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Spline approximation to the solution of the Volterra integral equation of the second kind

Author: Arun N. Netravali
Journal: Math. Comp. 27 (1973), 99-106
MSC: Primary 65R05
MathSciNet review: 0366068
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Abstract: A cubic spline approximation in $ {C^2}$ to the solution of a general Volterra integral equation of the second kind is constructed. Under certain conditions, convergence of the approximation and its first two derivatives is proved and error bounds are obtained. The question of stability is not examined.

References [Enhancements On Off] (What's this?)

  • [1] A. N. Netravali & R. J. P. deFigueiredo, "Spline approximation to the solution of the linear Fredholm integral equation of the second kind". (Submitted for publication.)
  • [2] J. G. Jones, On the numerical solution of convolution integral equations and systems of such equations, Math. Comp. 15 (1961), 131–142. MR 0122001,
  • [3] B. Noble, The numerical solution of nonlinear integral equations and related topics, Nonlinear Integral Equations (Proc. Advanced Seminar Conducted by Math. Research Center, U.S. Army, Univ. Wisconsin, Madison, Wis., 1963) Univ. Wisconsin Press, Madison, Wis., 1964, pp. 215–318. MR 0173369
  • [4] M. K. Jain and K. D. Sharma, Numerical solution of linear differential equations and Volterra’s integral equation using Lobatto quadrature formula, Comput. J. 10 (1967), 101–107. MR 0211610,
  • [5] L. Fox and E. T. Goodwin, The numerical solution of non-singular linear integral equations, Philos. Trans. Roy. Soc. London. Ser. A. 245 (1953), 501–534. MR 0054355,
  • [6] P. Linz, The Numerical Solution of Volterra Integral Equation by Finite Difference Methods, MRC Technical Report #827, Madison, Wis., 1967.
  • [7] P. Pouzet, Étude, en vue de leur traitement numérique d'équations intégrales et intégro-différentielles du type de Volterra pour des problèmes de conditions initiales, Thesis, University of Strasbourg, 1962.
  • [8] Pierre Pouzet, Mèthode d’intégration numérique des équations intégrales et intégro-différentielles du type de Volterra de seconde espèce. Formules de Runge-Kutta, Symposium on the numerical treatment of ordinary differential equations, integral and integro-differential equations (Rome, 1960) Birkhäuser, Basel, 1960, pp. 362–368 (French). MR 0127556
  • [9] M. Laudet & H. Oulès, "Sur l'intégration numérique des équations intégrales du type de Volterra," Symposium on the Numerical Treatment of Ordinary Differential Equations, Integral and Integro-Differential Equations (Rome, 1960), Birkhäuser, Basel, 1960, pp. 117-121. MR 23 #B597.
  • [10] B. A. Bel′tjukov, An analogue of the Runge-Kutta method for the solution of nonlinear integral equations of Volterra type, Differencial′nye Uravnenija 1 (1965), 545–556 (Russian). MR 0195277
  • [11] Alan Feldstein and John R. Sopka, Numerical methods for nonlinear Volterra integro-differential equations, SIAM J. Numer. Anal. 11 (1974), 826–846. MR 0375816,
  • [12] Frank R. Loscalzo and Thomas D. Talbot, Spline function approximations for solutions of ordinary differential equations, SIAM J. Numer. Anal. 4 (1967), 433–445. MR 0221756,
  • [13] F. R. Loscalzo & I. J. Schoenberg, On the Use of Spline Functions for the Approximation of Solutions of Ordinary Differential Equations, Tech. Summary Report #723, Math. Res. Center, U.S. Army, University of Wisconsin, Madison, Wis., 1967.
  • [14] Richard S. Varga, Error bounds for spline interpolation, Approximations with Special Emphasis on Spline Functions (Proc. Sympos.Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 367–388. MR 0252915
  • [15] H. S. Hung, The Numerical Solution of Differential and Integral Equations by Spline Functions, Tech. Summary Report #1053, Math. Res. Center, U.S. Army, University of Wisconsin, Madison, Wis., March 1970.
  • [16] J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The theory of splines and their applications, Academic Press, New York-London, 1967. MR 0239327
  • [17] Garrett Birkhoff and Carl R. De Boor, Piecewise polynomial interpolation and approximation, Approximation of Functions (Proc. Sympos. General Motors Res. Lab., 1964 ), Elsevier Publ. Co., Amsterdam, 1965, pp. 164–190. MR 0189219
  • [18] Milton Lees, Approximate solutions of parabolic equations, J. Soc. Indust. Appl. Math. 7 (1959), 167–183. MR 0110212
  • [19] E. N. Nilson, Cubic splines on uniform meshes, Comm. ACM 13 (1970), 255–258. MR 0283959,
  • [20] A. N. Netravali, Signal Processing Techniques Based on Spline Functions, Ph.D. Dissertation, Rice University, Houston, Texas, 1970.

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Article copyright: © Copyright 1973 American Mathematical Society

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