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On the Existence Solution of a Class Boundary Integral Equations

Year 2013, Volume: 26 Issue: 2, 165 - 171, 14.01.2013

Abstract

This paper is devoted to investigating the existence and uniqueness solutions class boundary integral equations over a regular closed surface. This paper provides sufficient conditions for the existence and uniqueness solution in the space of continuous functions of class boundary integral equations.

References

  • Gakhov, F.D., “Boundary Value Problems”, Oxford and Addison-Wesley, Reading, Pergamon, MA, (1966).
  • Lu, J.-K., “Boundary Value Problems for Analytic Functions”, World Scientific, Singapore-New Jersey-London-Hong Kong, (1993).
  • Muskhelishvili, N.I., “Singular Integral Equations”, Nauka, Moscow, (1968).
  • Belotserkovskii, S.M., Lifanov, I.K., “Numerical Methods for the Singular Integral Equations, Nauka, Moscow, (1985).
  • Gabdulkhaev, B.G., “Optimal Approximation to Linear Problem”, Kazan University Publications, Kazan, (1980).
  • Gabdulkhaev, B.G., “Finite Approximations of Singular Integrals, Direct Solution Methods of Singular Integral and Integro-Diffferential Equations”, Itogi Nauki i Tekniki, VINITI AN SSSR, Math. Analysis, 18: 251-301, (1980).
  • Gabdulkhaev, B.G., Gorlov, V.E., “On the Optimal Algorithm of the Approximate Solutions of Singular Integral Equations”, Izv. Vuzov Math. 11: 13- 31, (1976).
  • Ivanov, V.V., “The Theory of Approximate Methods and its Application to the Numerical Solution of Singular Integral Equations”, Naukova Dumka, Kiev, (1968).
  • Jinyuan, D., “The Collocation Methods and Singular Integral Equations with Cauchy Kernels”, Acta Math. Sc. 20(B3): 289-302, (2000).
  • Mustafaev, N.M., “On the Approximate Solution of the Singular Integral Equation that is Defined on Closed Smooth Curve”, Singular Integral Operators, AGU Publications, 91-99, Baku, (1987).
  • Mustafa, N., “Fixed Point Theory and Approximate Solutions of Non-linear Singular Integral Equations”, Complex Variables and Elliptic Equations, 53(11): 1047-1058, (2008).
  • Panasyuk, V.V., Savruk , M.P. and Nazarchuk, Z.T., “Singular Integral Equations Methods in Two- Dimensional Diffraction Problem”, Naukova Dumka, Kiev, (1984).
  • Parton, V.Z. and Perlin, P.I., “Integral Equations of Elasticity Theory”, Nauka, Moscow, (1977).
  • Prösdorf, S., “Some Class Singular Integral Equations”, Mir, Moscow, (1979). [15] Prösdorf,
  • “Projektionsverfahren und die Naherungsweise Losung Singularer Gleichungen”, Leipziq, (1977).
  • B., Seychuk , V.N., “Direct Methods of the Solutions of Singular Integral Equations that are Defined on Lyapunov Curve”, Ph.D. Thesis, Kishinev University, Kishinev, (1987).
  • Zolotaryevskii, V.A., “On the Approximate Solution of Singular Integral Equations”, Math. Res., Shtiintsa, 9(3): 82-94, Kishinev, (1974).
  • Zoloratyevskii, V.A. and Seychuk, V.N., “The Solution of the Singular Integral Equation that are Defined on Lyapunov Curve by Collocation Method”, Differ. Eqn., 19(6): 1056-1064, (1983).
  • Mustafa, N. and Ardil, C., “On the Approximate Solution of a Nonlinear Singular Integral Equation”, International
  • Mathematical Sciences (WASET), 3(1): 1-7, (2009). of Journal Computational
  • and Mustafa, N. and Khalilov, E., “The Collocation Method for the Solution of Boundary Integral Equations”, Applicable Analysis, 88(12): 1665-1675, (2009).
  • Mustafa, N., “On the Approximate Solution of Singular Integral Equations with Negative Index”, Complex Variables and Elliptic Equations, 55(7): 621- 631, (2010).
  • Mustafa, N., Çağlar, M., “Approximate Solution of Singular Integral Equations with Negative Index, Gazi University Journal of Science, 23(4): 449-455, (2010).
  • Engleder, S., Steinbach, O., “Modified Boundary Integral Formulations for the Helmholtz Equation”, J. Math. Anal. Appl., 331: 396-407, (2007).
  • Jijun, J., “Determination of Dirichlet-to-Neumann Map for a Mixed Boundary Problem”, Appl. Math.and Comput., 161: 843-864, (2005).
  • Kleinman, R.E., Roach, G. F., “Boundary Integral Equations for the Three-Dimensional Helmholtz’s Equation”, SIAM Rev., 16: 214-236, (1974).
  • Mai-Duy, N., Tran-Cong, T., Tanner, R.I., “A Domain-Type Boundary-Integral- Equation Method for Two-Dimensional Biharmonic Dirichlet Problem”, Eng,. Anal. Boundary Elements, 30: 809-817, (2006). [27] Camp, C.V., Gipson, G.S., “A Boundary Element for Viscous Flows at Low Reynold’s Numbers”, Eng. Anal. Boundary Elements, 6 (3): 144-151, (1989).
  • Ingham, D.B., Kelmanson, M.A., In: Brebbia, C. A., Orszag, S. A., editors. “Boundary Integral Equation Analysis of Singular, Potential and Biharmonic Problems”, Lecture Notes in Engineering, Springer, Berlin, (1984).
  • Colton, D.L., Kress, R., “Integral Equation Methods in Scattering Theory”, John Willey & Sons, Inc., New York, (1983).
  • Jones, D.S., “Integral Equations for the Exterior Acoustic Problem”, Q. J. Mech. Appl. Math., 27: 129- 142, (1974).
  • Kleinman, R.E. and Roach, G.F., “On Modified Green Functions in Exterior Problems for the Helmholtz’s Equation”, Proc. R. Soc. Lond., A 383: 313-333, (1982).
  • Ursell, F., “On the Exterior Problems of Acoustics”, 11, Proc. Cambridge Phil. Soc., 84: 545- 548, (1978).
  • Mikhlin, S.G., “Partial Differential Linear Equations”, Vischaya Skola, Moscow, (1977).
  • Kolmogorov, A.N., Fomin, S.V., “Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, (1981).
Year 2013, Volume: 26 Issue: 2, 165 - 171, 14.01.2013

Abstract

References

  • Gakhov, F.D., “Boundary Value Problems”, Oxford and Addison-Wesley, Reading, Pergamon, MA, (1966).
  • Lu, J.-K., “Boundary Value Problems for Analytic Functions”, World Scientific, Singapore-New Jersey-London-Hong Kong, (1993).
  • Muskhelishvili, N.I., “Singular Integral Equations”, Nauka, Moscow, (1968).
  • Belotserkovskii, S.M., Lifanov, I.K., “Numerical Methods for the Singular Integral Equations, Nauka, Moscow, (1985).
  • Gabdulkhaev, B.G., “Optimal Approximation to Linear Problem”, Kazan University Publications, Kazan, (1980).
  • Gabdulkhaev, B.G., “Finite Approximations of Singular Integrals, Direct Solution Methods of Singular Integral and Integro-Diffferential Equations”, Itogi Nauki i Tekniki, VINITI AN SSSR, Math. Analysis, 18: 251-301, (1980).
  • Gabdulkhaev, B.G., Gorlov, V.E., “On the Optimal Algorithm of the Approximate Solutions of Singular Integral Equations”, Izv. Vuzov Math. 11: 13- 31, (1976).
  • Ivanov, V.V., “The Theory of Approximate Methods and its Application to the Numerical Solution of Singular Integral Equations”, Naukova Dumka, Kiev, (1968).
  • Jinyuan, D., “The Collocation Methods and Singular Integral Equations with Cauchy Kernels”, Acta Math. Sc. 20(B3): 289-302, (2000).
  • Mustafaev, N.M., “On the Approximate Solution of the Singular Integral Equation that is Defined on Closed Smooth Curve”, Singular Integral Operators, AGU Publications, 91-99, Baku, (1987).
  • Mustafa, N., “Fixed Point Theory and Approximate Solutions of Non-linear Singular Integral Equations”, Complex Variables and Elliptic Equations, 53(11): 1047-1058, (2008).
  • Panasyuk, V.V., Savruk , M.P. and Nazarchuk, Z.T., “Singular Integral Equations Methods in Two- Dimensional Diffraction Problem”, Naukova Dumka, Kiev, (1984).
  • Parton, V.Z. and Perlin, P.I., “Integral Equations of Elasticity Theory”, Nauka, Moscow, (1977).
  • Prösdorf, S., “Some Class Singular Integral Equations”, Mir, Moscow, (1979). [15] Prösdorf,
  • “Projektionsverfahren und die Naherungsweise Losung Singularer Gleichungen”, Leipziq, (1977).
  • B., Seychuk , V.N., “Direct Methods of the Solutions of Singular Integral Equations that are Defined on Lyapunov Curve”, Ph.D. Thesis, Kishinev University, Kishinev, (1987).
  • Zolotaryevskii, V.A., “On the Approximate Solution of Singular Integral Equations”, Math. Res., Shtiintsa, 9(3): 82-94, Kishinev, (1974).
  • Zoloratyevskii, V.A. and Seychuk, V.N., “The Solution of the Singular Integral Equation that are Defined on Lyapunov Curve by Collocation Method”, Differ. Eqn., 19(6): 1056-1064, (1983).
  • Mustafa, N. and Ardil, C., “On the Approximate Solution of a Nonlinear Singular Integral Equation”, International
  • Mathematical Sciences (WASET), 3(1): 1-7, (2009). of Journal Computational
  • and Mustafa, N. and Khalilov, E., “The Collocation Method for the Solution of Boundary Integral Equations”, Applicable Analysis, 88(12): 1665-1675, (2009).
  • Mustafa, N., “On the Approximate Solution of Singular Integral Equations with Negative Index”, Complex Variables and Elliptic Equations, 55(7): 621- 631, (2010).
  • Mustafa, N., Çağlar, M., “Approximate Solution of Singular Integral Equations with Negative Index, Gazi University Journal of Science, 23(4): 449-455, (2010).
  • Engleder, S., Steinbach, O., “Modified Boundary Integral Formulations for the Helmholtz Equation”, J. Math. Anal. Appl., 331: 396-407, (2007).
  • Jijun, J., “Determination of Dirichlet-to-Neumann Map for a Mixed Boundary Problem”, Appl. Math.and Comput., 161: 843-864, (2005).
  • Kleinman, R.E., Roach, G. F., “Boundary Integral Equations for the Three-Dimensional Helmholtz’s Equation”, SIAM Rev., 16: 214-236, (1974).
  • Mai-Duy, N., Tran-Cong, T., Tanner, R.I., “A Domain-Type Boundary-Integral- Equation Method for Two-Dimensional Biharmonic Dirichlet Problem”, Eng,. Anal. Boundary Elements, 30: 809-817, (2006). [27] Camp, C.V., Gipson, G.S., “A Boundary Element for Viscous Flows at Low Reynold’s Numbers”, Eng. Anal. Boundary Elements, 6 (3): 144-151, (1989).
  • Ingham, D.B., Kelmanson, M.A., In: Brebbia, C. A., Orszag, S. A., editors. “Boundary Integral Equation Analysis of Singular, Potential and Biharmonic Problems”, Lecture Notes in Engineering, Springer, Berlin, (1984).
  • Colton, D.L., Kress, R., “Integral Equation Methods in Scattering Theory”, John Willey & Sons, Inc., New York, (1983).
  • Jones, D.S., “Integral Equations for the Exterior Acoustic Problem”, Q. J. Mech. Appl. Math., 27: 129- 142, (1974).
  • Kleinman, R.E. and Roach, G.F., “On Modified Green Functions in Exterior Problems for the Helmholtz’s Equation”, Proc. R. Soc. Lond., A 383: 313-333, (1982).
  • Ursell, F., “On the Exterior Problems of Acoustics”, 11, Proc. Cambridge Phil. Soc., 84: 545- 548, (1978).
  • Mikhlin, S.G., “Partial Differential Linear Equations”, Vischaya Skola, Moscow, (1977).
  • Kolmogorov, A.N., Fomin, S.V., “Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, (1981).
There are 34 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Nizami Mustafa

Publication Date January 14, 2013
Published in Issue Year 2013 Volume: 26 Issue: 2

Cite

APA Mustafa, N. (2013). On the Existence Solution of a Class Boundary Integral Equations. Gazi University Journal of Science, 26(2), 165-171.
AMA Mustafa N. On the Existence Solution of a Class Boundary Integral Equations. Gazi University Journal of Science. July 2013;26(2):165-171.
Chicago Mustafa, Nizami. “On the Existence Solution of a Class Boundary Integral Equations”. Gazi University Journal of Science 26, no. 2 (July 2013): 165-71.
EndNote Mustafa N (July 1, 2013) On the Existence Solution of a Class Boundary Integral Equations. Gazi University Journal of Science 26 2 165–171.
IEEE N. Mustafa, “On the Existence Solution of a Class Boundary Integral Equations”, Gazi University Journal of Science, vol. 26, no. 2, pp. 165–171, 2013.
ISNAD Mustafa, Nizami. “On the Existence Solution of a Class Boundary Integral Equations”. Gazi University Journal of Science 26/2 (July 2013), 165-171.
JAMA Mustafa N. On the Existence Solution of a Class Boundary Integral Equations. Gazi University Journal of Science. 2013;26:165–171.
MLA Mustafa, Nizami. “On the Existence Solution of a Class Boundary Integral Equations”. Gazi University Journal of Science, vol. 26, no. 2, 2013, pp. 165-71.
Vancouver Mustafa N. On the Existence Solution of a Class Boundary Integral Equations. Gazi University Journal of Science. 2013;26(2):165-71.