Chord techniques and Newton's method for discrete bifurcation problems

Summary The numerical treatment of discrete bifurcation problems (2) with chord methods or Newton's method is a question of constructing appropriate initial approximations to prevent the sequence from converging to the trivial solution. This problem is being discussed under conditions which are satisfied for quite a few examples arising in applications (see Sect. 3).     

Counting solutions in bifurcation problems

In this paper, we use the topological degree to obtain some sharp lower bounds for the number of solutions of the parameter slices of the semi-bounded components of the set of nontrivial solutions of an abstract nonlinear equation with a trivial state. By a semi-bounded component one means a component that is bounded in one direction of the parameter. The spectrum of linearization of the equation at the trivial state is not assumed to be discrete. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 149–167, 2006.     

Laguerre series direct method for variational problems

A direct method for solving variational problems via Laguerre series is presented. First, an operational matrix for the integration of Laguerre polynomials is introduced. The variational problems are reduced to the solution of algebraic equations. An illustrative example is given.     

Optimal control problems via a direct method

Summary In this paper we deal with the minimization problem of a cost functional associated to a nonlinear boundary value control problem of a general form, defined in the fixed time interval [0, 1]. Specifically, we first give conditions which ensure that the nonlinear boundary value control problem is solvable and we study the structure of the relative solution set. Then, based on the properties of this set, we establish conditions ensuring both the existence of quasisolutions and that of solutions of the minimization problem under consideration. Such conditions will depend also on the choice of the control space L^r([0, 1],R ^m) where 1 r + Partially supported by the research project M.P.I. (40%) «Teoria del controllo dei sistemi dinamici».     

Shifted Legendre direct method for variational problems

The shifted Legendre polynomial series is employed to solve variational problems. The solution is carried out by using an operational matrix for integrating the shifted Legendre polynomial vector. Variational problems are reduced to solving algebraic equations. Two illustrative examples are given, and the computational results obtained by Legendre series direct method are compared with the exact solutions.     

Applications of Galerkin's method to bifurcation and two-point boundary value problems

The modeling of physical systems inherently involves constructing a mathematical approximation from observable data and/or a priori assumptions. This study refines some recent work on causal interpolation and causal approximation as system modeling techniques. Sufficient conditions for causal interpolators to approximate continuous causal systems are established. State realizations for minimal norm causal interpolators are also established.     

Application of lower-bound direct method to large-scale problems

In this paper, a lower-bound direct method is presented for the large-scale problems. This method is based on the statical shakedown analysis and nonlinear mathematical programming. Here, the interior point method combined with the method of difference of convex functions (IPDCA) is used to solve large-scale problems. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Leitmann’s direct method for fractional optimization problems

Based on a method introduced by Leitmann [G. Leitmann, A note on absolute extrema of certain integrals, Int. J. Non-Linear Mech. 2 (1967) 55-59], we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

A direct method of linearization for continuous minimax problems

We consider the problem of minimizing a nondifferentiable function that is the pointwise maximum over a compact family of continuously differentiable functions. We suppose that a certain convex approximation to the objective function can be evaluated. An iterative method is given which uses as successive search directions approximate solutions of semi-infinite quadratic programming problems calculated via a new generalized proximity algorithm. Inexact line searches ensure global convergence of the method to stationary points.This work was supported by Project No. CPBP-02.15/2.1.1.     

On the bifurcation diagram of discrete analogues for ordinary bifurcation problems

Ordinary bifurcation problems of the form (1) typically have at most one nontrivial, nonnegative solution for λ > 0. The paper shows that this is in general not true for discrete analogues to (1) no matter how small the step width h > 0 is chosen.     

Numerical solution of Hopf bifurcation problems

We present two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ?. It has isolated solutions in the ?-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.     

A consistent direct transcription method for optimal control problems in multibody dynamics

The present work deals with optimal control problems governed by differential-algebraic equations (DAEs). In particular, the control effort, which is necessary for moving a multibody system from one configuration to another, will be minimized. The orientation of the rigid bodies will be described using directors, which facilitates the integration of the equations of motion with an energy-momentum consistent time-stepping scheme [1]. This type of structure-preserving integrators offer outstanding numerical stability and robustness properties in comparison to the often applied generalized coordinates formulation. In the context of optimal control, other kinds of consistent integrators have been applied previously in [2] and [3]. We will test the different formulations with two numerical examples, a 3-link manipulator and a satellite. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hopf bifurcation for non-densely defined Cauchy problems

In this paper, we establish a Hopf bifurcation theorem for abstract Cauchy problems in which the linear operator is not densely defined and is not a Hille?CYosida operator. The theorem is proved using the center manifold theory for non-densely defined Cauchy problems associated with the integrated semigroup theory. As applications, the main theorem is used to obtain a known Hopf bifurcation result for functional differential equations and a general Hopf bifurcation theorem for age-structured models.     

Half-stable solution branches for ordinary bifurcation problems

It is shown that second order bifurcation problems with a positive, autonomous nonlinearity have a smooth branch of positive solutions which tends to infinity. Moreover, this branch satisfies a stability rule saying that the solutions are stable if the branch turns to the right and unstable if it turns to the left.     

A direct general finite difference method for two-dimensional heat conduction problems

A numerical method, based on general finite difference forms, is introduced and applied for solving heat conduction problems in two-dimensional domains of any shape and compounded by subdomains with differing thermal properties.