On the Existence Solution of a Class Boundary Integral Equations

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

Nizami Mustafa

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.

Keywords

Boundary integral equation , Boundary value problem , Helmholtz equation.

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).