P. Mokhtary


An efficient discrete collocation method for solving Volterra type weakly singular integral equations with logarithmic kernels is investigated. One of features of these equations is that, in general the first erivative of solution behaves like as a logarithmic function, which is not continuous at the origin.
In this paper, to make a compatible approximate solution with the exact ones, we introduce a new collocation approach, which applies the M¨untz logarithmic polynomials(Muntz polynomials with logarithmic terms) as basis functions. Moreover, since implementation of this technique leads to integrals with logarithmic singularities that are often difficult to solve numerically, we apply a suitable quadrature method that allows the exact evaluation of integrals of polynomials with logarithmic weights. To this end, we first remind the well-known Jacobi–Gauss quadrature and then extend it to integrals with logarithmic weights. Convergence analysis of the proposed scheme are presented, and some numerical results are illustrated to demonstrate the efficiency and accuracy of the proposed method.

Article Details

1. Atkinson, K. E.The Numerical Solution of Integral Equations of the Second Kind, Cambridge, 1997.
2. Andreasen, M. G. and Mei, K. K.Comments on ”Scattering by conducting rectangular cylinders”, IEEE Trans, Antennas and Propagation, 11(1963), 52–56.
3. Andronov, I. V.Integro-differential equations of the convolution on a finite interval with kernel having a logarithmic singularity, J. Math. Sci. 79(1996), no. 4, 1161–1165.
4. Assari, P., Adibi, H., and Dehghan, M. A meshless discrete Galerkin(MDG) method for the numerical solution of integral equations with logarithmic kernels, J. Comput. Appl. Math, 267 (2014), 160–181.
5. Banaugh, P. P. and Goldsmith, W. Diffraction of steady acoustic waves by surfaces of arbitrary shape, J. Acoust. Soc. Am., 35 (1963), 1590–1601.
6. Banaugh, P. P. and Goldsmith, W. Diffraction of steady elastic waves by surfaces of arbitrary shape, J. Appl. Mech, 30 (1963), 589–597.
7. Ball, J. S. and Beebe, N. H. F.Efficient Gauss-related quadrature for two classes of logarithmic weight functions, ACM Transactions on Mathemat ical Software, 33 (2007), no. 3, Article No. 19.
8. Brunner, H. Collocation Methods for Volterra and Related Functional Equations, Cambridge University Press: Cambridge, 2004.
9. Canuto, C., Hussaini, M. Y.,Quarteroni, A., and Zang, T. A. Spectral Methods, Fundamentals in Single Domains, Berlin: Springer-Verlag, 2006.
10. Christiansen S. Numerical solution of an integral equation with a logarithmic kernel, BIT, 11 (1971), 276–287.
11. Dom´ inquez, V. High-order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs, J. Comput. Appl. Math., 161 (2003), 145–159.
12. Guseinov, E. A. and I´ linskii, A. S.Integral equations of the first kind with a logarithmic singularity in the kernel and their application in problems of diffraction by thin screens, U. S. S. R. Comput. Maths. Math. Phys.,27 (1987), no. 4, 58–63.
13. Gusenkova,A. A. and Pleshchinskii, N. B. Integral equations with logarithmic singularities in the kernels of boundary-value problems of plane elasticity theory for regions with a defect, J. Appl. Maths. Mechs., 64(2000), no. 3, 433–441.
14. Khader, A. H., Shamardan, A. B., Callebaut, D. K. and Sakran, M. R. A. Solving integral equations with logarithmic kernels by Chebyshev polynomials, Numer. Algorithms, 47 (2008), 81–93.
15. Mei, K. K. and Van Bladel, J. G. Low-frequency scattering by rectangular cylinders, IEEE Trans, Antennas and Propagation, 11 (1963), 52–56.
16. Mei, K. K. and Van Bladel, J. G. Scattering by perfectly-conducting rectangular cylinders , IEEE Trans, Antennas and Propagation, 11 (1963),185–192.
17. Mennouni, A.Airfoil polynomials for solving integro-diferential equations with logarithmic kernel, Appl. Math. Comput., 218 (2012), 11947–11951.
18. Milovanovi´ c, G. V. M¨untz orthogonal polynomials and their numerical evaluation, Internat. Ser. Numer. Math., 131 (1999), Birkh¨auser Verlag13asel/ Switzerland.
19. Mokhtary, P.Reconstruction of exponentially rate of convergence to Leg endre collocation solution of a class of fractional integro-differential equations, J. Comput. Appl. Math., 279 (2015), 145–158.
20. Mokhtary, P.Numerical treatment of a well-posed Chebyshev Tau method for Bagley-Torvik equation with high-order of accuracy, Numer. Algorithms, 72 (2016), 875–891.
21. Mokhtary, P.Discrete Galerkin method for fractional integro-differential equations, Acta. Math. Sci., 36 (2016), no. B(2), 560–578.
22. Mokhtary, P. Numerical analysis of an operational Jacobi Tau method forfractional weakly singular integro-differential equations, Appl. Numer. Math., 121 (2017), 52–67.
23. Mokhtary, P. and Ghoreishi, F. Convergence analysis of spectral Tau method for fractional Riccati differential equations, Bull. Iranian Math. Soc., 40 (2014), no. 5, 1275–1296.
24. Muschelischwili, N. I. Singula ¨re Integralgleichungen, Akademie-Verlag, Berlin, 1965.
25. Noble, B. Integral equation perturbation methods in low frequency diffraction in R. E. Langer electromagnetic waves, The University of Wis consin Press, Madison (1963), 323–360.
26. Rivlin, T. J.An Introduction to the Approximation of Functions, United States, 1969.
27. Shen, J., Tang, T., and Wang ,L.-L., Spectral Methods, Algorithms, Analysis and Applications, Springer, 2011.
28. Symm, G. T. An integral equation method in conformed mapping, Nu mer. Math., 9 (1966), 250–258.
29. Tang, T., McKee, S., and Diogo, T. Product integration methods for an integral equation with logarithmic singular kernel, Appl. Numer. Math., 9 (1992), 259–266.
30. Theocaris, P. S., Chrysakis, A. C., and Ioakimidis,N. I. Cauchy-type integrals and integral equations with logarithmic singularities, J. Eng. Math., 13 (1979), no. 1, 63–74.
How to Cite
مختاریپ. (2018). Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels. Iranian Journal of Numerical Analysis and Optimization, 8(2), 95-118. https://doi.org/10.22067/ijnao.v8i2.60778