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Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind

Year 2013, Volume: 10 Issue: 1, - , 01.05.2013

Abstract

The paper investigates a numerical method for the second kind Fredholm
integral equation with weakly singular kernel k(x, y), in particular, when k(x, y) = ln |x−y|,
and k(x, y) = |x − y|
−α, −1 ≤ x, y ≤ 1, 0 < α < 1. The solutions of such equations may
exhibit a singular behaviour in the neighbourhood of the endpoints x = ±1. We introduce
a new smoothing transformation based on the Kress transformation for solving weakly
singular Fredholm integral equations of the second kind, and then using the Hermite
smoothing transformation as a standard. With the transformation an equation which
is still weakly singular is obtained, but whose solution is smoother. The transformed
equation is then solved numerically by product integration methods with interpolating
polynomials. Two types of interpolating polynomials, namely the Gauss-Legendre and
Chebyshev polynomials, have been used. Numerical examples are presented to investigate
the performance of the former.

References

  • [1] A. Young, Approximate product-integration, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 224 (1954), 552–561.
  • [2] A.-M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore 1997.
  • [3] A. J. Jerri, Introduction to Integral Equations with Applications, Marcel Dekker, New York 1985.
  • [4] K. E. Atkinson, A Survey of Numerical Methods for Solution of Fredholm Integral Equation of Second Kind, SIAM, Philadelphia 1976.
  • [5] K. E. Atkinson, The Numerical Solution of Integral Equation of the Second Kind, Cambridge University Press,1997.
  • [6] C. T. H. Baker, The Numerical Treatment of Integral Equation, Oxford University Press,1977.
  • [7] C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numerische Mathematik 2 (1960), 197–205.
  • [8] C. Schneider, Product integration for weakly singular integral equations, Mathematics of Computation 36 (1981), 207–213.
  • [9] D. Elliott and S. Pr¨ossdorf, An algorithm for the approximate solution of integral equations of Mellin type, Numerische Mathematik 70 (1995), 427–452.
  • [10] G. Criscuolo, G. Mastroianni and G. Monegato, Convergence properties of a class of product formulas for weakly singular integral equations, Mathematics of Computation 55 (1990), 213–230.
  • [11] G. Monegato and L. Scuderi, High order methods for weakly singular integral equations with non smooth input function, Mathematics of Computation 67 (1998), 1493–1515.
  • [12] H. Kaneko and Y. Xu, Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind, Mathematics of Computation 62 (1994), 739–753.
  • [13] H. Kaneko and Y. Xu, Numerical solutions for weakly singular Fredholm integral equations of the second kind, Applied Numerical Mathematics 7 (1991), 167–177.
  • [14] I. G. Graham and G. A. Chandler, High order methods for linear functionals of solutions of second kind integral equations, SIAM Journal on Numerical Analysis 25 (1988), 1118–1137.
  • [15] I. H. Sloan, On choosing the points in product integration, Journal of Mathematical Physics 21 (1980), 1032–1039.
  • [16] I. G. Graham, Galerkin methods for second kind integral equations with singularities, Mathematics of Computation 39 (1982), 519–533.
  • [17] I. G. Graham, Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, Journal of Integral Equations 4 (1982), 1–30.
  • [18] J. M. Rosenberg, A Guide to Matlab: For Beginners and Experienced Users, Cambridge University Press, 2001.
  • [19] K. E. Atkinson, The numerical solution of Fredholm integral equation of second kind, SIAM Journal on Numerical Analysis 4 (1967), 337–348.
  • [20] L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge 1985.
  • [21] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia 1985.
  • [22] P. K. Kythe and P. Puri, Computational Methods for Linear Integral Equations, Birkhauser, Boston 2002.
  • [23] P. F. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, 1984.
  • [24] R. Kress, A Nystr¨om method for boundary integral equation in domains with corners, Numerische Mathematik 58 (1990), 145–161.
  • [25] R. Kress, Linear Integral Equations, Springer-Verlag, Berlin 1989.
Year 2013, Volume: 10 Issue: 1, - , 01.05.2013

Abstract

References

  • [1] A. Young, Approximate product-integration, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 224 (1954), 552–561.
  • [2] A.-M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore 1997.
  • [3] A. J. Jerri, Introduction to Integral Equations with Applications, Marcel Dekker, New York 1985.
  • [4] K. E. Atkinson, A Survey of Numerical Methods for Solution of Fredholm Integral Equation of Second Kind, SIAM, Philadelphia 1976.
  • [5] K. E. Atkinson, The Numerical Solution of Integral Equation of the Second Kind, Cambridge University Press,1997.
  • [6] C. T. H. Baker, The Numerical Treatment of Integral Equation, Oxford University Press,1977.
  • [7] C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numerische Mathematik 2 (1960), 197–205.
  • [8] C. Schneider, Product integration for weakly singular integral equations, Mathematics of Computation 36 (1981), 207–213.
  • [9] D. Elliott and S. Pr¨ossdorf, An algorithm for the approximate solution of integral equations of Mellin type, Numerische Mathematik 70 (1995), 427–452.
  • [10] G. Criscuolo, G. Mastroianni and G. Monegato, Convergence properties of a class of product formulas for weakly singular integral equations, Mathematics of Computation 55 (1990), 213–230.
  • [11] G. Monegato and L. Scuderi, High order methods for weakly singular integral equations with non smooth input function, Mathematics of Computation 67 (1998), 1493–1515.
  • [12] H. Kaneko and Y. Xu, Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind, Mathematics of Computation 62 (1994), 739–753.
  • [13] H. Kaneko and Y. Xu, Numerical solutions for weakly singular Fredholm integral equations of the second kind, Applied Numerical Mathematics 7 (1991), 167–177.
  • [14] I. G. Graham and G. A. Chandler, High order methods for linear functionals of solutions of second kind integral equations, SIAM Journal on Numerical Analysis 25 (1988), 1118–1137.
  • [15] I. H. Sloan, On choosing the points in product integration, Journal of Mathematical Physics 21 (1980), 1032–1039.
  • [16] I. G. Graham, Galerkin methods for second kind integral equations with singularities, Mathematics of Computation 39 (1982), 519–533.
  • [17] I. G. Graham, Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, Journal of Integral Equations 4 (1982), 1–30.
  • [18] J. M. Rosenberg, A Guide to Matlab: For Beginners and Experienced Users, Cambridge University Press, 2001.
  • [19] K. E. Atkinson, The numerical solution of Fredholm integral equation of second kind, SIAM Journal on Numerical Analysis 4 (1967), 337–348.
  • [20] L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge 1985.
  • [21] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia 1985.
  • [22] P. K. Kythe and P. Puri, Computational Methods for Linear Integral Equations, Birkhauser, Boston 2002.
  • [23] P. F. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, 1984.
  • [24] R. Kress, A Nystr¨om method for boundary integral equation in domains with corners, Numerische Mathematik 58 (1990), 145–161.
  • [25] R. Kress, Linear Integral Equations, Springer-Verlag, Berlin 1989.
There are 25 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Hassan M. S. Bawazir This is me

Ali Abd Rahman This is me

Nor’aini binti Aris This is me

Publication Date May 1, 2013
Published in Issue Year 2013 Volume: 10 Issue: 1

Cite

APA Bawazir, H. M. S., Rahman, A. A., & Aris, N. b. (2013). Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind. Cankaya University Journal of Science and Engineering, 10(1).
AMA Bawazir HMS, Rahman AA, Aris Nb. Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind. CUJSE. May 2013;10(1).
Chicago Bawazir, Hassan M. S., Ali Abd Rahman, and Nor’aini binti Aris. “Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind”. Cankaya University Journal of Science and Engineering 10, no. 1 (May 2013).
EndNote Bawazir HMS, Rahman AA, Aris Nb (May 1, 2013) Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind. Cankaya University Journal of Science and Engineering 10 1
IEEE H. M. S. Bawazir, A. A. Rahman, and N. b. Aris, “Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind”, CUJSE, vol. 10, no. 1, 2013.
ISNAD Bawazir, Hassan M. S. et al. “Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind”. Cankaya University Journal of Science and Engineering 10/1 (May 2013).
JAMA Bawazir HMS, Rahman AA, Aris Nb. Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind. CUJSE. 2013;10.
MLA Bawazir, Hassan M. S. et al. “Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind”. Cankaya University Journal of Science and Engineering, vol. 10, no. 1, 2013.
Vancouver Bawazir HMS, Rahman AA, Aris Nb. Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind. CUJSE. 2013;10(1).