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We study the shape optimization problem for the paper machine headbox which distributes a mixture of water and wood fibers in the paper making process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The mathematical formulation leads to the optimal control problem in which the control variable is the shape of the domain representing the header, the state problem is represented by the generalized Navier-Stokes system with nontrivial boundary conditions. This paper deals with numerical aspects of the problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of solutions for the second-order nonlinear delay differential equations. By the use of the Schauder fixed point theorem, the existence of the solutions on the half-line is derived. Via the Banach contraction principle, another result concerning the existence and uniqueness of solutions on the half-line is established. The main results in this paper extend some of the existing literatures. [ABSTRACT FROM AUTHOR]

The paper establishes existence, uniqueness and regularity for the solution of a phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions. Moreover, some estimates for the solution are given. The non-linear parabolic problem considered here can be used to modeling the solidification (liquidification) process to a matter. The non-homogeneous boundary conditions (depending both on time and space variables) can be understood as a boundary control. [ABSTRACT FROM AUTHOR]

In this paper we prove the convergence of an iterative scheme of fractional steps type for a non-homogeneous Cauchy-Neumann boundary optimal control problem governed by non-linear phase-field system, when the boundary control is dependent both on time and spatial variables. Moreover, necessary optimality conditions are established for the approximating process. The advantage of such approach leads to a numerical algorithm in order to approximate the original optimal control problem. [ABSTRACT FROM AUTHOR]

Consider the over-determined system Fx = b where $${{\bf F}\in\mathcal{R}^{m \times n}, m \geq n}$$ and rank ( F) = r ≤ n, the effective condition number is defined by $${{\rm Cond_{-}eff }= \frac {\|{\bf b}\|}{\sigma_r\|{\bf x}\|}}$$, where the singular values of F are given as σmax = σ1 ≥ σ2 ≥ . . . ≥ σ r > 0 and σ r+1 = . . . = σ n = 0. For the general perturbed system ( A+Δ A)( x+Δ x) = b+Δ b involving both ΔA and Δb, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz’s problem by the collocation Trefftz methods (CTM). Motz’s problem is the benchmark of singularity problems. We choose the general particular solutions $${v_L = \sum\nolimits_{k=0}^L d_k (\frac {r}{R_p})^{k+\frac 12}}$$ $${{\rm cos}(k +\frac 12)\theta}$$ with a radius parameter R p. The CTM is used to seek the coefficients d i by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter R p = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of R p is misleading. Under the optimal choice R p = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method. [ABSTRACT FROM AUTHOR]

Abstract  In this paper the problem of finding the shape of distributed piezoelectric modal sensors/actuators for plates with arbitrary boundary conditions is treated by an optimization approach. A binary function is used to model the design variable: the polarization profile of the piezoelectric layers. Contrary to what it could be expected, it is analytically proved that it is possible to find manufacturable polarization profiles taking on only two values, i.e. either positive or negative polarization. Several numerical examples are shown to corroborate that such topologies isolate particular vibration modes in the frequency domain. [ABSTRACT FROM AUTHOR]

The paper investigates model reduction techniques that are based on a nonlocal quasi-continuum-like approach. These techniques reduce a large optimization problem to either a system of nonlinear equations or another optimization problem that are expressed in a smaller number of degrees of freedom. The reduction is based on the observation that many of the components of the solution of the original optimization problem are well approximated by certain interpolation operators with respect to a restricted set of representative components. Under certain assumptions, the “optimize and interpolate” and the “interpolate and optimize” approaches result in a regular nonlinear equation and an optimization problem whose solutions are close to the solution of the original problem, respectively. The validity of these assumptions is investigated by using examples from potential-based and electronic structure-based calculations in Materials Science models. A methodology is presented for using quasi-continuum-like model reduction for real-space DFT computations in the absence of periodic boundary conditions. The methodology is illustrated using a basic Thomas–Fermi–Dirac case study. [ABSTRACT FROM AUTHOR]

Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a Sturm-Liouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equations: x″(t) + αx′(t) + βx(t) = f(t, x(t), x(α1(t)), …, x(αn(t))), a.e. t ϵ [0, T], Δx(tk) = Iκ(x(tκ), x′(tκ)), k = 1,…, m, Δx′(tκ) = Jκ(x(tκ), x′(tκ)), k = 1,…, m, x(i)(0) = x(i)(T), i = 0, 1. Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaeffer's fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, We study the existence and uniqueness of solutions for boundary value problems to the singular one-dimension p-Laplacian by using mixed monotone method. Our results improve several recent results established in the literature. [ABSTRACT FROM AUTHOR]

The study in this paper is a numerical integration of second-order three-point boundary value problems under two imposed nonlocal boundary conditions at t = t0, t = ξ, and t = t1 in a general setting, where t0 < ξ < t1. We construct a two-stage Lie-group shooting method for finding unknown initial conditions, which are obtained through an iterative solution of derived algebraic equations in terms of a weighting factor r ϵ (0, 1). The best r is selected by matching the target with a minimal discrepancy. Numerical examples are examined to confirm that the new approach has high efficiency and accuracy with a fast speed of convergence. Even for multiple solutions, the present method is also effective to find them. [ABSTRACT FROM AUTHOR]