An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

Authors:
Ivan P. Gavrilyuk and Volodymyr L. Makarov

Journal:
Math. Comp. **76** (2007), 1895-1923

MSC (2000):
Primary 65J15, 65M15; Secondary 34G20, 35K90

DOI:
https://doi.org/10.1090/S0025-5718-07-01987-4

Published electronically:
April 19, 2007

MathSciNet review:
2336273

Full-text PDF

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Abstract: An exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in Banach space is proposed. The algorithm is based on an equivalent Volterra integral equation including the operator exponential generated by the operator coefficient. The operator exponential is represented by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient, and then the integrals involved are approximated using the Chebyshev interpolation and an appropriate Sinc quadrature. Numerical examples are given which confirm theoretical results.

**1.**D.Z. Arov, I.P. Gavrilyuk and V.L. Makarov,*Representation and approximation of solutions of initial value problems for differential equations in Hilbert space based on the Cayley transform*, Progress in partial differential equations (C. Bandle et al. eds.), vol. 1, Pitman Res. Notes Math. Sci., 1995, pp. 40-50. MR**1416572 (97h:34074)****2.**A. Ashyralyev and P. Sobolevskii,*Well-Posedness of Parabolic Difference Equations*, Birkhäuser Verlag, Basel, 1994. MR**1299329 (95j:65094)****3.**N.Yu. Bakaev,*Stability estimates for a general discretization method*, Soviet Math. Dokl.**40**(1990), 11-15. MR**1035844 (91e:65104)****4.**T.Ju. Bohonova, I.P. Gavrilyuk, V.L. Makarov and V. Vasylyk,*Exponentially convergent Duhamel's like algorithms for differential equations with an operator coefficient possessing a variable domain in Banach space*, Reports on Numerical Mathematics, Friedrich-Schiller-Universität Jena (`http://www.minet.uni-jena.de/Math-Net/reports05/reports.html`#2005), 05-06 (2005), 1-25.**5.**C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang,*Spectral Methods in Fluid Dynamics*, Springer-Verlag, Berlin, Heidelberg, New York et al., 1988. MR**917480 (89m:76004)****6.**M.L. Fernandez, Ch. Lubich, C. Palencia and A. Schädle,*Fast Runge-Kutta approximation of inhomogeneous parabolic equations*, Numerische Mathematik**5**, (2005), 1-17.**7.**H. Fujita, N. Saito,and T. Suzuki,*Operator Theory and Numerical Methods*, Elsevier, Heidelberg, 2001.**8.**I.P. Gavrilyuk,*Strongly P-positive operators and explicit representation of the solutions of initial value problems for second order differential equations in Banach space*, Journ.of Math. Analysis and Appl.**1(88)**(2003), 327-349. MR**1704587 (2001j:34072)****9.**I.P. Gavrilyuk,*Algorithms without accuracy saturation and exponential convergent algorithms for operator equations*, Journal of Numerical and Applied Mathematics (ISSN 0868-6912)**236**(1999), 28-43.**10.**I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij,*-matrix approximation for elliptic solution operators in cylinder domains*, East-West Journal of Numerical Analysis**9**(2001), no. 1, 25-58. MR**1839197 (2002e:65064)****11.**I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij,*-matrix Approximation for the Operator Exponential with Applications*, Numer. Math.**92**(2002), 83-111. MR**1917366 (2003g:65061)****12.**I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij,*Data-sparse approximation to the operator-valued functions of elliptic operator*, Math. Comp.**73**(2004), 1297-1324. MR**2047088 (2005b:47086)****13.**I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij,*Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems*, Computing**74**(2005), 131-157. MR**2133692 (2006f:65049)****14.**I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij,*Data-sparse approximation of a class of operator-valued functions*, Math. Comp.**74**(2005), 681-708 MR**2114643 (2005i:65068)****15.**I.P. Gavrilyuk and V.L. Makarov,*Representation and approximation of the solution of an initial value problem for a first order differential eqation in Banach space*, Z. Anal. Anwend. (ZAA)**15**(1996), 495-527. MR**1394440 (97h:65076)****16.**I.P. Gavrilyuk and V.L. Makarov,*Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces*, Math. Comp.**74**(2005), 555-583. MR**2114638 (2005j:65053)****17.**I.P. Gavrilyuk and V.L. Makarov,*Exponentially convergent parallel discretization methods for the first order evolution equations*, Computational Methods in Applied Mathematics (CMAM)**1**, 4, (2001), 333-355. MR**1892950 (2003f:65174)****18.**I.P. Gavrilyuk and V.L. Makarov,*The Cayley transform and the solution of an initial value problem for a first order differential equation with an unbounded operator coefficient in Hilbert space*, Numer. Func. Anal. Optimiz.**15**, (1994), 583-598. MR**1281563 (95b:34096)****19.**I.P. Gavrilyuk and V.L. Makarov,*Exponentially convergent parallel discretization methods for the first order differential equations*, Doklady of the Ukrainian Academy of Scienses**3**, (2002), 1-6.**20.**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(5):2144-2171, 2005. MR**2192335 (2006m:65100)****21.**I.P. Gavrilyuk and V.L. Makarov,*An explicit boundary integral representation of the solution of the two-dimensional heat equation and its discretization*, J. Integral Equations Appl.**12**, (Spring 2000), 1, 63-83. MR**1760898 (2001c:65129)****22.**I.P. Gavrilyuk and V.L. Makarov,*An exponentially convergent algorithm for nonlinear differential equations in Banach spaces*, Reports on Numerical Mathematics, Friedrich-Shiller-Universität Jena (`http://www.minet.uni-jena.de/Math-Net/reports/`) 02/05 (2005), 1-22.**23.**I.P. Gavrilyuk and V.L. Makarov and V. Vasylyk,*A new estimate of the Sinc method for linear parabolic problems including the initial point*, Computational Methods in Applied Mathematics (CMAM)**4**, (2004), 2, 1-27. MR**2119621 (2005m:65214)****24.**J.A. Goldstein,*Semigroups of Linear Operators and Applications,*Oxford University Press, New York, Clarendon Press, Oxford, 1985.MR**790497 (87c:47056)****25.**D. González and C. Palencia,*Stability of time-stepping methods for abstract time-dependent parabolic problems*, SIAM J. Numer. Anal.**35**, (2004), 3, 973-989. MR**1619918 (99b:65072)****26.**D. Henry,*Geometrical Theory of Semilinear Parabolic Equations*, Springer-Verlag, Berlin-Heidelberg-New York, 1981. MR**610244 (83j:35084)****27.**M.A. Krasnosel'skij and P.E. Sobolevskij,*Fractional powers of operators acting in Banach spaces (in Russian)*, Doklady AN SSSR**129**, (1959), 3, 499-502. MR**0108733 (21:7447)****28.**K. Kwon and D. Sheen,*A parallel method for the numerical solution of integro-differential equation with positive memory*, Comput. Methods Appl. Mech. Engrg.**192**, (2003), 41-42, 4641-4658. MR**2012483 (2004k:65263)****29.**M. López-Fernández, C. Palencia and A. Schädle,*A spectral order method for inverting sectorial Laplace transforms*, ZIB-Report 05-26, April 2005.**30.**M. López-Fernández and C. Palencia,*On the numerical inversion of the Laplace transform of certain holomorphic mappings*, Applied Numerical Mathematics,**51**, (2004), 289-303. MR**2091405 (2005e:65210)****31.**M. López-Fernández, C. Palencia and A. Schädle,*Fast Runge-Kutta approximation of inhomogeneous parabolic differential equations*, Preprint 2005.**32.**J. Lund and K.L. Bowers,*Sinc methods for quadrature and differential equations*,SIAM, Philadelphia, 1992. MR**1171217 (93i:65004)****33.**A. Pazy,*Semigroups of linear operator and applications to partial differential equations*, Springer-Verlag, Berlin-Heidelberg-New York, 1983. MR**710486 (85g:47061)****34.**D. Sheen, I.H. Sloan and V. Thomée,*A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature*, Math.Comp.**69**, (2000), 177-195. MR**1648403 (2000i:65161)****35.**D. Sheen,I.H. Sloan and V.Thomée,*A parallel method for time-discretization of parabolic equations based on Laplace transformation and quadrature*, IMA Journal of Numerical Analysis,**23**, (2003), 269-299. MR**1975267 (2004b:65161)****36.**M.Z. Solomjak,*Application of the semi-group theory to investigation of differential equations in Banach spaces (in Russian)*, Doklady AN SSSR**122**, (1958), 2, 766-769. MR**0105029 (21:3775)****37.**F. Stenger,*Numerical methods based on Sinc and analytic functions.*Springer Verlag, 1993. MR**1226236 (94k:65003)****38.**G. Szegö,*Orthogonal Polynomials.*American Mathematical Society,New York, 1959. MR**0106295 (21:5029)****39.**G. Szegö,*Orthogonal Polynomials (with an Introduction and a Complement by J.L. Geronimus)*, State Publishing House of Physical and Mathematical Literature, Moscow, 1962.**40.**V. Thomée,*A high order parallel method for time discretisation of parabolic type equations based on Laplace transformation and quadrature*, Int. J. Numer Anal. Model.**2**(2005), 85-96. MR**2112660 (2005i:65159)****41.**V. Vasylyk,*Uniform exponentially convergent method for the first order evolution equation with unbounded operator coefficient*, Journal of Numerical and Applied Mathematics (ISSN 0868-6912),**1**, (2003), 99-104 (in Russian).

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Additional Information

**Ivan P. Gavrilyuk**

Affiliation:
Staatliche Studienakademie Thueringen-Berufsakademie Eisenach, University of Cooperative Edukation, Am Wartenberg 2, D-99817 Eisenach, Germany

Email:
ipg@ba-eisenach.de

**Volodymyr L. Makarov**

Affiliation:
National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschenkivska 3, 01601 Kiev, Ukraine

Email:
makarov@imath.kiev.ua

DOI:
https://doi.org/10.1090/S0025-5718-07-01987-4

Keywords:
Nonlinear evolution equation,
exponentially convergent algorithms,
Sinc-methods

Received by editor(s):
March 15, 2005

Received by editor(s) in revised form:
June 30, 2006

Published electronically:
April 19, 2007

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.