Linear Two-point Boundary Value Problems Research Articles

B-spline Collocation Methods for Solving Linear Two-Point Boundary Value Problems

B-spline Collocation Methods for Solving Linear Two-Point Boundary Value Problems

Well-posedness criteria for one family of boundary value problems

This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.

A quasi-linear local variational iteration method for orbit transfer problems

A novel Quasi-linear Local Variational Iteration Method (QLVIM) for solving two-point boundary value problems (TPBVPs) of strongly nonlinear systems is proposed. With quasi-linearization technique, the multidimensional nonlinear TPBVP is transformed into a series of iterative linear TPBVPs, and then into initial value problems (IVPs) which appear in pairs. Then these IVPs are solved by Local Variational Iteration Method (LVIM). Combining the rapid convergence of quasi-linearization with the high computational efficiency of LVIM, the proposed QLVIM can solve various orbit transfer problems in aerospace engineering efficiently and accurately. The high-performance of this method is verified in solving perturbed Lambert's problems and a circular restricted three-body orbit transfer problem. Comparisons with various methods show that the QLVIM has the advantages of high convergence speed, low computation cost and longer solvable time span. Rather than a refined initial guess of starting velocity, the QLVIM only needs a rough initial guess of the trajectory. The results also prove that the QLVIM is capable of solving long-time-span orbit transfer problem. A theoretical proof of its quadratic convergence is given at last.

New Spline Method for Solving Linear Two-Point Boundary Value Problems

In this research, second order linear two-point boundary value problems are treated using new method based on hybrid cubic B-spline. The values of the free parameter,Gamma , are chosen via optimization. The value of the free parameter plays an important role in giving accurate results. Optimization of this parameter is carried out. This method is tested on four examples and a comparison with cubic B-spline, trigonometric cubic B-spline and extended cubic B-spline methods has been carried out. The examples suggest that this method produces more accurate results than the other three methods. The numerical results are presented to illustrate the efficiency of our method.

New Cubic B-Spline Approximation for Solving Linear Two-Point Boundary-Value Problems

In this study, we introduce a new cubic B-spline (CBS) approximation method to solve linear two-point boundary value problems (BVPs). This method is based on cubic B-spline basis functions with a new approximation for the second-order derivative. The theoretical new approximation for a second-order derivative and the error analysis have been successfully derived. We found that the second-order new approximation was O(h3) accurate. By using this new second-order approximation, the proposed method was O(h5) accurate. Four numerical problems consisting of linear ordinary differential equations and trigonometric equations with different step sizes were performed to validate the accuracy of the proposed methods. The numerical results were compared with the least squares method, finite difference method, finite element method, finite volume method, B-spline interpolation method, extended cubic B-spline interpolation method and the exact solutions. By finding the maximum errors, the results consistently showed that the proposed method gave the best approximations among the existing methods. We also found that our proposed method involved simple implementation and straightforward computations. Hence, based on the results and the efficiency of our method, we can say that our method is reliable and a promising method for solving linear two-point BVPs.

Second Chebyshev wavelets (SCWs) method for solving finite-time fractional linear quadratic optimal control problems

In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.

ON A METHOD FOR SOLVING A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we consider a boundary value problem for a family of linear differential equations that obey a family of nonlinear two-point boundary conditions. For each fixed value of the family parameter, the boundary value problem under study is a nonlinear two-point boundary value problem for a system of ordinary differential equations. Non-local boundary value problems for systems of partial differential equations, in particular, non-local boundary value problems for systems of hyperbolic equations with mixed derivatives, can be reduced to the family of boundary value problems for ordinary differential equations. Therefore, the establishment of solvability conditions and the development of methods for solving a family of boundary value problems for differential equations are actual problems. In this paper, using the ideas of the parametrization method of D. S. Dzhumabaev, which was originally developed to establish the signs of unambiguous solvability of a linear two-point boundary value problem for a system of ordinary equations, a method for finding a numerical solution to the problem under consideration is proposed.

On One Algorithm To Find a Solution to a Linear Two-Point Boundary Value Problem

On One Algorithm To Find a Solution to a Linear Two-Point Boundary Value Problem

A Modification of the Parameterization Method for a Linear Boundary Value Problem for a Fredholm Integro-Differential Equation

A modification of the parameterization method is proposed to solve a linear two-point boundary value problem for a Fredholm integro-differential equation. The domain of the problem is partitioned and additional parameters are set as the values of the solution at interior points of the partition subintervals. Definition of a regular pair consisting of a partition and chosen interior points is given. The original problem is transformed into a multipoint boundary value problem with parameters. For fixed values of parameters, we get a special Cauchy problem for a system of integro-differential equations on the subintervals. Using the solution to this problem, the boundary condition and continuity conditions of solutions at the interior mesh points of the partition, we construct a system of linear algebraic equations in parameters. It is established that the solvability of the problem under consideration is equivalent to that of the constructed system.

Fractal Cubic Spline Methods for Singular Boundary-Value Problems

In this article, we propose a new numerical method based on fractal cubic spline for the numerical approximate solutions of linear and nonlinear singular two-point boundary-value problems. Error analysis of the proposed method is examined and it is shown that proposed method has second-order convergence. Numerical examples are provided to validate the theoretical results.

On the Numerical Solution of Fourth-Order Linear Two-Point Boundary Value Problems

This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method reformulates the equation as a collection of second-kind integral equations defined on local subdomains. Each such equation can be stably discretized and solved. The boundary values of these local solutions are matched by solving a banded linear system. The method of deferred corrections is then used to increase the accuracy of the scheme. Deferred corrections require applying the integral operator to a function on the entire domain, for which we provide an algorithm with linear cost. We illustrate the performance of our method on several numerical examples.

Approximate Solutions of Some Linear Boundary Value Problems Using the Homotopy Analysis Method Combined with Sumudu Transform

This paper considers a combined form of the Sumudu transform with the modified homotopy analysis method (MHAM) to find approximate and analytical solutions for linear two-point boundary value problems. This method is called the modified Sumudu transform homotopy analysis method (MSTHAM). The suggested technique relies on the freedom of the homotopy analysis method (HAM) by introducing an appropriate initial approximation and furthermore, the residual error will be canceled in some points of the interval (RECP). Only a first order approximation of MSTHAM will be required, as compared to STHAM, which needs more iterations for the same cases of study. The proposed method has been applied to solve two examples and the results have been compared with those obtained by the standard STHAM and the exact solutions, where it is found that the proposed solutions are of high accuracy and, therefore, MSTHAM is extremely efficient, simple and can be applied to other types of problems.

A Unified Approach for Solving Linear and Nonlinear Odd-Order Two-Point Boundary Value Problems

In this article, we propose new spectral solutions for odd-order two-point boundary value problems. A numerical algorithm based on the use of collocation methods is implemented and analyzed. A new matrix of derivatives of certain Legendre basis polynomials is derived to serve in deriving the proposed algorithm. Our algorithm has the advantage that it is applicable to both linear and nonlinear odd-order two-point boundary value problems. The convergence and error analysis of the suggested method are investigated. Five illustrative examples supported with comparisons are displayed to ensure the efficiency and applicability of the proposed algorithm.

Approximate Design of Optimal Controller for Continuous Disturbance Nonlinear Active Magnetic Bearing System

A successive approximation algorithm for optimal controller of nonlinear active magnetic bearing system is presented. By constructing a series of linear two-point boundary value problems, the original optimal control problem is transformed into a series of inhomogeneous linear two-point boundary value problems. The optimal control law consists of exact linear term and nonlinear compensation term. The method only needs to solve the nonlinear compensation terms of the adjoint vector differential equation iteratively.

Collocation methods for general Riemann-Liouville two-point boundary value problems

General Riemann-Liouville linear two-point boundary value problems of order αp, where n − 1 < αp < n for some positive integer n, are investigated on the interval [0,b]. It is shown first that the natural degree of regularity to impose on the solution y of the problem is $y\in C^{n-2}[0,b]$ and $D^{\alpha _{p}-1}y\in C[0,b]$ , with further restrictions on the behavior of the derivatives of y(n− 2) (these regularity conditions differ significantly from the natural regularity conditions in the corresponding Caputo problem). From this regularity, it is deduced that the most general choice of boundary conditions possible is $y(0) = y^{\prime }(0) = {\dots } = y^{(n-2)}(0) = 0$ and ${\sum }_{j = 0}^{n_{1}}\beta _{j}y^{(j)}(b_{1}) =\gamma $ for some constants βj and γ, with b1 ∈ (0,b] and $n_{1}\in \{0, 1, \dots , n-1\}$ . A wide class of transformations of the problem into weakly singular Volterra integral equations (VIEs) is then investigated; the aim is to choose the transformation that will yield the most accurate results when the VIE is solved using a collocation method with piecewise polynomials. Error estimates are derived for this method and for its iterated variant. Numerical results are given to support the theoretical conclusions.

An iterative method for suboptimal control of a class of nonlinear time-delayed systems

ABSTRACTThis work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.

On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations

A linear two-point boundary value problem for a system of loaded differential equations with impulse effect is investigated. Values in the previous impulse points are taken into consideration in the conditions of impulse effect. The considered problem is reduced to an equivalent multi-point boundary value problem for the system of ordinary differential equations with parameters. A numerical implementation of parametrization method is offered using the Runge–Kutta method of 4th-order accuracy for solving the Cauchy problems for ordinary differential equations. The constructed numerical algorithms are illustrated by examples.

Accurate Computing of Higher Vibration Modes of Thin Flexible Structures

Motivated by the need for high-fidelity modeling and accurate control of future hypersonic vehicles subjected to complex aerothermomechanical loads and many other such applications, a novel scheme is proposed to accurately compute higher modes of vibration for one-dimensional structures by coupling the classic Ritz method as a predictor and the linear two-point boundary value problem solver SUPORE as a corrector. The Ritz method is used to compute a preliminary estimate of the natural frequencies for the desired modes. These estimates are then used as initial guesses by SUPORE, which employs superposition and reorthonormalization to accurately solve the governing boundary value problem. Compared to a high-fidelity Ritz approximation or a highly refined finite element modeling procedure, this is a simple, computationally less expensive, and yet highly accurate method to compute mode shapes for applications that require higher modes. This scheme is used to compute mode shapes within an absolute and relative error tolerance of for the idealized case of a free–free Euler–Bernoulli beam and a typical air-breathing hypersonic vehicle modeled as a thin-walled structure and taking into account deformations due to bending, shear, and torsion.

Optimal vibration control of nonlinear systems with multiple time-delays: An application to vehicle suspension

ABSTRACTThis paper addresses optimal vibration control problem for nonlinear systems with state, control, and measurement delays. By using the functional transformations, the original system is reduced to an equivalent delay-free one. The approximating sequence method is applied to simplify and solve the nonlinear optimal vibration control (NOVC) problem, which is reduced to a linear two-point boundary value problem of approximating sequences. The nonlinear optimal control law is obtained from the convergence solutions by solving a Riccati equation, a Sylvester equation, and the state and adjoint vector differential equations of approximating sequences. The proposed control strategy is applied to design a vibration controller for an active vehicle suspension. The nonlinear vehicle suspension system with delays is modelled, as well as the road disturbance model is established. Comparing with a passive controller and a non-delay-compensation controller (NDCC), the simulations for the designed NOVC are demonstrated. The simulation results illustrate the effectiveness and the simplicity of the proposed approach.

An efficient recursive shooting method for the optimal control of time-varying systems with state time-delay

This paper presents a recursive shooting method for solving the optimal control problem of linear time-varying systems with state time-delay. In this approach, the original time-delay optimal control problem is first transformed into a sequence of linear two-point boundary value problems (TPBVPs) without delay and advance terms. Then, by using a shooting method for the solution of latter sequence in a recursive manner, an optimal control law is achieved which consists feedback and forward terms. The feedback term is determined by solving a matrix Riccati differential equation. The forward term is an infinite sum of adjoint vectors, which can be obtained by solving the above-mentioned sequence of linear non-delay TPBVPs. The convergence analysis of the proposed approach is also provided. Finally, some comparative results are included to illustrate the effectiveness of the proposed approach.