Solution of linear two-point boundary value problems via a collocation method and application to optimal control

The boundary value problems of ordinary differential equations play an important role in many fields. Here, we implement the He-Laplace method for the solution of linear and nonlinear two-point boundary value problems. The aim of this paper is to compare the performance of the He-Laplace method with shooting method. As a result, for the same number of terms, our method provides relatively more accurate results with rapid convergence than other methods.

Solution of linear two-point boundary value problems and optimal control of time-varying systems by shifted Chebyshev approximations

An invariant imbedding algorithm for the solution of inhomogeneous linear two-point boundary value problems

Linear two-point boundary value problems defined on an infinite domain are converted to initial-value problems using invariant imbedding. The resulting Riccati equations are integrated numerically until the desired accuracy is obtained. Several criteria for determining the appropriate length of integration are presented. Several example problems are presented.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

We discuss algorithmic matters of a computer code for solving linear two-point boundary-value problems. The method of solution uses superposition coupled with an orthonormalization procedure and a variable-step Runge–Kutta–Fehlberg integration scheme. Each time the linearly independent solutions start to lose their numerical independence, the vectors are reorthonormalized before integration proceeds. The underlying principle of the algorithm is then to piece together the intermediate (orthogonalized) solutions, defined on the various subintervals, to obtain the desired solution.

A quantitative measure of well-conditioning for linear two-point boundary value problems

Solution of a nonlinear boundary value problem for a second-order ordinary differential equation by the method of finite differences

An Automatic Continuation Strategy for the Solution of Singularly Perturbed Linear Two-Point Boundary Value Problems

A computer program is described which has been developed for obtaining approximate solutions to linear initial and boundary value problems involving differential equations. The program can be used for ordinary curve fitting as well. For each problem, input to the program includes:1. The equations (in symbolic form) to be satisfied - the differential equations, equations describing auxiliary conditions such as boundary conditions, etc.2. A numerical description of the regions in which each of the equations are to be satisfied.3. Sets of functions (in symbolic form) to be used in linear combinations to approximate the solution functions.Given the above input, the program generates an approximation to the solutuon of the specified problem in terms of the specified functions which is optimum in the least squares sense. A number of examples illustrating the use of the program are included in the paper.

This paper considers the global optimal control for nonlinear interconnected large-scale systems with respect to linear quadratic functional. A successive approximation procedure is proposed to design the feedforward feedback nearly optimal control law. By introducing the Successive Approximation Approach of the nonlinear differential equation theory, a nonhomogeneous linear two-point boundary value problem sequence is constructed to approximate the nonlinear large-scale two-point boundary value problem, which is the necessary condition of the nonlinear control problem. By iteratively solving this nonhomogeneous linear approximating sequence, a global nearly optimal control law that composed of a linear closed-loop feedback compensatory term and an open-loop feedforward compensatory term is designed. An algorithm is presented to design certain qualified nearly optimal control law by truncating an infinite iteration result. An illustrative example shows the validity of the algorithm.

This paper extends results of Yamamoto et al. (Numer. Funct. Anal. Optimiz. 2008; 29:213–224) to the boundary value problem where the sign of r(x) is indefinite. Let H ν A ν U ν = f ν be the finite difference equations on partitions , ν = 1,2,… with as ν → ∞, where H ν and A ν are diagonal and tridiagonal matrices, respectively, and f ν are vectors generated by discretization of f(x). Then equivalent conditions for the boundary value problem to have a unique solution u ∈ C 2[a, b] are given in terms of and .

Consider the boundary value problem ℒu ≡ −(pu′)′ + qu′ + ru = f, a ≤ x ≤ b, u(a) = u(b) = 0. Let H ν A ν U = f and be its finite difference equations and piecewise linear finite element equations on partitions , ν = 1, 2,… with , as ν → ∞, where H ν are n ν × n ν diagonal matrices and A ν as well as are n ν × n ν tridiagonal. It is shown that the following three conditions are equivalent: (i) The boundary value problem has a unique solution u ∈ C 2[a, b]. (ii) For sufficiently large ν ≥ ν0, the inverse exists and , ∀ i, j with a constant M > 0 independent of h ν. (iii) For sufficiently large ν ≥ , exists and , ∀ i, j with a constant independent of h ν. It is also shown by a numerical example that the finite difference method with uniform nodes x i+1 = x i + h, 0 ≤ i ≤ n, h = (b − a)/(n + 1) applied to the boundary value problem with no solution gives a ghost solution for every n.

We present a new discrete scheme for solving boundary value problems for linear ordinary differential equations with constant coefficients or variable coefficients. Our approach works directly on the approximation of polynomials which coefficients related to higher derivatives. In particular, we can transform a general system to essentially diagonally dominant form according to this discrete schemes. Preliminary numerical results show the effectiveness of this method.

The least-squares finite element method for first order systems corresponding to second order linear two-point boundary value problems is considered. A theoretical analysis and error estimates are developed and the estimates are seen to be consistent with our previous numerical studies in Carey and Shen [5]. The method is not subject to the LBB condition and we consider, in particular, the estimates when the polynomial degree differs for the corresponding variables. Superconvergence estimates are also developed.