Collocating convolutions
Author:
Frank Stenger
Journal:
Math. Comp. 64 (1995), 211235
MSC:
Primary 65D30; Secondary 41A35, 65N35, 65R10
MathSciNet review:
1270624
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: An explicit method is derived for collocating either of the convolution integrals or , where , a subinterval of . The collocation formulas take the form or , where g is an mvector of values of the function g evaluated at the "Sinc points", and are explicitly described square matrices of order m, and , for arbitrary . The components of the resulting vectors p (resp., q) approximate the values of p (resp., q) at the Sinc points, and may then be used in a Sinc interpolation formula to approximate p and q at arbitrary points on (a, b). The procedure offers a new method of approximating the solutions to (definite or indefinite) convolutiontype integrals or integral equations as well as solutions of partial differential equations that are expressed in terms of convolutiontype integrals or integral equations via the use of Green's functions. If u is the solution of a partial differential equation expressed as a vdimensional convolution integral over a rectangular region B, and if u is analytic and of class on the interior of each line segment in B, then the complexity of computing an approximation of u by the method of this paper is .
 [1]
Hermann
Brunner, Discretization of Volterra integral
equations of the first kind, Math. Comp.
31 (1977), no. 139, 708–716. MR 0451794
(56 #10076), http://dx.doi.org/10.1090/S00255718197704517946
 [2]
, A survey of recent advances in the numerical solution of Volterra integral and integrodifferential equations, J. Comput. Appl. Math. 8 (1982), 147163.
 [3]
Frank
de Hoog and Richard
Weiss, High order methods for Volterra integral equations of the
first kind, SIAM J. Numer. Anal. 10 (1973),
647–664. MR 0373354
(51 #9554)
 [4]
Seymour
Haber, Two formulas for numerical indefinite
integration, Math. Comp.
60 (1993), no. 201, 279–296. MR 1149292
(93d:65026), http://dx.doi.org/10.1090/S00255718199311492929
 [5]
Ralph
Baker Kearfott, A sinc approximation for the
indefinite integral, Math. Comp.
41 (1983), no. 164, 559–572. MR 717703
(85g:65029), http://dx.doi.org/10.1090/S0025571819830717703X
 [6]
Peter
Linz, A survey of methods for the solution of Volterra integral
equations of the first kind, Application and numerical solution of
integral equations (Proc. Sem., Australian Nat. Univ., Canberra, 1978)
Monographs Textbooks Mech. Solids Fluids: Mech. Anal., vol. 6,
Nijhoff, The Hague, 1980, pp. 183–194. MR 582990
(81m:65199)
 [7]
Peter
Linz, Analytical and numerical methods for Volterra equations,
SIAM Studies in Applied Mathematics, vol. 7, Society for Industrial
and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 796318
(86m:65163)
 [8]
C.
Lubich, Convolution quadrature and discretized operational
calculus. I, Numer. Math. 52 (1988), no. 2,
129–145. MR
923707 (89g:65018), http://dx.doi.org/10.1007/BF01398686
 [9]
C.
Lubich, Convolution quadrature and discretized operational
calculus. II, Numer. Math. 52 (1988), no. 4,
413–425. MR
932708 (89g:65019), http://dx.doi.org/10.1007/BF01462237
 [10]
Ch.
Lubich and A.
Ostermann, RungeKutta methods for parabolic
equations and convolution quadrature, Math.
Comp. 60 (1993), no. 201, 105–131. MR 1153166
(93d:65082), http://dx.doi.org/10.1090/S00255718199311531667
 [11]
John
Lund and Kenneth
L. Bowers, Sinc methods for quadrature and differential
equations, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 1992. MR 1171217
(93i:65004)
 [12]
Bruce
V. Riley, A sinccollocation method for weakly singular Volterra
integral equations, Computation and control (Bozeman, MT, 1988)
Progr. Systems Control Theory, vol. 1, Birkhäuser Boston, Boston,
MA, 1989, pp. 263–275. MR 1046856
(91j:45019)
 [13]
Frank
Stenger, Numerical methods based on sinc and analytic
functions, Springer Series in Computational Mathematics, vol. 20,
SpringerVerlag, New York, 1993. MR 1226236
(94k:65003)
 [14]
Frank
Stenger, Numerical methods based on Whittaker cardinal, or sinc
functions, SIAM Rev. 23 (1981), no. 2,
165–224. MR
618638 (83g:65027), http://dx.doi.org/10.1137/1023037
 [15]
M. Stromberg, Solution of shock problems by methods using sinc functions, Ph.D. thesis, University of Utah, 1988.
 [16]
, Approximate solution of quasilinear equations of conservation law type, Computation and Control (K. L. Bowers and J. Lund, eds.), Birkhäuser, Basel, 1989, pp. 316331.
 [17]
D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, NJ, 1936.
 [18]
Andrew
Young, The application of approximate product integration to the
numerical solution of integral equations, Proc. Roy. Soc. London Ser.
A. 224 (1954), 561–573. MR 0063779
(16,179b)
 [1]
 H. Brunner, Discretization of Volterra integral equations of the first kind, Math. Comp. 31 (1977), 708716. MR 0451794 (56:10076)
 [2]
 , A survey of recent advances in the numerical solution of Volterra integral and integrodifferential equations, J. Comput. Appl. Math. 8 (1982), 147163.
 [3]
 F. de Hoog and R. Weiss, High order methods for Volterra integral equations of the first kind, SIAM J. Numer. Anal. 10 (1973), 647664. MR 0373354 (51:9554)
 [4]
 S. Haber, Two formulas for numerical indefinite integration, Math. Comp. 60 (1993), 279296. MR 1149292 (93d:65026)
 [5]
 R. B. Kearfott, A sinc approximation for the indefinite integral, Math. Comp. 41 (1983), 559572. MR 717703 (85g:65029)
 [6]
 P. Linz, A survey of methods for the solution of Volterra integral equations of the first kind, The Application and Numerical Solution of Integral Equations (R. S. Anderssen, Frank R. de Hoog, and Mark Lukas, eds.), Sijthoff & Noordhoff, Germantown, MD, 1980, pp. 183194. MR 582990 (81m:65199)
 [7]
 , Analytical and numerical methods for Volterra equations, SIAM, Philadelphia, PA, 1985. MR 796318 (86m:65163)
 [8]
 C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), 129145. MR 923707 (89g:65018)
 [9]
 , Convolution quadrature and discretized operational calculus. II, Numer. Math. 52 (1988), 413425. MR 932708 (89g:65019)
 [10]
 Ch. Lubich and A. Ostermann, RungeKutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993), 105131. MR 1153166 (93d:65082)
 [11]
 J. Lund and K. L. Bowers, Sinc methods for quadrature and differential equations, SIAM, Philadelphia, PA, 1992. MR 1171217 (93i:65004)
 [12]
 B. Riley, A sinc collocation method for weakly singular integral equations, Computation and Control (K. L. Bowers and J. Lund, eds.), Birkhäuser, Basel, 1989, pp. 263275. MR 1046856 (91j:45019)
 [13]
 F. Stenger, Numerical methods based on sinc and analytic functions, SpringerVerlag, New York, 1993. MR 1226236 (94k:65003)
 [14]
 , Numerical methods based on Whittaker cardinal, or sinc functions, SIAM Rev. 23 (1981), 165224. MR 618638 (83g:65027)
 [15]
 M. Stromberg, Solution of shock problems by methods using sinc functions, Ph.D. thesis, University of Utah, 1988.
 [16]
 , Approximate solution of quasilinear equations of conservation law type, Computation and Control (K. L. Bowers and J. Lund, eds.), Birkhäuser, Basel, 1989, pp. 316331.
 [17]
 D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, NJ, 1936.
 [18]
 A. Young, The application of approximate product integration to the numerical solution of integral equations, Proc. Roy. Soc. London Ser. A 224 (1954), 561573. MR 0063779 (16:179b)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65D30,
41A35,
65N35,
65R10
Retrieve articles in all journals
with MSC:
65D30,
41A35,
65N35,
65R10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199512706247
PII:
S 00255718(1995)12706247
Keywords:
Indefinite integral convolution
Article copyright:
© Copyright 1995
American Mathematical Society
