A Parallel Boundary Value Technique for Singularly Perturbed Two-Point Boundary Value Problems

A class of singularly perturbed two-point boundary-value problems (BVPs) for second-order ordinary differential equations (DEs) is considered here. In order to obtain numerical solution to these problems, an iterative non-overlapping domain decomposition method is suggested. The BVPs are independent in each subdomain and one can use parallel computers to solve these BVPs. One of the characteristics of the method is that the number of processors available is a free parameter of the method. Practical experiments on a Silicon Graphics Origin 200, with 4 MIPS R10000 processors have been performed, showing the reliability and performance of the proposed parallel schemes. Error estimates for the solution and numerical examples are provided.     

A numerical method for solving singularly perturbed systems of nonlinear two-point boundary-value problems

In this paper, a numerical method is proposed to solve singularly perturbed systems of nonlinear two-point boundary-value problems. First, Newton's iteration is used to linearize such problems, reducing these to a sequence of linear singularly perturbed two-point boundary-value problems. Then,a difference scheme is applied to solve the linear systems. The difference scheme is accurate up to O(h ²). Test examples are included to demonstrate the efficiency of the method.     

A numerical algorithm for singular perturbation problems exhibiting weak boundary layers

A class of singularly perturbed two-point boundary-value problems (BVP) for second-order ordinary differential equations is considered here. To avoid the numerical difficulties in the solution to these problems, we divide the domain into two subdomains. The first BVP is a layer domain problem and the second BVP is a regular domain problem. Error estimates are derived for the numerical solution. Numerical examples are provided in support of the proposed method.     

An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ? multiplying the highest derivative, and suitable boundary conditions. In this paper a computational method for solving this system is presented. In this method we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the numerical method, which is constructed for this problem and which involves an appropriate piecewise-uniform mesh.     

A new superconvergent method for systems of nonlinear singular boundary value problems

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h ⁶) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h ⁸) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.     

Application of a modified quasilinearization technique to totally singular optimal control problems†

This paper presents an algorithm for the computation of totally singular optimal control problems. The algorithm is an extension of a modified quasilinearization technique which was originally developed for the solution of non-linear two-point boundary-value problems. A numerical example is included for illustration.     

Non-polynomial sextic spline solution of singularly perturbed boundary-value problems

In this paper, we develop a generalized scheme based on non-polynomial sextic spline for the numerical solution of second-order singularly perturbed two-point boundary-value problems. The proposed method is second, fourth- and sixth-order accurate. Convergence analysis of the fourth-order method is briefly discussed. We show that the approximate solution obtained by the proposed method is better than existing spline methods. Numerical examples are given to illustrate the efficiency of our methods.     

An indirect numerical method for the solution of a class of optimal control problems with singular arcs

An indirect numerical method is presented that solves a class of optimal control problems that have a singular arc occurring after an initial nonsingular arc. This method iterates on the subset of initial costate variables that enforce the junction conditions for switching to a singular arc, and the time of switching off of the singular arc to a final nonsingular arc, to reduce a terminal error function of the final conditions to zero. This results in the solution to the two-point boundary-value problem obtained using the minimum principle and some necessary conditions for singular arcs. The main advantage of this method is that the exact solution to the two-point boundary-value problem is obtained. The main disadvantage is that the sequence of controls for the problem must be known to apply this method. Two illustrative examples are presented.     

Asymptotic initial-value method for a system of singularly perturbed second-order ordinary differential equations of convection–diffusion type

In this paper, a numerical method is suggested to solve a class of boundary value problems (BVPs) for a weakly coupled system of singularly perturbed second-order ordinary differential equations of convection–diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the BVP is constructed by using the basic ideas of a well known perturbation method namely Wentzal, Kramers and Brillouin (WKB). Then, some initial value problems (IVPs) are constructed such that their solutions are the terms of this asymptotic expansion. These problems happen to be singularly perturbed problems and, therefore, exponentially fitted finite difference schemes are used to solve these problems. As the BVP is converted into a set of IVPs and an asymptotic expansion approximation is used, the present method is termed as asymptotic initial-value method. The necessary error estimates are derived and examples provided to illustrate the method.     

Open-loop Stackelberg strategies in a linear-quadratic differential game with time delay

The open-loop Stackelberg strategies in a linear-quadratic differential game with time delay are constructed in the form of necessary and sufficient conditions. The evolution of a game with time delay is described by coupled differential equations composed of lumped and distributed parameter subsystems. Calculation of solutions is facilitated by transforming two-point boundary-value problems to terminal boundary-value ones by the introduction of Riccati-type differential equations.     

Computing the positive solutions of the discrete third-order three-point right focal boundary-value problems

The existence of positive solutions to the discrete third-order three-point boundary-value problems (BVPs) was recently established in Ji and Yang [Positive solutions of discrete third-order three-point right focal boundary value problems, J. Differ. Equat. Appl. 15 (2009), pp. 185–195]. In this paper, we propose an algorithm for the computation of such positive solutions. The method is based on the power method for the dominant eigenvalue and the Crout-like factorization algorithm for the sparse system of linear equations. At each iteration of the method, it calls for a linear solver with linear computational complexity. The proposed method is extremely effective for large-scale problems. A numerical example is also included to demonstrate the effectiveness of the algorithm when applied to the third-order three-point BVPs of differential equation.     

An initial-value technique to solve third-order reaction–diffusion singularly perturbed boundary-value problems

The aim of this paper is to build an efficient initial-value technique for solving a third-order reaction–diffusion singularly perturbed boundary-value problem. Using this technique, a third-order reaction–diffusion singularly perturbed boundary-value problem is reduced to three approximate unperturbed initial-value problems and then Runge–Kutta fourth-order method is used to solve these unperturbed problems numerically.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.     

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

We propose a fully discrete ε-uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. Thus, we show that it is possible to develop a ε-uniform method, totally in the context of finite differences, without solving any differential equation exactly. We further study the convergence properties of the numerical method proposed and prove that it nodally converges to the true solution for any ε. Finally, a set of numerical experiments is carried out to validate the theoretical results computationally.     

Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems

The paper describes computable local a posteriori error estimates for the numerical solution of convection-dominated boundary-value problems. Being applied to singularly perturbed elliptic equations, the obtained estimates are uniform w.r.t. the small parameter. Moreover, if quadrature errors are neglected the numerical approximation of the theoretical error bounds preserves the relation signs in the estimates.     

A method for the algebraic solution of a class of non-stationary,discrete-time,two-point boundary-value problems

Considerable attention has been given, even recently, to the solution of two-point boundary-value problems, by means of algorithms that cannot always be considered satisfactory. The present work suggests, for the discrete case, a method which, with reference to certain hypotheses, allows the algebraic solution of a large class of these problems. Specifically, by applying Casorati'a linear operator, it is possible to pass from a single system of 2n complete linear difference equations with time-varying coefficients to two distinct systems, each of n second-order difference equations. Thus after appropriate transformation of the boundary values it is possible to solve independently each of the two above systems. Also the technique proposed can be usefully employed both to solve systems of non-linear, two-point boundary-value difference equations (using the quasi-linearization method) and when it is desired, or it is only possible, to determine just a part of the unknown vector of a system of difference equations.     

Piecewise-linearized and linearized θ-methods for ordinary and partial differential equations

Initial- and boundary-value problems appear frequently in many branches of physics. In this paper, several numerical methods, based on linearization techniques, for solving these problems are reviewed. First, piecewise-linearized methods and linearized θ-methods are considered for the solution of initial-value problems in ordinary differential equations. Second, piecewise-linearized techniques for two-point boundary-value problems in ordinary differential equations are developed and used in conjunction with a shooting method. In order to overcome the lack of convergence associated with shooting, piecewise-linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes are presented. Third, methods of lines in either space or time for the solution of one-dimensional convection-reaction-diffusion problems that transform the original problem into an initial- or boundary-value one are reviewed. Methods of lines in time that result in boundary-value problems at each time step can be solved by means of the techniques described here, whereas methods of lines in space that yield initial-value problems and employ either piecewise-linearized techniques or linearized θ-methods in time are also developed. Finally, for multidimensional problems, approximate factorization methods are first used to transform the multidimensional problem into a sequence of one-dimensional ones which are then solved by means of the linearized and piecewise-linearized methods presented here.     

Controllability and Optimization of Pseudohyperbolic Systems

The problems of singular optimization and controllability of systems described by the boundary-value problems for pseudohyperbolic equations are studied. The existence and uniqueness theorems for solutions to basic boundary-value problems for pseudohyperolic equations in the nonclassic Sobolev-like spaces under wide assumptions as to right-hand sides are proved. Based on these results, final impulse-point controllability of the systems and existence of optimal control are studied.     

The recursive reduced-order solution of an open-loop controlproblem of linear singularly perturbed systems

A reduced-order method with an arbitrary degree of accuracy is obtained for solving the linear-quadratic optimal open-loop control problem. The original two-point boundary value problem is transformed into the pure-slow and pure-fast reduced-order completely decoupled initial value problems. By doing this, the stiffness of the singularly perturbed two-point boundary value problem is converted into the problem of an ill-defined linear system of algebraic equations