Abstract We represent the solution u ⁢ ( t ) {u(t)} of an initial value problem (IVP) for the first-order differential equation with an operator coefficient as a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials. In the case of a boundary value problem (BVP) for the second-order differential equation with an operator coefficient, we represent its solution using the Cayley transform and the Meixner-type polynomials. The approximate solution is the truncated sum of N (the discretization parameter) summands. We give the error estimate of these approximations depending on N and the distance of t to the initial point of the time interval or of the spatial argument x to the boundary of the spatial domain.

中文翻译：

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抽象微分方程凯莱变换方法的加权估计

摘要 我们使用对应算子系数的 Cayley 变换和 Laguerre 方程来表示具有算子系数的一阶微分方程的初值问题 (IVP) 的解 u ⁢ ( t ) {u(t)}多项式。对于具有算子系数的二阶微分方程的边值问题 (BVP)，我们使用 Cayley 变换和 Meixner 型多项式表示其解。近似解是 N 个（离散化参数）被加数的截断和。我们根据 N 和 t 到时间间隔初始点的距离或空间参数 x 到空间域边界的距离来给出这些近似值的误差估计。