UDC 519.6
1 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
 mayko@knu.ua
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SUPER-EXPONENTIAL RATE OF CONVERGENCE OF THE CAYLEY
TRANSFORM METHOD FOR AN ABSTRACT DIFFERENTIAL EQUATION
Abstract. A boundary-value problem (BVP) for an abstract differential equation with an operator coefficient in the Hilbert space is investigated.
The exact solution is presented as an infinite series by means of the Cayley transform of the operator
coefficient A and the Meixner type polynomials
in the independent variable x.
The approximate solution is given by the truncated sum of that series with
N summands.
The error estimates (with the weight function) depending not only on the discretization
parameter N but also on the distance
of the point x to the boundary of the interval are proven. They demonstrate that our algorithm has the super-exponential rate of convergence.
Keywords: boundary-value problem (BVP), Hilbert space, operator coefficient, Cayley transform, weighted estimates, super-exponentially convergent algorithm.
FULL TEXT
REFERENCES
- Galba E.F. On the accuracy order of a difference scheme for the Poisson equation with a mixed boundary condition. In: "Optimization of computer software algorithms." Kiev: V.M. Glushkov Institute of Cybernetics of AS USSR, 1985. P. 30–34.
- Makarov V. On a priori estimate of difference schemes giving an account of the boundary effect. Comptes rendus de l’Acad'emie Bulgare des Sciences (Proc. the Bulgarian Academy of Sciences). 1989. V. 42. N 5. P. 41–44.
- Molchanov I.N., Galba E.F. On the convergence of a difference scheme approximating the Dirichlet problem for an elliptic equation with piecewise constant coefficients. In: "Numerical methods and technology for developing software packages." Kiev: V.M. Glushkov Institute of Cybernetics, Academy of Sciences of the Ukrainian SSR, 1990. P. 161–165.
- Makarov V.L., Demkiv L.I. Estimation of the accuracy of difference schemes for parabolic equations that take into account the initial boundary effect. Dop. NAS of Ukraine. 2003. N 2. P. 26–32.
- Makarov V., Demkiv L. Accuracy estimates of differences schemes for quasi-linear parabolic equations taking into account the initial-boundary effect. Computational Methods in Applied Mathematics. 2003. Vol. 3, N 4. P. 579–595.
- Mayko N.V. The boundary effect in the error estimate of the finite-difference scheme for the two-dimensional heat equation. Journal of Computational and Applied Mathematics. 2013. N 3(113). P. 91–106.
- Maiko N.V., Ryabichev V.L. Estimation of the accuracy of the difference scheme for the two-dimensional Poisson equation taking into account the effect of boundary conditions. Kibernetika i sistemnyj analiz. 2016. Vol. 52, N 5. P. 113–124.
- Maiko N.V. Improved accuracy estimates for a difference scheme for a two-dimensional parabolic equation, taking into account the effect of boundary and initial conditions. Kibernetika i sistemnyj analiz. 2017. Vol. 53, N 1. P. 99–107.
- Maiko N.V. An estimate with the weight of accuracy of a difference scheme of an increased order of approximation for the two-dimensional Poisson equation taking into account the effect of the Dirichlet boundary condition. Kibernetika i sistemnyj analiz. 2018. Vol. 54, N 1. P. 145–153.
- Maiko N.V. Scheme of increased accuracy for the two-dimensional Poisson equation in a rectangle taking into account the influence of the Dirichlet boundary condition. Kibernetika i sistemnyj analiz. 2018. Vol. 54, N 4. P. 122–134.
- Makarov V.L., Mayko N.V. The boundary effect in the accuracy estimate for the grid solution of the fractional differential equation. Computational Methods in Applied Mathematics.Vol. 19, Iss. 2. P. 379–394.
- Makarov VL, Maiko NV Boundary effect in estimating the accuracy of a grid method for solving a differential equation with a fractional derivative. Kibernetika i sistemnyj analiz. 2019. Vol. 55, N 1. P. 80–95.
- Gavrilyuk I.P., Makarov V.L, Mayko N.V. Weighted estimates of the Cayley transform method for abstract differential equations. Computational Methods in Applied Mathematics. (To appear.)
- Makarov V.L. Meixner polynomials and their properties. Dop. NAS of Ukraine. 2019. N 7. P. 3–8.
