Ordinary Differential Equation Boundary Value Research Articles

An adaptive implementation of interpolation methods for boundary value ordinary differential equations

Various interpolation-based schemes are used to construct a variable order algorithm with local error control for the numerical solution of two-point boundary value problems. Results of computational experiments are presented to demonstrate the empirical relationship between prescribed local error and the resultant global error in the computed solution.

On accurate solution of renal models

A method for solving two-point boundary-value ordinary differential equations arising in modeling of the renal concentrating mechanism is given. It requires six function evaluations in each subinterval for an O( h 7) local error, where h is the size of each subinterval. No information outside the subinterval is used. Results of computational experiments comparing this method with other known methods are given.

A collocation solver for systems of boundary-value differential/algebraic equations

Abstract A spline collocation procedure has been implemented to solve systems of multi-point boundary-value problems for mixed order differential/algebraic equations (BVP-DAEs). This implementation is based on the modification of an existing code “COLSYS”, which solves boundary-value ordinary differential equations (BVP-ODEs) alone. With minor changes to the original COLSYS code, the resulting procedure offers robust solutions to systems of DAEs, in addition to ODEs. A numerical example with application in chemical engineering is shown to demonstrate the superior ability of this procedure over other software.

Improving the accuracy of finite difference methods for solving boundary value ordinary differential equations

It is shown that by introducing one or three interpolated values in each subinterval the local truncation error of finite difference (box, gap and deferred correction) methods can be decreased by two to four orders of magnitude when solving two point boundary value ordinary differential equations.

Studies in approximation methods—III: Boundary value ordinary differential equations

A numerical comparison is made between the orthogonal collocation and Runge-Kutta techniques for solving boundary value ordinary differential equations. Using various forms of the catalyst effectiveness factor system, it is shown that orthogonal collocation is the computationally superior method; this result also holds for a specific class of initial value problems. A multiple element approach in which the elements are located in an optimal manner is also developed. This optimal procedure holds promise for efficiently solving complicated boundary value problems.

Newton’s Method for Problems with Equality Constraints

Consider an operator $Q:X_1 \to X_2 $ where $X_1 $ and $X_2 $ are normed linear spaces. Let $\delta _1 , \cdots ,\delta _m $ be functionals on $X_1 $ and let $H_2 $ be a finite-dimensional affine subspace of $X_2 $. In this paper we consider the problem \[ Q(x) \in H_2 \quad {\text{subject to}}\quad \delta _i (x) = \beta _i, \quad i = 1, \cdots ,m.\] A generalized form of Newton’s method which approximates a solution of this problem by a sequence of the form \[ x_{n + 1} = x_n - \Gamma _n p_n (x_n ),\qquad n = 0,1, \cdots , \] is presented. It is shown that many important constrained problems including the ordinary differential equation boundary value problem and the Euler–Poisson equation from the calculus of variations fall very naturally within this theory.

Mildly nonlinear elliptic partial differential equations and their numerical solutoin. II

Abstract : High speed digital computer methods are studied for obtaining solutions of difference equation analogues of mildly nonlinear elliptic boundary value problems. The problem is formulated in terms of finding a vector X which satisfies AX = - f(X) + Y, where A is an irreducible M matrix, Y is a given vector, and f is a given function. Two general iteration techniques and related convergence theorems are explored. The methods and ideas also extend to a large class of ordinary differential equation boundary value problems. (Author)