Approximate Solution of Singular Integral Equations with Negative Index

Yıl 2010, Cilt: 23 Sayı: 4, 449 - 455, 19.03.2010

Nizami Mustafa Murat Çağlar

Öz

This paper is devoted to investigating a class of linear singular integral equations with a negative index on a closed, simple and smooth curve. In this paper, we propose the quadrature method for the approximate solution of the linear singular integral equations with negative index. Sufficient conditions are given for the convergence of this method in HÖlder space.

 

Key Words: Linear singular integral equation, Index of singular integral equation, Quadrature method.

Anahtar Kelimeler

Linear singular integral equation, Index of singular integral equation, Quadrature method.

Kaynakça

• [1] Gakhov, F.D., “Boundary Problems,” Nauka, Moscow, (1977).
• [2] Lu, Jian-Ke., “Boundary Value Problems for Analytic Functions”, World Scientific, SingaporeNew-Jersey-London-Hong- Kong, (1993).
• [3] Muskhelishvili, N.I., “Singular Integral Equation”, Nauka, Moscow, (1968).
• [4] Belotserkovskii, S.M., Lifanov, I.K., “Numerical Methods for the Singular Integral Equations”, Nauka, Moscow, (1985).
• [5] Gabdulkhaev, B.G., “Optimal Approximation to Linear Problem”, Kazan University Publications, Kazan, (1980).
• [6] Gabdulkhaev, B.G., 1980, “Finite Approximations of Singular Integrals, Direct Solution Methods of Singular Integral and IntegroDifferentialEquations”, Itogi Nauki I Tekniki, VINITI AN SSSR, Mat. Analysis, 18: 25-31, (1980).
• [7] Gabdulkhaev, B.G., Gorlov, V.E., “On the Optimal Algorithm of the Approximate Solutions of Singular Integral Equations”, Izv. Vuzov Mat. 11: 13-31, (1976).
• [8] Ivanov, V.V., “The Theory of Approximate Methods and its Application to the Numerical Solution of Singular Integral Equations”, Naukova Dumka, Kiev, (1968).
• [9] Kalandia, A.I., “Mathematical Methods of the TwoDimensional Elastics”, Nauka, Moscow, (1973)
• [10]Mustafaev, N.M., “On the Approximate Solution of the Singular Integral Equation that is Defined on Closed Smooth Curve”, Singular Integral Operators, AGU, Publications, Baku, 1: 91-99, (1987).
• [11] Mustafaev, N.M., “Approximate Formulas for Singular Integrals and Their Application to the Approximate Solution of Singular Integral Equations that are Defined on Closed Smooth Curve”, Phd. Thesis, AN Az. SSR, Institute of Math. and Mech., Baku, 1-130 (1991).
• [12] Mustafa, N. and Yazar, M.I.,“On the Approximate Solution of a Nonlinear Singular Integral Equation with a Cauchy Kernel”, Far East J. Appl. Math. 27(1): 101-119 (2007).
• [13] Mustafa, N., “On the Approximate Solution of Nonlinear Operator Equations”, Far East J. Appl. Math. 27(1):121-13 (2007).
• [14] Mustafa, N., “Fixed point theory and approximate solutions of non-linear singular integral equations”, Complex Variables and Elliptic Equations, 53(11):1047-1058 (2008).
• [15]Mustafa, N., Ardil, C., “On the Approximate olution of a Nonlinear Singular Integral Equation”, International Journal of Computational and Mathematical Science, 3(1): 1-7 (2009).
• [16] Mustafa, N. and Khalilov, E.H., “The collocation method for the solution of boundary integral Equations”, Applicable Analysis, 88(12): 1665- 1675, (2009).
• [17] Panasyuk, V.V., Savruk, M.P. and Nazarchuk, Z.T., “Singular Integral Equations Methods in two Dimensional Difraksion Problem”, Naukova Dumka, Kiev, (1984).
• [18]Parton, V.Z., Perlin, P.I., “Integral Equations of Elasticity Theory”, Nauka, Moscow, (1977). [19]Prösdorf, Z., “Some Class Singular Integral Equations”, Mir, Moscow, (1979).
• [20]Prösdorf, S., Silberman, B., “Projektionsverfahren und die naherungsweise Losung Singularer” Gleichungen, Leipziq, (1977).
• [21]Saleh, M.H., “Basis of Quadrature Method for onlinear Singular Integral Equations with Hilbert Kernel in the Space H φ,k ”, In Az. NIINTI, 279: 1- 40 (1984).
• [22]Seychuk, V.N., “Direct Methods of the Solutions of Singular Integral Equations that are defined on Lyapunov Curve”, Phd. Thesis, Kishinev University, Kishinev, 1-123 (1987).
• [23]Zolotaryevskii, V.A., “On the Approximate Solution of Singular Integral Equations”, Mathematical Research, Kishinev, Shtiintsa, 9(3): 82-94 (1974).
• [24]Zolotaryevskii, V.A., Seychuk, V.N., “The Solution of the Singular Integral Equation That is Defined on Lyapunov Curve by Collocation Method”, Dif. Equations, 19(6):1056-1064 (1983).
• [25]Musaev, B.I., “On Approximate Solution of the Singular Integral Equations”, ANAz. SSR, Institute of Physics Preprint 17:1-16 (1986).
• [26] Musaev, B.I., “On Approximate Solution of the Singular Integral Equations”, Izv. AN Az. SSR, Fizik-Teknik and Science, 5: 15-21 (1986).
• [27] Musaev, B.I., “On the approximate solution of singular integral equations with negative index by Bubnov-Galerkin and collocation methods”, Soviet Math. Dokl. 35(2): 411-416 (1987).
• [28] Alper, S.Ya., “On the Regular Approach in Region of Complex-Variable Functions”, Izv. AN SSSR, Mat. 6: 423-444 (1955).
• [29] Mustafaev, N.M., “Error of the Approximation to the Singular Integral Equation that is defined on Closed Smooth Curve”, In Az. NIINTI, 338(B88): 1-37 (1988).