A Progressive Quasi-Bounding Method and Its Applications to Boundary Value Problems

Boundary value problems (BVPs) have wide applications in computer graphics and mechanical engineering. This paper presents a quasi-bounding method for progressively solving several boundary value problems, where initial solutions are easy to be obtained for rapidly solving the accurate solution. Given an equation $F(u,t)=0,t\in [a,b]$, and several boundary value constraints as well, two systems consisting of $n+1$ equations are derived for rapidly searching two polynomials $f_{n,1}(t)$ and $f_{n,2}(t)$ of degree $n$, which satisfy $F(f_{n,1},t)\le$ $0\le F(f_{n,2},t),t\in [a,b]$ in the cases when certain conditions are satisfied. From the middle value theorem, the solution $u^{\star }(t)$ is bounded by $f_{n,1}(t)$ and $f_{n,2}(t)$, $t\in [a,b]$. These two bounding polynomials $f_{n,i}(t),$ $i=1,2$, are taken as initial values for progressive refinements of approximation error in two ways, i.e., B-Spline form of the same degree with more knots, and Bézier form of a higher degree. Numerical experiments show that the new method can be applied to more generalized BVPs, and achieves better computational stability, much better approximation with less error and better computational efficiency than prevailing methods, even by using a small degree $n$.