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Solving integral equations with logarithmic kernels by Chebyshev polynomials

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Abstract

In this paper, a finite Chebyshev expansion is developed to solve Volterra integral equations with logarithmic singularities in their kernels. The error analysis is derived. Numerical results are given showing a marked improvement in comparison with the piecewise polynomial collocation method given in literature.

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References

  1. Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 45, 417–437 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brunner, H., van der Houwen, P.J.: The Numerical Solution of Volterra Equations. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  3. Brunner, H., Pedas, A., Vainikko, G.: The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations. Math. Comput. 68, 1079–1095 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brunner, H., Pedas, A, Vainikko, G.: Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. Report A425, Helsinki University of Technology (2000)

  5. Dixon, J., Mckee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 11, 535–544 (1986)

    Article  MathSciNet  Google Scholar 

  6. El-Gendi, S.E.: Chebyshev solution of differential, integral and integro-differential equations. Comput. J. 12, 282–287 (1969)

    MATH  MathSciNet  Google Scholar 

  7. Fox, L., Parker, I.B.: Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London (1968)

    Google Scholar 

  8. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic, New York (1980)

    MATH  Google Scholar 

  9. Khater, A.H., Shamardan, A.B., Callebaut, D.K., Sakran, M.R.A.: Chebyshev solution of integral equations with singular kernel. Int. J. Comput. Math. Numer. Simul. 1, (2007)

  10. Orsi, A.P.: Product integration for Volterra integral equations of the second kind with weakly singular kernels. Math. Comput. 65, 1201–1212 (1996)

    Article  MATH  Google Scholar 

  11. Pedas, A., Vainikko, G.: Superconvergence of piecewise polynomial collocations for nonlinear weakly singular integral equations. J. Integral Equ. Appl. 9, 379–406 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    A. H. Khater

  2. Department of Mathematics, Faculty of Science, El-Minia University, El-Minia, Egypt

    A. B. Shamardan

  3. Department Natuurkunde, CGB, University of Antwerp (UA), 2020, Antwerp, Belgium

    D. K. Callebaut

  4. Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt

    M. R. A. Sakran

Authors
  1. A. H. Khater
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  2. A. B. Shamardan
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  3. D. K. Callebaut
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  4. M. R. A. Sakran
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Correspondence to A. H. Khater.

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Khater, A.H., Shamardan, A.B., Callebaut, D.K. et al. Solving integral equations with logarithmic kernels by Chebyshev polynomials. Numer Algor 47, 81–93 (2008). https://doi.org/10.1007/s11075-007-9148-5

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  • DOI: https://doi.org/10.1007/s11075-007-9148-5

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