Abstract
In this paper a class of Birkhoff type interpolation problem on arbitrary knots is studied. This class of Birkhoff interpolation is a generalization of a Birkhoff problem, which is included in one of the articles in the referenced list. Because the behavior of first and second order derivatives are more important than the higher order, therefore, for a basic function, a better approximation will be obtained than higher order derivatives. The low error value calculated for some functions using this class represents the relative importance of this class of Birkhoff interpolation problem. One of the uses of this class from the Birkhoff polynomials is to review and propose a quadrature formula for this class. One of the ways to reduce the error for interpolation polynomials of this class of Birkhoff interpolation problem is to select the Chebyshev polynomial zeros as nodal points, which can be used to solve numerical differential equations.
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References
Costabile, F.A., Longo, E.A.: A new collocation method for a BVP. Appl. Ind. Math. 82, 289–297 (2005)
Costabile, F.A., Longo, E.A.: A Birkhoff interpolation problem and application. Calcolo 47, 49–63 (2010)
Costabile, F.A., Napoli, A.: A class of Birkhoff–Lagrange-collocation methods for high order boundary value problems. Appl. Numer. Math. 116(6), 129–140 (2017)
Davis, P.J.: Interpolation and Approximation. Dover, New York (1975)
Greenberg, M.D.: Application of Green’s functions in Science and Engineering. Prentice-Hall, Upper Saddle River (1971)
Lorentz, G.G., Jetter, K., Riemensheider, S.D.: Birkhoff Interpolation. Addison-Wesley, Reading (1983)
Melnikov, Y.A.: Green’s functions and Infinite Products. Springer, Berlin (2011)
Nazarzadeh, A., Rahsepar Fard, K.H., Mahmoodi, A.: Another case of incidence matrix for bivariate Birkhoff interpolation. J. Numer. Appl. Math. 2(122), 55–70 (2016)
Shi, Y.G.: An extremal approach to Birkhoff quadrature formulas. J. Comput. Math. 195, 459–466 (2001)
Shi, Y.G.: Theory of Birkhoff Interpolation. Nova Science Publishers Inc, New York (2003)
Stakgold, I., Hoist, M.: Green’s function and Boundary Value Problems. Wiley, Hoboken (2011)
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Mahmoodi, A., Nazarzadeh, A. A Class of Birkhoff Type Interpolation and Applications. Results Math 73, 43 (2018). https://doi.org/10.1007/s00025-018-0803-z
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DOI: https://doi.org/10.1007/s00025-018-0803-z