Abstract

We consider the boundary value problems (BVPs) for linear second-order ODEs with a strongly positive operator coefficient in a Banach space. The solutions are given in the form of the infinite series by means of the Cayley transform of the operator, the Meixner type polynomials of the independent variable, the operator Green function, and the Fourier series representation for the right-hand side of the equation. The approximate solution of each problem is a partial sum of N (or expressed through N) summands. We prove the weighted error estimates depending on the discretization parameter N, the distance of the independent variable to the boundary points of the interval, and some smoothness properties of the input data.

摘要

我们考虑在Banach空间中具有强正算子系数的线性二阶ODE的边值问题（BVP）。通过算子的Cayley变换，自变量的Meixner型多项式，算子Green函数以及方程右侧的傅里叶级数表示形式，以无穷级数形式给出解。每个问题的近似解决方案是N个求和项（或用N表示）的部分和。我们根据离散化参数N，自变量到区间边界点的距离以及输入数据的某些平滑性证明了加权误差估计。