied. No higher powers than quadratic were considered. Lie’s theorem provides a superposition rule exists if certain operators generate a finite-dimensional Lie algebra. The Lie algebras possible were divided into twenty categories. For each case the general form of the coupled differential equations is obtained. The coupled differential equations are then separated where possible. Solutions are obtained expressed in terms of a finite number of particular solutions.

Systems of ordinary differential equations with nonlinear superposition principles

This work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations. x ̇ = η(x,t). (x, η ϵ R n, t ϵ R) . in terms of a finite number of particular solutions. The point of departure is Lie's criterion according to which such a rule exists if and only if the vector fields η( x, t). ∇ generate a finite dimensional Lie algebra. We provide three different constructive methods for deriving superposition rules and apply them to systems of coupled Riccati equations of the projective and conformal types based, respectively, on the Lie algebra sl( n + 1, R) and o( p + 1, n − p + 1).