# Exponentially Convergent Method For An Abstract Nonlocal Problem With Integral Nonlinearity

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We consider a nonlocal problem for the first-order differential equation with unbounded operator coefficient in a Banach space and a nonlinear integral nonlocal condition. We propose an exponentially convergent method for the numerical solution of this problem and justified this method under the assumptions that the indicated operator coefficient *A* is sectorial and that certain conditions for the existence and uniqueness of the solution are satisfied. This method is based on the reduction of the posed problem to an abstract Hammerstein-type equation, discretization of this equation by the method of collocation, and its subsequent solution by the method of simple iterations. Each iteration of the method is based on the Sinc-quadrature approximation of the exponential operator function represented by the Dunford–Cauchy integral over the hyperbola enveloping the spectrum of *A.* The integral part of the nonlocal condition is approximated by using the Clenshaw–Curtis quadrature formula.

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### References

- 1.V. B. Vasylyk and V. L. Makarov, “Exponentially convergent method for the first-order differential equation in a Banach space with integral nonlocal condition,”
*Ukr. Mat. Zh.*,**66**, No. 8, 1029–1040 (2014);**English translation***: Ukr. Math. J.*,**66**, No. 8, 1152–1164 (2014).Google Scholar - 2.S. G. Krein,
*Linear Differential Equations in Banach Spaces*[in Russian], Nauka, Moscow (1967).Google Scholar - 3.A. Bica, M. Curila, and S. Curila, “About a numerical method of successive interpolations for functional Hammerstein integral equations,”
*J. Comput. Appl. Math.*,**236**, No. 7, 2005–2024 (2012).MathSciNetCrossRefMATHGoogle Scholar - 4.I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, “Data-sparse approximation to the operator-valued functions of elliptic operator,”
*Math. Comput.*,**73**, 1297–1324 (2004).MathSciNetCrossRefMATHGoogle Scholar - 5.I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, “Data-sparse approximation of a class of operator-valued functions,”
*Math. Comput.*,**74**, 681–708 (2005).MathSciNetCrossRefMATHGoogle Scholar - 6.I. P. Gavrilyuk, V. L. Makarov, D. O. Sytnyk, and V. B. Vasylyk, “Exponentially convergent method for the
*m*-point nonlocal problem for a first order differential equation in Banach space,”*Numer. Funct. Anal. Optim.*,**31**, No. 1-3, 1–21 (2010).MathSciNetCrossRefMATHGoogle Scholar - 7.I. P. Gavrilyuk and V. L. Makarov, “Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces,”
*SIAM J. Numer. Anal.*,**43**, No. 5, 2144–2171 (2005).MathSciNetCrossRefMATHGoogle Scholar - 8.I. P. Gavrilyuk and V. L. Makarov, “An exponential convergent algorithm for nonlinear differential equations in Banach spaces,”
*Math. Comput.*,**76**, 1895–1923 (2007).CrossRefMATHGoogle Scholar - 9.I. P. Gavrilyuk, V. L. Makarov, and V. B. Vasylyk, “Exponentially convergent approximation to the elliptic solution operator,”
*Comput. Meth. Appl. Math.*,**6**, No. 4, 386–404 (2006).MathSciNetMATHGoogle Scholar - 10.T. Ju. Bohonova, I. P. Gavrilyuk, V. L. Makarov, and V. B. Vasylyk, “Exponentially convergent Duhamel-like algorithms for differential equations with an operator coefficient possessing a variable domain in a Banach space,”
*SIAM J. Numer. Anal.*,**46**, No. 1, 365–396 (2008).MathSciNetCrossRefMATHGoogle Scholar - 11.I. Gavrilyuk, V. Makarov, and V. Vasylyk,
*Exponentially Convergent Algorithms for Abstract Differential Equations*, Springer, Basel (2011).CrossRefMATHGoogle Scholar - 12.C. P. Gupta, “Functional analysis and applications,” in:
*L. Nachbin (editor), Proceedings of the Symposium of Analysis, Universidade Federal de Pernambuco Recife (Pernambuco, Brasil, July 9–29, 1972)*, Springer, Berlin (1974), pp. 184–238.Google Scholar - 13.D. Henry,
*Geometrical Theory of Semilinear Parabolic Equations*, Springer, Berlin (1981).CrossRefMATHGoogle Scholar - 14.P. Hess, “On nonlinear equations of Hammerstein type in Banach spaces,”
*Proc. Amer. Math. Soc.*,**30**, No. 2, 308–312 (1971).MathSciNetCrossRefMATHGoogle Scholar - 15.T. Kato, “On linear differential equations in Banach spaces,”
*Comm. Pure Appl. Math.*,**9**, 479–486 (1956).MathSciNetCrossRefMATHGoogle Scholar - 16.M. López-Fernández, C. Palencia, and A. Schädle, “A spectral order method for inverting sectorial Laplace transforms,”
*SIAM J. Numer. Anal.*,**44**, 1332–1350 (2006).MathSciNetCrossRefMATHGoogle Scholar - 17.W. McLean and V. Thomee, “Time discretization of an evolution equation via Laplace transforms,”
*IMA J. Numer. Anal.*,**24**, 439–463 (2004).MathSciNetCrossRefMATHGoogle Scholar - 18.Ph. Clément, H. J. A. M. Heijmans, S. Angenent, et al.,
*One-Parameter Semigroups*, North-Holland Publ. Comp., Amsterdam (1987).MATHGoogle Scholar - 19.P. M. Fitzpatrick and W. V. Petryshyn, “Galerkin methods in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations,”
*Trans. Amer. Math. Soc.*,**238**, 321–340 (1978).MathSciNetCrossRefMATHGoogle Scholar - 20.B. Some, “Some recent numerical methods for solving nonlinear Hammerstein integral equations,”
*Math. Comput. Modelling*,**18**, No. 9, 55–62 (1993).MathSciNetCrossRefMATHGoogle Scholar - 21.S. Kumar and I. H. Sloan, “A new collocation-type method for Hammerstein integral equations,”
*Math. Comput.*,**48**, No. 178, 585–593 (1987).MathSciNetCrossRefMATHGoogle Scholar - 22.L. N. Trefethen,
*Approximation Theory and Approximation Practice*, Society for Industrial and Applied Mathematics, Philadelphia (2013).MATHGoogle Scholar - 23.V. Vasylyk, “Exponentially convergent method for the
*m*-point nonlocal problem for an elliptic differential equation in Banach space,”*J. Numer. Appl. Math.*,**105**, No. 2, 124–135 (2011).Google Scholar - 24.V. Vasylyk, “Nonlocal problem for an evolution first-order equation in Banach space,”
*J. Numer. Appl. Math.*,**109**, No. 3, 139–149 (2012).Google Scholar - 25.V. Vasylyk, “Exponentially convergent method for integral nonlocal problem for the elliptic equation in Banach space,”
*J. Numer. Appl. Math.*,**110**, No. 3, 119–130 (2013).Google Scholar - 26.J. A. C. Weideman, “Improved contour integral methods for parabolic PDEs,”
*IMA J. Numer. Anal.*,**30**, No. 1, 334–350 (2010).MathSciNetCrossRefMATHGoogle Scholar