Two-point Boundary Value Problem Research Articles

Spectral Solutions of Even-Order BVPs Based on New Operational Matrix of Derivatives of Generalized Jacobi Polynomials

The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given.

Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.

Fixed Points of (α, β, F*) and (α, β, F**)-Weak Geraghty Contractions with an Application

This study aims to provide some new classes of (α,β,F*)-weak Geraghty contraction and (α,β,F**)-weak Geraghty contraction, which are self-generalized contractions on any metric space. Furthermore, we find that the mappings satisfying the definition of such contractions have a unique fixed point if the underlying space is complete. In addition, we provide an application showing the uniqueness of the solution of the two-point boundary value problem.

Neighboring extremal nonlinear model predictive control of a rigid body on SO(3)

AbstractThe issue of implementing nonlinear model predictive control (NMPC) on mechanical systems evolving on special orthogonal group (SO(3)) is taken into consideration in the first place. Necessary conditions of optimality are extracted based on Lie group variational integrators, leading to a two-point boundary value problem (TPBVP) which is solved using sensitivity derivatives and indirect shooting methods. Fast Newton-like methods referred to as fast solvers which are commonly used to solve the TPBVP are established based on the repetition of a nonlinear process. The numerical schemes employed to alleviate the computation burden consist of eliminating some constraint-related but non-essential terms in the trend of sensitivity derivatives calculation and for solving the TPBVP equations. As another claim, assuming that a first attempt to resolve the NMPC problem is accessible, the problem subjected to some changes in its initial conditions (due to some re-planning schemes) can be resolved cost-effectively based on it. Instead of solving the whole optimization process from the scratch, the optimal control inputs and states of the system are updated based on the neighboring extremal (NE) method. For this purpose, two approaches are considered: applying NE method on the first solution that leads to a neighboring optimal solution, or assisting this latter by updating the NMPC-related optimization using exact TPBVP equations at some predefined intermediate steps. It is shown through an example that the first method is not accurate enough due to error accumulations. In contrast, the second method preserves the accuracy while reducing the computation time significantly.

A hybrid optimisation method for intercepting satellite trajectory based on differential game

AbstractThis study addresses orbit design and optimisation for the situation of satellite interception in which the target spacecraft is capable of manoeuvring using continuous magnitude restricted thrust. For the purpose of designing a long-range continuous thrust interception orbit, the orbit motion equations of two satellites with J2 perturbation are constructed. This problem is assumed to be a typical pursuit-evasion problem in differential game theory; using boundary constraint conditions and a performance index function that includes time and fuel consumption, the saddle point solution corresponding to the bilateral optimal is derived, and then this pursuit-evasion problem is transformed into a two-point boundary value problem. A hybrid optimisation method using a genetic algorithm (GA) and sequential quadratic programming (SQP) is derived to obtain the optimal control strategy. The proposed model and algorithm are proved to be feasible for the given simulation cases.

A Dimension-reduction method for the finite-horizon spacecraft pursuit-evasion game

<p style='text-indent:20px;'>The finite-horizon two-person zero-sum differential game is a significant technology to solve the finite-horizon spacecraft pursuit-evasion game (SPE game). Considering that the saddle point solution of the differential game usually results in solving a high-dimensional (24 dimensional in this paper) two-point boundary value problem (TPBVP) that is challengeable, a dimension-reduction method is proposed in this paper to simplify the solution of the 24-dimensional TPBVP related to the finite-horizon SPE game and to improve the efficiency of the saddle point solution. In this method, firstly, the 24-dimensional TPBVP can be simplified to a 12-dimensional TPBVP by using the linearization of the spacecraft dynamics; then the adjoint variables associated with the relative state variables between the pursuer and evader can be expressed in the form of state transition; after that, based on the necessary conditions for the saddle point solution and the adjoint variables in the form of state transition, the 12-dimensional TPBVP can be transformed into the solving of 6-dimensional nonlinear equations; finally, a hybrid numerical algorithm is developed to solve the nonlinear equations so as to obtain the saddle point solution. The simulation results show that the proposed method can effectively obtain the saddle point solution and is robust to the interception time, the orbital altitude and the initial relative states between the pursuer and evader.</p>

Boundary value problems for a second-order differential equation with involution in the second derivative and their solvability

<abstract> <p>We consider the two-point boundary value problems for a nonlinear one-dimensional second-order differential equation with involution in the second derivative and in lower terms. The questions of existence and uniqueness of the classical solution of two-point boundary value problems are studied. The definition of the Green's function is generalized for the case of boundary value problems for the second-order linear differential equation with involution, indicating the points of discontinuities and the magnitude of discontinuities of the first derivative. Uniform estimates for the Green's function of the linear part of boundary value problems are established. Using the contraction mapping principle and the Schauder fixed point theorem, theorems on the existence and uniqueness of solutions to the boundary value problems are proved. The results obtained in this paper cover the boundary value problems for one-dimensional differential equations with and without involution in the lower terms.</p> </abstract>

Realization of the brachistochronic motion of Chaplygin sleigh in a vertical plane with an unilateral nonholonomic constraint

The paper considers the procedure for determining the brachistochronic motion of the Chaplygin sleigh in a vertical plane, where the blade is such that it prevents the motion of the contact point in one direction only. The position of the sleigh mass center and orientation at the final positions is specified, as well as the initial value of mechanical energy. The simplest formulation of a corresponding optimal control problem is given and it is solved by applying Pontryagin?s maximum principle. For some cases, analytical solutions of differential equations of the two-point boundary value problem (TPBVP) of the maximum principle were found. Numerical integration was carried out for other cases using the shooting method, where the assessment of missing terminal conditions was given and it was shown that the solution obtained represents the global minimum time for the brachistochronic motion. The method of the brachistochronic motion by means of a single holonomic and a single unilateral nonholonomic mechanical constraint is presented.

New sharp estimates of the interval length of the uniqueness results for several two-point fractional boundary value problems

<abstract><p>This paper investigates the existence and uniqueness of solutions for several two-point fractional BVPs, including hybrid fractional BVP, sequential fractional BVP and so on. Using the Banach contraction mapping theorem, some sharp conditions that depend on the length of the given interval are presented, which ensure the uniqueness of solutions for the considered BVPs. Illustrative examples are also constructed. The results obtained in this study are noteworthy extensions of earlier works.</p></abstract>

Some results for two classes of two-point local fractional proportional boundary value problems

In this paper, we consider two classes of boundary value problems in the frame of local proportional fractional derivatives. For both of these classes, we obtain the associated Green?s functions and discuss their properties. Using these properties, we go about the uniqueness of the solutions. In addition, we establish Lyapunov-type and Hartman-Wintner-type inequalities and build sharp estimated for the unique solutions of the considered equations.

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem for One Nonlinear Functional - Differential Equation of Even Order

In this article, we consider a two-point boundary value problem for one nonlinear functional differential equation of even order with strong non-linearity on segment [0,1] with homogeneous boundary conditions. With the use of special topological means, sufficient conditions for the existence and uniqueness of a positive solution to the problem under consideration. Existence of a positive solution proved using the well-known Krasnoselsky theorem on a fixed point in a cone, uniqueness is respectively established using the contraction mapping principle. A non-trivial example is given, illustrating the fulfillment of sufficient conditions unique solvability of the problem.

Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation

<p style='text-indent:20px;'>We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments.</p>

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Denumerably many Positive Solutions for Iterative System of Boundary Value Problems with N-Singularities on Time Scales

In this paper we consider a iterative system of two-point boundary value problems with integral boundary conditions having n singularities and involve an increasing homeomorphism, positive homomorphism operator. By applying Hölder’s inequality and Krasnoselskii’s cone fixed point theorem in a Banach space, we derive sufficient conditions for the existence of denumerably many positive solutions. Finally we provide an example to check validity of our obtained results.

Performance characteristics assessment of hollow fiber membrane-based liquid desiccant dehumidifier for drying application

• Proposed novel integral technique based analytical model to assess HFLDD performance. • Analyzed the influence of inlet and design parameters on HFLDD performance. • Effect of Le and St numbers across and along the HFLDD is evaluated. • Case study on conventional/desiccant dryer performance is analyzed in humid climate. • Desiccant dryer found to be best alternative for humid climatic conditions. In the present study, a novel integral technique based two-point boundary value problem is developed using the simplified polynomial equations to assess the hollow fiber membrane-based liquid desiccant dehumidifier performance for drying application. The developed numerical technique is validated with the experimental data available in the literature and observed a maximum deviation of ± 9.7%. The liquid desiccant chosen is lithium chloride. By considering thermal and moisture effectiveness as performance parameters and choosing the air inlet temperature, air specific humidity, desiccant temperature, and liquid to gas ratio as input parameters, the influence of Lewis number and Stanton number on dehumidifier performance is analyzed. The performance analysis concluded that a high liquid to gas ratio, desiccant temperature, and specific humidity enhance the dehumidifier performance. Further, the variation of performance parameters along the length and thermal mass of the membrane column is also assessed. Moreover, a case study on conventional and desiccant dryers performance in humid climates is assessed and observed that the desiccant dryer is the best alternative compared to the conventional dryer. In addition, for the given operating range and inlet condition, the maximum possible vapour absorption rate (δ VAR ) and energy exchange for the conventional dryer is found to be 93 g/hr and 0.8 kW, whereas, for the desiccant dryer, it is 164 g/hr, and 1.4kW respectively.

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

An exact bifurcation diagram for ap-qLaplacian boundary value problem

We study positive solutions to thep–qLaplacian two-point boundary value problem:{−μ[(u′)p−1]′−[(u′)q−1]′=λu(1−u)on (0,1)u(0)=0=u(1)whenp=4andq=2. Hereλ>0is a parameter andμ≥0is a weight parameter influencing the higher-order diffusion term. Whenμ=0(the Laplacian case) the exact bifurcation diagram for a positive solution is well-known, namely, whenλ≤π2there are no positive solutions, and forλ>π2there exists a unique positive solutionuλ,μsuch that‖uλ,μ‖∞→0asλ→π2and‖uλ,μ‖∞→1asλ→∞. Here, we will prove that for allμ>0similar bifurcation diagrams preserve, and they all bifurcate from(λ,u)=(π2,0). Our results are established via the method of sub-super solutions and a quadrature method. We also present computational evaluations of these bifurcation diagrams for various values ofμand illustrate how they evolve whenμvaries.

A sinc-collocation approximation solution for strongly nonlinear class of weakly singular two-point boundary value problems

In this study, an efficient collocation method based on Sinc function coupled with double exponential transformation is developed. This approach is used for solving a class of strongly nonlinear regular or weekly singular two-point BVPs with homogeneous or non homogeneous boundary conditions. The properties of the Sinc-collocation scheme were used to reduce the computations of the problem to the nonlinear system of equations. To use the Newton method in solving the nonlinear system, its vectormatrix form was obtained. The convergence analysis of the method is discussed. The analysis show that the method is convergent exponential. In order to investigate the capability and accuracy of the method, it is applied to solve several existing problems chosen from the open literature. The numerical results compared with other existing methods. The obtained results indicate high capacity and rapid convergence of the proposed method.