The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind
Tóm tắt
In the present paper we investigate the regularizing properties of the product trapezoidal method for solving weakly singular first kind Volterra integral equations with perturbed right-hand sides. Some numerical results are also presented.
Tài liệu tham khảo
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 10th edn. National Bureau of Standards, Dover, New York (1972)
Anderssen, R.S., de Hoog, F.R.: Abel integral equations. In: Golberg, M. (ed.) Numerical Solution of Integral Equations, pp. 373–410. Plenum Press, New York (1990)
Atkinson, K.E.: An existence theorem for Abel integral equations. SIAM J. Math. Anal. 5(5), 729–736 (1974)
Branca, H.W.: The nonlinear Volterra equation of Abel’s kind and its numerical treatment. Computing 20, 307–324 (1978)
Brunner, H., van der Houwen, P.J.: The Numerical Solution of Volterra Equations. Elsevier, Amsterdam (1986)
Bughgeim, A.L.: Volterra Equations and Inverse Problems. VSP/de Gruyter, Zeist/Berlin (1999)
Durbin, J.: Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov–Smirnov test. J. Appl. Probab. 8, 431–453 (1971)
Eggermont, P.P.B.: A new analysis of the trapezoidal-discretization method for the numerical solution of Abel-type integral equations. J. Integral Equations 3, 317–332 (1981)
Erdős, P., Feller, W., Pollard, H.: A property of power series with positive coefficients. Bull. Amer. Math. Soc. 55, 201–204 (1949)
Gorenflo, R., Vessella, S.: Abel Integral Equations. Springer-Verlag, New York (1991)
Hackbusch, W.: Integral Equations. Birkhäuser, Basel (1995)
Hardy, G.H.: Divergent Series. Oxford University Press, Oxford (1948)
Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley, New York (1974)
Kaluza, T.: Über die Koeffizienten reziproker Funktionen. Math. Z. 28, 161–170 (1928)
Lamm, P.: A survey of regularization methods for first-kind Volterra equations. In: Colton, D., Engl, H.W., Louis, A.K., McLaughlin, J.R., Rundell, W., (eds.) Surveys on Solution Methods for Inverse Problems, pp. 53–82. Springer, Vienna, New York (2000)
Lerche, I., Zeitler, E.: Projections, reconstructions and orthogonal functions. J. Math. Anal. Appl. 56, 634–649 (1976)
Plato, R.: Resolvent estimates for Abel integral operators and the regularization of associated first kind integral equations. J. Integral Equations Appl. 9(3), 253–278 (1997)
Plato, R.: Concise Numerical Mathematics. AMS, Providence, Rhode Island (2003)
Plato, R.: Fractional multistep methods for weakly singular Volterra equations of the first kind with noisy data. Numer. Funct. Anal. Optim. 26(2), 249–269 (2005)
Rogozin, B.A.: Asymptotics of the coefficients in the Levi–Wiener theorems on absolutely convergent trigonometric series. Sib. Math. J. 14, 917–923 (1973)
Rogozin, B.A.: Asymptotic behavior of the coefficients of functions of power series and Fourier series. Sib. Math. J. 17, 492–498 (1976)
Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)
Szegö, G.: Bemerkungen zu einer Arbeit von Herrn Fejér über die Legendreschen Polynome. Math. Z. 25, 172–187 (1926)
Weiss, R.: Product integration for the generalized Abel equation. Math. Comput. 26, 177–190 (1972)