Deficient discrete cubic spline solution for a system of second order boundary value problems

In this paper we use deficient discrete cubic spline to obtain approximate solution of a system of second order boundary value problems. It is shown that the method is of order 2 when a parameter takes a specific value. A well known numerical example is presented to illustrate our method as well as to compare the performance with other numerical methods proposed in the literature.     

Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variations

In the parameter variation method, a scalar parameterk, k[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method.The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables).     

Optimality conditions for discrete calculus of variations problems

Nonlinear discrete calculus of variations problems with variable endpoints and with equality type constraints on trajectories are considered. We derive new nontrivial first- and second-order necessary optimality conditions.     

Automatic solution of optimal-control problems. I. simplest problem in the calculus of variations

An automatic method for obtaining the numerical solution for the simplest problem in the calculus of variations is described. The nonlinear two-point boundary-value Euler-Lagrange equation is solved using the Newton-Raphson method for obtaining successive approximations of the solution. The derivatives required for the solution of the problem are computed automatically using the table method. The user of the program need only input the integrand of the objective function in the calculus-of-variations problem and specify the boundary conditions. None of the derivatives usually associated with the Euler-Lagrange equation and the Newton-Raphson method need be calculated by hand. An example is given with numerical results. The automatic solution of the simplest problem in the calculus of variations in this paper is considered to be the first step in the automatic solution of more general optimal-control problems.     

On the existence of a certain cubic spline

We propose a constructive proof of the existence and uniqueness of a defect-1 parabolic interpolation non-periodic spline that does not require additional boundary-value conditions. An estimate of the interpolation error is obtained. Bibliography: 3 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 124–129.     

One-dimensional non-coercive problems of the calculus of variations

We establish a necessary and sufficient condition for the existence of the minimum of the functional dt in the class in terms of a limitation on the slope d. We derive some applications regarding quasi-coercive and non-coercive integrands.     

An explicit solution for the cubic spline interpolation for functions of a single variable

An algorithm for computing the cubic spline interpolation coefficients without solving the matrix equation involved is presented in this paper. It requires only O(n) multiplication or division operations for computing the inverse of the matrix compared to O(n²) or larger number of operations in the Gauss elimination method.     

A fourth-order cubic spline method for linear second-order two-point boundary-value problems

A cubic spline method for linear second-order two-point boundary-valueproblems is analysed. The method is a Petrov-Galerkin methodusing a cubic spline trial space, a piecewise-linear test space,and a simple quadrature rule for the integration, and may beconsidered a discrete version of the H¹-Galerkin method. Themethod is fully discrete, allows an arbitrary mesh, yields alinear system with bandwidth five, and under suitable conditionsis shown to have an 0(h^4– rate of convergence in the W_p¹norm for i = 0, 1, 2, 1p. The H¹-Galerkin method and orthogonalspline collocation with Hermite cubics are also discussed.     

Lagrange multipliers for regular and singular problems in the calculus of variations

A Lagrange multiplier rule is presented for a variational problem of Bolza type under hypotheses that allow certain components of the coefficient matrices involved in the functional being minimized to fail to be integrable near an endpoint of the interval on which the relevant functions are defined. The problem is also addressed when all coefficients are of classL ², but not necessarily bounded. Applications are made to ascertain properties of functions providing equality to certain singular and regular integral inequalities appearing in the literature.     

A new class of problems in the calculus of variations

This paper investigates an infinite-horizon problem in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak.     

A class of homogenization problems in the calculus of variations

We study a class of integral functionals for which the integrand f^e(x, u, ?u) is an oscillatory function of both x and u. Our method is based on the concept of Γ-convergenee. Technical difficulties arise because f^e(x, u, ?u) is not convex or equi-continuous in u with respect to e. Two somewhat different approaches, based respectively on abstract convergence theorems and the study of affine functions, are exploited together to overcome these technical difficulties. As an application, we give another proof of a homogenization result of P. L. Lions, G. Papanicolaou, and S. R. S. Varadhan for Hamilton-Jacobi equations.     

A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems

Bickley [5] had suggested the use of cubic splines for the solution of general linear two-point boundary-value problems. It is well known since then that this method gives only order h² uniformly convergent approximations. But cubic spline interpolation itself is a fourth-order process. We present a new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems: y″ = f(x, y, y′), a < x < b, α₀y(a) − α₀y′(a) = A, β₀y(b) + β₁y′(b) = B. We generate the solution at the nodal points by a fourth-order method and then use ‘conditions of continuity’ to obtain smoothed approximations for the second derivatives of the solution needed for the construction of the cubic spline solution. We show that our method provides order h⁴ uniformly convergent approximations over [a, b]. The fourth order of the presented method is demonstrated computationally by two examples.     

Third-order variable-mesh cubic spline methods for nonlinear two-point singularly perturbed boundary-value problems

A family of third-order variable-mesh methods for singularly perturbed two-point boundary-value problems of the form y=f(x,y,y),y(a)=A, y(b)=B is derived. The convergence analysis is given, and the method is shown to have third-order convergence properties. Several test examples are solved to demonstrate the efficiency of the method.