Weighted accuracy estimates of the Cayley transform method for abstract boundary-value problems in a Banach space

1Makarov, VL
Mayko, NV
1Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2020, 5:3-9
Section: Mathematics
Language: Ukrainian

We study the first BVP for linear second-order differential equations with a strongly positive operator coefficient in a Banach space. The exact solutions of these BVPs are represented in the form of infinite series by means of the Cayley transform of the operator coefficient, the Meixner-type polynomials in the independent variable, and the Fourier series representation of the right-hand side of the equation. The approximate solution of each problem is a partial sum of the corresponding series (with the discretization parameter N). We prove the weighted accuracy estimates taking the boundary effect into account. These estimates demonstrate that the proposed methods have the power rate of convergence or the exponential rate of convergence in accordance with the smoothness properties of the input data.

Keywords: Banach space, boundary effect, boundary-value problem (BVP), Cayley transform, exponential rate of convergence, method without saturation of accuracy

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