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An introduction to iterative techniques for potential problems

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Potential Theory Surveys and Problems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1344))

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References

  1. R. S. Anderssen et al: The application and numerical solution of integral equations. Sijthoff and Noordhoff, The Netherlands, 1980.

    Book  MATH  Google Scholar 

  2. T. S. Angell, R. E. Kleinman, G. F. Roach: Iterative methods for potential problems. (This volume), 1987.

    Google Scholar 

  3. K. E. Atkinson: The solution of non-unique linear integral equations. Numer. Math. 10, 117–124, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  4. Ju. M. Berezanskii: Expansions in eigenfunctions of self adjoint operators. A.M.S. Translations Math. Mono., 17, Providence, 1968.

    Google Scholar 

  5. H. Bialy: Iterative behandlung linearer Funktrangleighungen. Arch. Rat. Mech. Anal. 4, 166–176, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Courant, D. Hilbert: Methods of mathematical physics. Interscience, New York, 1953.

    MATH  Google Scholar 

  7. H. W. Engl: A successive approximation method for solving equations of second kind with arbitrary spectral radius. Jour. Int. Equ. 8, 239–247, 1985.

    MathSciNet  MATH  Google Scholar 

  8. G. I. Eskin: Boundary value problems for elliptic pseudo-differential equations. A.M.S. Translations of Math. Mono. 52, Providence, 1981.

    Google Scholar 

  9. P. R. Garabedian: Partial Differential Equations. Wiley, New York, (341–344), 1964.

    MATH  Google Scholar 

  10. I. C. Gohberg, I. A. Fel’dman: Convolution equations and projection methods for their solution. A.M.S. Translations of Math. Mono. 41, Providence, 1974.

    Google Scholar 

  11. P. Grisvard: Elliptic problems in non-smooth domains. Monographs and studies in mathematics 24, Pitmans, London, 1985

    MATH  Google Scholar 

  12. C. W. Groetsch: Theory of Tikhonov regularisation for Fredholm equations of the first kind. Research Notes in Mathematics, 105, Pitmans, London, 1984.

    MATH  Google Scholar 

  13. S. Heikkila, G. F. Roach: On equivalent norms and the contraction mapping principle. Jour. Nonlinear Analysis TMA 8, No.10, 1185–1193, 1985.

    MathSciNet  Google Scholar 

  14. J. L. Howland: Symmeterising kernels and the integral equations of the first kind of classical potential theory. Proc. Amer. Math. Soc. 19, 1–7, 1968.

    Article  MathSciNet  MATH  Google Scholar 

  15. G. C. Hsaio, R. C. McCamy: Solution of boundary value problems by integral equations of the first kind. SIAM. Rev. 15, 687–705, 1973.

    Article  MathSciNet  Google Scholar 

  16. G. C. Hsaio, W. Wendland; On Galerkin’s method for a class of integral equations of the first kind. Applicable Anal. 6, 155–157, 1977.

    Article  Google Scholar 

  17. G. C. Hsaio, G. F. Roach: On the relationship between boundary value problems. Jour. Math. Anal. Appl. 68, (2), 557–566, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. S. Jones: Integral equations for the exterior acoustic problem. Q. Jour. Mech. Appl. Math. XXVII, 1, 129–142, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. J. Kammerer, M. Z. Nashed: Iterative methods for the best approximate solutions of linear integral equations of the first and second kinds. Jour. Math. Anal. and Applic. 40, (3), 547–573, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. V. Kantorovich, G. F. Akilov: Functional Analysis in Normed Spaces. Pergamon, London, 1964.

    MATH  Google Scholar 

  21. T. Kato: Perturbation theory for linear operators. Springer Verlag, Berlin, 1966.

    Book  MATH  Google Scholar 

  22. O. D. Kellogg: Fundations of Potential Theory. Dover Publications, New York, 1953.

    Google Scholar 

  23. R. E. Kleinman: Iterative solutions of boundary value problems. Function Theoretic Methods for Partial Differential Equations. Springer LNM 561, 198–313, 1976.

    MathSciNet  Google Scholar 

  24. R. E. Kleinman, G. F. Roach: Boundary integral equations for the three dimensional Helmholtz equation. SIAM Reviews 16, (2), 214–236, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. E. Kleinman, G. F. Roach: On modified Green’s functions in exterior problems for the Helmholtz equation. Proc. Roy. Soc. Lond. A 383, 313–332, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. E. Kleinman, G.F. Roach: Operators of minimal norm via modified Green’s functions. Proc. Roy. Soc. Edin. 94 A, 163–178, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  27. V. A. Kondratiev: Boundary value problems for elliptic equations in domains with conical and angular points. Trudy Moskovkogo Mat. Obschetsva 16, 209–292. Transactions Moscow Mat. Soc. 16, 227–313, 1967.

    Google Scholar 

  28. J. Kral: Integral operators in potential theory. Lecture Notes in Mathematics 823. Springer Verlag, 1980.

    Google Scholar 

  29. M. A. Krasnosel’skii et al: Approximate solution of operator equations. Wolters-Noordhoff, Groningen, 1969.

    Google Scholar 

  30. R. Kress, G. F. Roach: On the convergence of successive approximations for an integral equation in a Green’s function approach to the Dirichlet problem. Jour. Math. Anal. and Applic. 55, (1), 102–111, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. L. Lions, E. Magenes: Non homogeneous boundary value problems and applications. Springer Verlag, Berlin, 1972.

    Book  MATH  Google Scholar 

  32. A. T. Lonseth: Approximate solution of Fredholm-type integral equations. Bull.Amer. Math. Soc. 60, 415–430, 1954.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. G. Mikhlin, K. L. Smolitskiy: Approximate Methods for Solution of Differential and Integral Equations. American Elsevier, New York, 1967.

    Google Scholar 

  34. C. Miranda: Partial differential equations of elliptic type (2nd Ed.). Springer Verlag, Berlin, 1970.

    Book  MATH  Google Scholar 

  35. J. Mooney, G. F. Roach: Iterative bounds for the stable solutions of convex nonlinear boundary value problems. Proc. Roy. Soc. Edin. 76A, 81–94, 1976.

    MathSciNet  MATH  Google Scholar 

  36. W. M. Patterson: Iterative methods for the solution of a linear operator equation in Hilbert space. Lecture Notes in Mathematics No. 394. Springer Verlag, Berlin, 1974.

    MATH  Google Scholar 

  37. W. V. Petryshyn: On a general iterative method for the approximate solution of linear operator equations. Maths. Comp. 17, 1–10, 1963.

    Article  MathSciNet  MATH  Google Scholar 

  38. W. V. Petryshyn: On generalised inverses and on the uniform convergence of (I-βK)n with application to iterative methods. Jour. Math. Anal. Appl. 18, 417–439, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  39. L. B. Rall: Error bounds for iterative solutions of Fredholm integral equations. Pacific J. Maths. 5, 977–986, 1955.

    Article  MathSciNet  MATH  Google Scholar 

  40. G. F. Roach: On the approximate solution of elliptic self adjoint boundary value problems. Arch. Rat. Mech. Anal. 17, (3), 243–254, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  41. G. F. Roach: Approximate Green’s functions and the solution of related integral equations. Arch. Rat. Mech. Anal. 36. (1), 79–88, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  42. G. F. Roach: Green’s functions (2nd Ed.). Cambridge University Press, Cambridge, 1970.

    MATH  Google Scholar 

  43. G. F. Roach: On the commutative properties of boundary integral operators. Proc. Amer. Math. Soc. 73, (2), 219–227, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  44. G. F. Roach: Weighted norms and the contraction mapping principle. Univ. Göttingen, NAM Bericht No. 10, 1–11, 1974.

    Google Scholar 

  45. I. Stakgold: Boundary value problems of mathematical physics. Macmillan, New York, 1966.

    MATH  Google Scholar 

  46. A. E. Taylor: Introduction to Functional Analysis. Wiley, New York, 1958.

    MATH  Google Scholar 

  47. A. Walther, B. Dejon: General report on the numerical treatment of integral and integro-differential equations. Symposium on the Numerical Treatment of ODE, Int. Equations, Integro-differential equations (645–671), Birkhauser, Basle, 1960.

    Google Scholar 

  48. W. L. Wendland, E. Stephan, G. Hsaio: On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Method in Applied Sci. 1, 265–321, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Wiarda: Integralgleichungen. Teubner, Leipzig, 1930.

    MATH  Google Scholar 

  50. K. Yosida: Functional Analysis. Springer Verlag, Berlin, 1968.

    Book  MATH  Google Scholar 

  51. A. C. Zaanen: Linear Analysis. North Holland, Amsterdam, 1964.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Strathclyde, 26 Richmond Street, G1 1XH, Glasgow

    G. F. Roach

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  1. G. F. Roach
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Josef Král Jaroslav Lukeš Ivan Netuka Jiří Veselý

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© 1988 Springer-Verlag

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Roach, G.F. (1988). An introduction to iterative techniques for potential problems. In: Král, J., Lukeš, J., Netuka, I., Veselý, J. (eds) Potential Theory Surveys and Problems. Lecture Notes in Mathematics, vol 1344. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103350

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  • DOI: https://doi.org/10.1007/BFb0103350

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  • Print ISBN: 978-3-540-50210-4

  • Online ISBN: 978-3-540-45952-1

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