Nonlinear Two-point Boundary Value Problems Research Articles

Criteria for the Existence of an Isolated Solution of a Nonlinear Boundary-Value Problem

A nonlinear two-point boundary-value problem for an ordinary differential equation is studied by the method of parametrization. We construct systems of nonlinear algebraic equations that enable us to find the initial approximation to the solution to the posed problem. In terms of the properties of constructed systems, we establish necessary and sufficient conditions for the existence of an isolated solution to the analyzed boundary-value problem.

Alternative methods for solving nonlinear two-point boundary value problems

Abstract In this sequel, the numerical solution of nonlinear two-point boundary value problems (NTBVPs) for ordinary differential equations (ODEs) is found by Bezier curve method (BCM) and orthonormal Bernstein polynomials (OBPs). OBPs will be constructed by Gram-Schmidt technique. Stated methods are more easier and applicable for linear and nonlinear problems. Some numerical examples are solved and they are stated the accurate findings.

Numerical study on gas flow through a micro–nano porous medium based on finite difference schemes on quasi-uniform grids

In this paper, the unsteady isothermal flow of a gas through a semi-infinite micro–nano porous medium described by a non-linear two-point boundary value problem on a semi-infinite interval has been considered. We solve this problem by a nonstandard finite difference method defined on quasi-uniform grids in order to derive a new numerical approximation. By introducing a stencil that is constructed in such a way that the boundary conditions at infinity are exactly assigned, the proposed method is effectively used to determine the numerical solution. In addition, a mesh refinement and the Richardson’s extrapolation allow to improve the accuracy of the numerical solution and to define a posteriori estimator for the global error of the proposed numerical scheme. We determine the accurate initial slope dudx(0)=−1.1917906497194208 calculated for α=0.5 with good capturing the essential behavior of u(x). This clearly demonstrates that the numerical solutions presented in this paper result highly accurate and in excellent agreement with the existing solutions available in the literature.

Analytical bounds for the electromechanical buckling of a compressed nanocantilever

• We obtain novel and accurate analytical bounds for the pull-in parameters of a compressed nanocantilever. • The interaction between electrostatic, surface, and axial forces is investigated. • The analytical predictions closely agree with numerical results obtained by shooting method. • The analytical bounds are very useful for validating numerical and approximated methods. • The approach may also include the effects of surface elasticity and residual stresses. An analytical approach is presented for the accurate definition of lower and upper bounds for the pull-in voltage and tip displacement of a micro- or nanocantilever beam subject to compressive axial load, electrostatic actuation and intermolecular surface forces. The problem is formulated as a nonlinear two-point boundary value problem and has been transformed into an equivalent nonlinear integral equation. Initially, new analytical estimates are found for the beam deflection, which are then employed for assessing novel and accurate bounds from both sides for the pull-in parameters, taking into account for the effects of the compressive axial load. The analytical predictions are found to closely agree with the numerical results provided by the shooting method. The effects of surface elasticity and residual stresses, which are of significant importance when the physical dimensions of structures descend to nanosize, are also included in the proposed approach.

A Legendre collocation method for distributed-order fractional optimal control problems

In many dynamic processes, the fractional differential operators not only appear as discrete fractional, but they also possess a continuous nature in a sense that their order is distributed over a given range. This paper is concerned with the optimization of systems whose governing equations contain a fractional derivative of distributed order, in the Caputo sense. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration-by-parts formula, the necessary optimality conditions are derived in terms of a nonlinear two-point distributed-order fractional boundary value problem. A Legendre spectral collocation method is developed for solving such problem. The solution method involves the use of three-term recurrence relations for both the left- and right-sided fractional integrals of the shifted Legendre polynomials. The convergence of the proposed method is discussed. The optimal profiles show the performance of the numerical solution and the effect of the fractional derivatives in the optimal results.

INVENTORY CONTROL WITH FIXED COST AND PRICE OPTIMIZATION IN CONTINUOUS TIME

We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper[8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see[11]), i.e., the (s,S,p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s,S,p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Selecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.

On the General Solution to the Bratu and Generalized Bratu Equations

This work shows that the Bratu equation belongs to a general class of Lienard-type equations for which the general solution may be exactly and explicitly computed within the framework of the generalized Sundman transformation. In this perspective the exact solution of the Bratu nonlinear two-point boundary value problem as well as of some well-known Bratu-type problems have been determined.

Approximate optimal iterative approach for a class of oscillatory neuron models

ABSTRACTControl of oscillatory neuron models becomes a growing interest due to the application of implanted stimulus electrodes in mitigating pathological behaviours. We present an approximate optimal iterative control method for minimum-current control of the FitzHugh–Nagumo model with external disturbance. We first focus on revealing optimality conditions based on the Pontryagin's maximum principle and constructing a related nonlinear two-point boundary value problem. Then, we transform the problem into two iterative sequences of linear differential equations. The control law is finally derived and composed of feedback and compensation terms which can be approximated using approximate iterative approach. Simulations illustrate the results.

Series solution of Painlevé II in electrodiffusion: conjectured convergence

A perturbation series solution is constructed with the use of Airy functions, for a nonlinear two-point boundary-value problem arising in an established model of steady electrodiffusion in one dimension, with two ionic species carrying equal and opposite charges. The solution includes a formal determination of the associated electric field, which is known to satisfy a form of the Painlevé II differential equation. Comparisons with the numerical solution of the boundary-value problem show excellent agreement following termination of the series after a sufficient number of terms, for a much wider range of values of the parameters in the model than suggested by previously presented analysis, or admitted by previously presented approximation schemes. These surprising results suggest that for a wide variety of cases, a convergent series expansion for the Painlevé transcendent describing the electric field has been obtained. A suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases. It is conjectured that the series does converge for this subset.

A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u ″ = f (t, u), which arise in a wide variety of engineering applications. We prove the L 2 stability of the LDG scheme and optimal L 2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.

Digital-Control-Based Approximation of Optimal Wave Disturbances Attenuation for Nonlinear Offshore Platforms

The irregular wave disturbance attenuation problem for jacket-type offshore platforms involving the nonlinear characteristics is studied. The main contribution is that a digital-control-based approximation of optimal wave disturbances attenuation controller (AOWDAC) is proposed based on iteration control theory, which consists of a feedback item of offshore state, a feedforward item of wave force and a nonlinear compensated component with iterative sequences. More specifically, by discussing the discrete model of nonlinear offshore platform subject to wave forces generated from the Joint North Sea Wave Project (JONSWAP) wave spectrum and linearized wave theory, the original wave disturbances attenuation problem is formulated as the nonlinear two-point-boundary-value (TPBV) problem. By introducing two vector sequences of system states and nonlinear compensated item, the solution of introduced nonlinear TPBV problem is obtained. Then, a numerical algorithm is designed to realize the feasibility of AOWDAC based on the deviation of performance index between the adjacent iteration processes. Finally, applied the proposed AOWDAC to a jacket-type offshore platform in Bohai Bay, the vibration amplitudes of the displacement and the velocity, and the required energy consumption can be reduced significantly.

A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel

The aim of this paper is to propose a new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration by parts, the necessary optimality conditions are derived in terms of a nonlinear two-point fractional boundary value problem. Based on the convolution formula and generalized discrete Gronwall’s inequality, the numerical scheme for solving this problem is developed and its convergence is proved. Numerical simulations and comparative results show that the suggested technique is efficient and provides satisfactory results.

A nonlinear two-point boundary-value problem in geophysics

We study a recently derived model for gyres, equivalent to a a two-point boundary-value problem for ocean flows with no azimuthal variations. For a large class of oceanic vorticities we establish the existence of solutions using an approach based on the topological transversality theorem.

NUMERICAL SOLUTION OF UNSTEADY FLOW BETWEEN PARALLEL PLATES USING HAAR WAVELET-QUASILINEARIZATION METHOD

The present study is concerned with the analysis of unsteady flow of viscous incompressible fluid between parallel rectangular and circular plates. The Navier-Stokes equations modelling the flow are transformed into a fourth order non-linear ordinary differential equation using a similarity transformation. The resulting equation is solved by a new efficient numerical scheme based on Haar wavelet-quasilinearization method. The solution representing pressure gradient is obtained for large values of Reynolds number. A comparison with numerical results generated by MATLAB routine bvp4c demonstrates that the proposed method is more efficient and promising technique for solving highly nonlinear two-point boundary value problems. The absolute error and rate of convergence of numerical results are computed to test the applicability and accuracy of the method

Optimal state feedback sampled-data control design for Markov jump linear systems

ABSTRACTThis paper is entirely devoted to analyse and solve the optimal state feedback sampled-data control design problem for continuous-time Markov jump linear systems. This is an important unsolved problem in the context of optimal control theory. To this end, necessary and sufficient optimality conditions are characterised in terms of a specific nonlinear two-point boundary value problem. A global convergent algorithm able to solve iteratively the optimality conditions is provided. Moreover, some mathematical properties of the solution of a differential Riccati equation are raised in order to translate the previous conditions into linear matrix inequalities. This result allows a mode independent feedback control structure for the Markovian system under analysis. A practical application borrowed from the literature is included for illustration.

An efficient parallel processing optimal control scheme for a class of nonlinear composite systems

This article presents an efficient parallel processing approach for solving the optimal control problem of nonlinear composite systems. In this approach, the original high-order coupled nonlinear two-point boundary value problem (TPBVP) derived from the Pontryagin's maximum principle is first transformed into a sequence of lower-order decoupled linear time-invariant TPBVPs. Then, an optimal control law which consists of both feedback and forward terms is achieved by using the modal series method for the derived sequence. The feedback term specified by local states of each subsystem is determined by solving a matrix Riccati differential equation. The forward term for each subsystem derived from its local information is an infinite sum of adjoint vectors. The convergence analysis and parallel processing capability of the proposed approach are also provided. To achieve an accurate feedforward-feedback suboptimal control, we apply a fast iterative algorithm with low computational effort. Finally, some comparative results are included to illustrate the effectiveness of the proposed approach.

Nonlinear convection in nano Maxwell fluid with nonlinear thermal radiation: A three-dimensional study

The combined effects of nonlinear thermal convection and radiation in 3D boundary layer flow of non-Newtonian nanofluid are scrutinized numerically. The flow is induced by the stretching of a flat plate in two lateral directions. The mechanism of heat and mass transport under thermophoretic and Brownian motion is elaborated via implementation of the thermal convective condition. The prevailing two-point nonlinear boundary value problem is reduced to a two-point ordinary differential problem by employing suitable similarity transformations. The solutions are computed by the implementation of homotopic scheme. At the end, a comprehensive parametric study has been conducted to analyze the typical trend of the solutions. It is found that the nanoparticle volume fraction and temperature profiles are stronger for the case of solar radiation in comparison with problem without radiation.

Girsanov's transformation based variance reduced Monte Carlo simulation schemes for reliability estimation in nonlinear stochastic dynamics

The study considers the problem of simulation based time variant reliability analysis of nonlinear randomly excited dynamical systems. Attention is focused on importance sampling strategies based on the application of Girsanov's transformation method. Controls which minimize the distance function, as in the first order reliability method (FORM), are shown to minimize a bound on the sampling variance of the estimator for the probability of failure. Two schemes based on the application of calculus of variations for selecting control signals are proposed: the first obtains the control force as the solution of a two-point nonlinear boundary value problem, and, the second explores the application of the Volterra series in characterizing the controls. The relative merits of these schemes, vis-à-vis the method based on ideas from the FORM, are discussed. Illustrative examples, involving archetypal single degree of freedom (dof) nonlinear oscillators, and a multi-degree of freedom nonlinear dynamical system, are presented. The credentials of the proposed procedures are established by comparing the solutions with pertinent results from direct Monte Carlo simulations.

Multilevel anti-derivative wavelets with augmentation for nonlinear boundary value problems

The multilevel augmentation method with the anti-derivatives of the Daubechies wavelets is presented for solving nonlinear two-point boundary value problems. The anti-derivatives of the Daubechies wavelets are applied as the multilevel bases for the subspaces of approximate solutions. This process results in a full nonlinear system that can be solved by the multilevel augmentation method for reducing computational cost. The convergence rate of the present method is shown. It is the order of 2^{s}, 0leq sleq p when p is the order of the Daubechies wavelets. Various examples of the Dirichlet boundary conditions are shown to confirm the theoretical results.

Discrete approximate optimal vibration control for nonlinear vehicle active suspension

This paper presents the approximate optimal vibration control methodology for discrete nonlinear vehicle active suspension subject to persistent road disturbances. Based on a dynamic model of nonlinear vehicle active suspension and a linear exogenous system model of the persistent road disturbances, a nonlinear two-point boundary value (TPBV) problem is introduced. By introducing a sensitive parameter, the original TPBV problem is reformed as a series of TPBV problems without nonlinear items. A discrete approximate optimal vibration controller (DAOVC) can be obtained by solving a Riccati equation, Stein equation, and nonlinear compensation item, respectively. An iteration algorithm is designed to realize the computational realizability of DAOVC based on the control performance. It is demonstrated that the control performance of ride comfort, road holding ability, and suspension deflection under the DAOVC are much smaller than the ones under the classical feedforward and feedback optimal vibration controller (FFOVC) and open-loop vehicle suspension system.