Solution Of Two-point Boundary Value Problems Research Articles

Numerical Solution of a Two-Point Boundary-Value Problem for a System of Linear Differential Equations with Nonlinear Two-Dimensional Spectral Parameter

We study the possibility of application of the implicit-function method to the solution of two-point boundary-value problems for systems of linear differential equations with nonlinear two-dimensional spectral parameters appearing in the coefficients of this system and boundary conditions. We also substantiate the convergence of the approximate solutions of discretized problems to the exact solutions of the original problem.

Comparison of the Direct Method and the Maximum Principle in the Problem of the Aircraft Program Control Design

The article deals with the problem of program control design for a dynamic object defined by a nonlinear system of differential equations. Known methods of optimal control require the two-point boundary value problem solution, which in general is coupled with fundamental difficulties. Therefore, this paper proposes a technique that uses the direct method, in which the functional is minimized directly using a population-based algorithm. The use of direct methods is based on the assumption that control signals may be defined by a finite set of parameters. Then a scalar functional is formed, the numerical value of which measures the quality of the obtained solutions. In this case, the search for optimal control is reduced to the problem of single-criterion multi-parameter optimization. The practical importance of this approach is that it eliminates the need to solve a two-point boundary value problem. However, this results in another difficulty, since the approximation of control, in general, requires a large number of parameters. It is known that in this case, the effectiveness of conventional gradient numerical optimization methods decreases markedly. Therefore, it is proposed to take the next step and apply genetic or population-based optimization algorithms that have confirmed their performance in solving this class of problems. For this purpose the paper uses one of the modifications of the particle swarm algorithm. The technique is applied to a test problem describing the spatial movement of a maneuverable aircraft. The direct method is compared with two classical solutions based on the condition that the partial control derivatives of the Hamilton function are equal to zero and with the condition of Hamilton function maximum over controls (Pontryagin’s maximum principle). The presented results show the high degree of similarity between obtained controls for all considered methods of selecting the target functional. At the same time, the accuracy of classical algorithms turns out to be slightly worse, and they show a higher sensitivity to the quality of the initial approximation. Thus, the obtained results confirm the approximate equivalence of the direct method and the classical methods of program control design, at least for the class of problems under consideration. The practical significance of this research is that the use of the direct method is much simpler than solving a two-point boundary value problem necessary for classical algorithms.

THEORETICAL AND NUMERICAL STUDIES OF TWO POINT BOUNDARY VALUE PROBLEMS USING TRIGONOMETRIC AND EXPONENTIAL CUBIC B-SPLINES

A collocation method with di erent types of B-spline functions such as trigonometric and exponential cubic B-splinesfunctions are proposed from many researchers for studying the numerical solutions of many physical nonlinear partialdi erential equations and ordinary di erential equations. In this paper, a collocation nite di erence scheme based ontrigonometric and exponential cubic B-splines functions are investigated and applied for the numerical solution of two-pointboundary value problems. The convergence of trigonometric and exponential cubic B-splines is proved using diagonallydominant matrices. The numerical examples show that our method is very e ective and the maximum absolute error isacceptable.

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty

ABSTRACTThe difficulty of solving the min–max optimal control problems (M-MOCPs) with uncertainty using generalised Euler–Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler–Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.

PERFORMANCE OF MODIFIED NON-LINEAR SHOOTING METHOD FOR SIMULATION OF 2ND ORDER TWO-POINT BVPS

In this research article, numerical solution of nonlinear 2nd order two-point boundary value problems (TPBVPs) is discussed by the help of nonlinear shooting method (NLSM), and through the modified nonlinear shooting method (MNLSM). In MNLSM, fourth order Runge-Kutta method for systems is replaced by Adams Bashforth Moulton method which is a predictor-corrector scheme. Results acquired numerically through NLSM and MNLSM of TPBVPs are discussed and analyzed. Results of the tested problems obtained numerically indicate that the performance of MNLSM is rapid and provided desirable results of TPBVPs, meanwhile MNLSM required less time to implement as comparable to the NLSM for the solution of TPBVPs.

Complexity of certain nonlinear two-point BVPs with Neumann boundary conditions

We study the solution of two-point boundary-value problems for second order ODEs with boundary conditions imposed on the first derivative of the solution. The right-hand side function g is assumed to be r times (r≥1) continuously differentiable with the rth derivative being a Hölder function with exponent ϱ∈(0,1]. The boundary conditions are defined through a continuously differentiable function f. We define an algorithm for solving the problem with error of order m−(r+ϱ) and cost of order mlogm evaluations of g and f and arithmetic operations, where m∈N. We prove that this algorithm is optimal up to the logarithmic factor in the cost. This yields that the worst-case ε-complexity of the problem (i.e., the minimal cost of solving the problem with the worst-case error at most ε>0) is essentially Θ((1/ε)1/(r+ϱ)), up to a log1/ε factor in the upper bound. The same bounds hold for r+ϱ≥2 even if we additionally assume convexity of g. For r=1, ϱ∈(0,1] and convex functions g, the information ε-complexity is shown to be Θ((1/ε)1/2).

A variant of mountain pass theorem

An existence result for a critical point of mountain-pass type, where the classical Palais--Smale condition is not required, is presented. A multiple-critical-point result is then obtained. As an application, the existence of two positive classical solutions for two-point boundary-value problems, without assuming any asymptotic condition on the nonlinearity, is obtained.

Canonical Solution of Two-Point Boundary-Value Problems

Canonical Solution of Two-Point Boundary-Value Problems

An iterative method for time optimal control of dynamic systems

An iterative method for time optimal control of dynamic systemsAn iterative method for time optimal control of a general type of dynamic systems is proposed, subject to limited control inputs. This method uses the indirect solution of open-loop optimal control problem. The necessary conditions for optimality are derived from Pontryagin's minimum principle and the obtained equations lead to a nonlinear two point boundary value problem (TPBVP). Since there are many difficulties in finding the switching points and in solving the resulted TPBVP, a simple iterative method based on solving the minimum energy solution is proposed. The method does not need finding the switching point so that the resulted TPBVP can be solved by usual algorithms such as shooting and collocation. Also, since the solution of TPBVPs is sensitive to initial guess, a short procedure for making the proper initial guess is introduced. To this end, the accuracy and efficiency of the proposed method is demonstrated using time optimal solution of some systems: harmonic oscillator, robotic arm, double spring-mass problem with coulomb friction and F-8 aircraft.

On the construction of asymptotic solutions of two-point boundary-value problems for degenerate singularly perturbed systems of differential equations

We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.

Existence, multiplicity and non-existence of positive solutions for two-point boundary-value problems with strong singularity

We study the existence, multiplicity and non-existence of positive solutions for the singular two-point boundary-value problemswhere $\varphi_{p}(s)=|s|^{p-2}s$, $p>1$, λ is a non-negative real parameter and f ∈ C((0, 1) × [0,∞), (0,∞)). Here, f(t, u) may be singular at t = 0 and/or 1. To obtain the main results we use the global continuation theorem and fixed-point index theory.

Maximum payload path planning for redundant manipulator using indirect solution of optimal control problem

The aim of this paper is to determine the maximum load-carrying capacity and the corresponding joint optimal path for kinematically redundant manipulators to track a given end effector path, subject to actuator torque limits. The path-planning problem is solved by using the indirect solution of open-loop optimal control problem and the scheme makes full use of redundancy to increase the load-carrying capacity. In this method, after decomposing the joint coordinates into redundant and nonredundant parts, dynamic equations are derived and the inverse kinematic is solved in terms of the redundant terms. Then, the Hamiltonian function is formed and the necessary conditions for optimality are derived from Pontryagin’s minimum principle. The obtained equations establish a two-point boundary value problem (TPBVP) which is solved by numerical techniques. An efficient algorithm based on the TPBVP solution is presented to determine the maximum payload and the corresponding joint optimal path. Simulations are performed for two types of mobile base manipulator in order to show how the proposed method is applied to find the maximum payload of redundant manipulator and how extra degrees of freedom are used to increase the maximum payload value.

Some fixed point theorems and existence of positive solutions of two-point boundary-value problems

Three theorems are obtained for the existence of at least one or three fixed points for a completely continuous mapping, which extend the Krasnoselskii’s compression–expansion theorem in cones. Based on them two theorems for the existence of positive solutions of two-point boundary-value problems are proved under a quite relaxed condition compared with the existing literature.

Efficient approximation of implicitly defined functions: General theorems and classical benchmark studies

The traditional techniques of approximation theory in the form of kernel interpolation and cubic spline approximation are used to obtain representations and estimates for functions implicitly defined as solutions of two-point boundary-value problems. We place this benchmark analysis in the following more general context: the approximation of operator fixed points, not known in advance, through a balanced combination of discretization and iteration. We have chosen to make use of the pendulum and elastica equations, linked by the Kirchhoff analogy, to illustrate these ideas. In the study of these important classical models, it is approximation theory, not numerical analysis, which is the required theory; a significant example from micro-biology is cited related to nucleosome repositioning. In addition, other suggested uses of approximation theory emerge. In particular, the determination of approximations via symbolic calculation programs such as Mathematica is proposed to facilitate exact error estimation. No numerical linear inversion is required to compute the approximations in any case. The basic premise of the paper is that approximations should be exactly computable in function form (up to round-off error), with error estimated in a smooth averaged norm. Functional analysis is employed as an effective organizing principle to achieve this ‘a priori’ estimation.

Triple positive solutions for two-point boundary-value problems with one-dimensional p-Laplacian

We consider the following boundary-value problems: . Using a generalization of Leggett–Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary-value problems.

Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.

High-Order interpolants for solutionsof two-point boundary value problems using MIRK methods

In this paper, high-order interpolants are presented for constructing continuous solutionsto a system of two-point boundary value differential equations between widely spaced but accurate dependent variable values. These interpolants are local and symmetric, requiring data only within a single mesh interval and they require a small number of right-hand side evaluations of the defining ODE system to achieve the required order of accuracy. Internal derivative information in the mono-implicit Runge-Kutta formulae is exploited to reduce the number of additional right-hand side evaluations necessary to define the interpolant to the required order of accuracy. When the underlying ODE system is second order, very economical and accurate interpolants are found. All of the interpolants are suitable for grid refinement algorithms in automatic adaptive two-point boundary value packages such as TWPBVP.

Weak caratheodory solution of two-pointboundary value problems in banach spaces

Weak caratheodory solution of two-pointboundary value problems in banach spaces

A discrete Taylor series method for the solution of two-point boundary-value problems

A method for solving ordinary differential equations of the two-point boundary-value problems (TPBVP) is presented. The ordinary differential equations of the TPBVP are reduced to linear integral equations. The solution to the linear integral equation reformulation is expanded in the discrete Taylor series and a Gauss-quadrature integration rule is provided for approximating definite integrals. Application of this method results in the transformation of the integral equations into a linear system of algebraic equations in the Taylor coefficients. Illustrative examples are included to demonstrate the validity and applicability of the proposed method.

A Motivational Example for the Numerical Solution of Two-Point Boundary-Value Problems

This paper presents an example of a two-point boundary-value problem which can be used to motivate the study of numerical techniques for solving such problems by undergraduates. The problem is referred to as the putting problem and illustrates the shooting method for two-point boundary-value problems. The differential equations are developed and numerical results are given.