A simple Taylor-series expansion method for a class of second kind integral equations. (English) Zbl 0936.65146

The authors consider the integral equation \[ x(s)- \lambda \int^1_0 k(s,t)x(t) dt= y(s),\quad s\in [0,1], \] with kernel \(k\) either a continuous and (rapidly) decreasing convolution kernel, or containing a weak (algebraic) singularity. Motivated by certain limitations of the Taylor series expansion method proposed in M. Perlmutter and R. Siegel [ASME J. Heat Transfer, 85, 55-62 (1963)], they show how the desired Taylor coefficient functions can be found by solving a linear algebraic system which does not involve the use of boundary conditions.
There is no convergence analysis, and no error estimates are given. Instead, the method is applied to three examples of integral equations with smooth kernels (arising in radiative heat transfer and electrostatics), and a constructed example with weakly singular kernel (where, not surprisingly, the method does not do well near the endpoints of the interval of integration).


65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Full Text: DOI


[1] Alpert, B; Beylkin, G; Coifman, R; Rokhlin, V, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM J. sci. comput., 14, 159-184, (1993) · Zbl 0771.65088
[2] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997. · Zbl 0899.65077
[3] Choi, B.C; Churchill, S.W, A technique for obtaining approximate solutions for a class of integral equations arising in radiative heat transfer, Internat J. heat fluid flow, 6, 1, 42-48, (1985)
[4] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985. · Zbl 0592.65093
[5] W. Hackbusch, Multi-grid Methods and Applications, Springer, Berlin, 1985. · Zbl 0595.65106
[6] E. Hopf, Mathematical Problems of Radiative Equilibrium, Cambridge University Press, Cambridge, 1934. · JFM 60.0809.01
[7] Kaper, H.G; Kellogg, R.B, Asymptotic behavior of the solution of the integral transport equation in slab geometry, SIAM J. appl. math., 32, 191-200, (1977) · Zbl 0351.45001
[8] Love, E.R, The electrostatic field of two equal circular co-axial conducting discs, Quart J. mech. appl. math., 2, 428-451, (1949) · Zbl 0040.12105
[9] S.G. Mikhlin, Integral Equations, 2nd Edition, Pergamon Press, London, 1964. · Zbl 0117.31902
[10] Perlmutter, M; Siegel, R, Effect of specularly reflecting grey surface on thermal radiation through a tube and from its heated wall, ASME J. heat transfer, 85, 1, 55-62, (1963)
[11] L. Reichel, Parallel iterative methods for the solution of Fredholm integral equations of the second kind, in: M.T. Heath (Ed.), Hypercube Multiprocessors, SIAM, Philadelphia, 1987, pp. 520-529.
[12] Reichel, L, Fast solution methods for Fredholm integral equations of the second kind, Numer. math., 57, 719-736, (1989) · Zbl 0706.65143
[13] Schneider, C, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral equations oper. theory, 2, 62-68, (1979) · Zbl 0403.45002
[14] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, 3rd Edition, Hemisphere Publishing Co., Washington, DC, 1992.
[15] E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, revised edition, Wadsworth Publishing Co., Inc, Belmont, CA, 1970. · Zbl 0102.41201
[16] Vainikko, G; Pedas, A, The properties of solutions of weakly singular integral equations, J. austral. math. soc., B 22, 419-430, (1981) · Zbl 0475.65085
[17] Yan, Y, A fast numerical solution for a second kind boundary integral equation with a logarithmic kernel, SIAM J. numer. anal., 31, 477-498, (1994) · Zbl 0805.65110
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.