A programmed algorithm for existence proofs for two-point boundary value problems
A new numerical algorithm is given for the approximate solution to nonlinear two-point boundary value problems for ordinary differential equations which has the property that if the algorithm may be carried through it establishes existence of a solution which is within known bounds. The system considered has the form (i) \(-U''(x)+F(x,U(x),U'(x))=0,\quad 0<x<1,\) (ii) \(-a_ 0U'(0)+F_ 0(U(0))=0,\quad a_ 1U'(1)+F_ 1(u(1))=0,\) \(U\in R,\) where \(F,F_ 0,F_ 1\) are twice continuously differentiable and for \(a_ i=0,\quad F_ i(y)=y-r_ i,\quad i=0,1.\) The algorithm (called the EB-algorithm) contains two parts: In the first part an approximate solution w(x) to equation (i) is obtained using polynomial splines. In the second part the algorithm computes constants \(\epsilon >0,\quad k_ 0>0,\) and a function \(\psi_ 0(x)\geq 0\) subject to a given set of constraints. If these quantities may be successfully computed then (i), (ii) have a solution U(x) such that \(| U(x)-w(x)| \leq (1+\epsilon)\psi_ 0(x),\quad | U'(x)-w'(x)| \leq (1+\epsilon)k_ 0.\) The key step in deriving the algorithm involves writing the equation for the first variation \(u(x)=U(x)-w(x)\) in the form \((iii)\quad Au(x)+N(x,u(x),u'(x))=0\) where A is linear, invertible, and inverse- positive (i.e. such that \(Au(x)\geq 0\) implies \(u(x)\geq 0).\) The theory of smooth comparison functions is applied to obtain an existence statement together with two-sided error bounds for u(x). The authors describe a computer program which they have written which effectively implements the EB-algorithm. The procedure for obtaining error bounds requires use of interval arithmetic and was written in the FORTRAN-SC programming language. Numerical results are presented which were obtained when the program was applied to five specific differential equations. Accuracy better than \(10^{-5}\) was obtained in all applications.
nonlinear two-point boundary value problems
scientific article; zbMATH DE number 4153801
scientific article