Abstract
Singular integral equation theory has broad applications to theoretical and practical investigations in mathematics, mathematical physics, hydrodynamic and elasticity theory. This fact motivated many researchers to work on this field and their studies have showed that finding approximate solutions of linear and nonlinear singular integral equations in Banach spaces provides many applications even if their definite solutions cannot be found or if there are difficulties in finding them. Thus, the central theme of the recent studies is to develop effective approximate solution methods for the linear and nonlinear singular integral equations in Banach spaces. This chapter has been devoted to investigating approximate solutions of linear and nonlinear singular integral equations in Banach spaces using technical methods such as collocation method, quadrature method, Newton–Kantorovich method, monotonic operators method, and fixed point theory depending on the type of the equations. We provide sufficient conditions for the convergence of these methods and investigate some properties.
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References
F.A. Abdullayev, E.H. Khalilov, Ground of the collocation method for a class of boundary integral equations. Differ. Equ. 49(1), 82–86 (2004)
V.M. Alexandrov, E.V. Kovalenko, Problems with Mixed Boundary Conditions in Continuum Mechanics (Nauka, Moscow 1986)
V.M. Alexandrov, I.I. Kudish, Asymptotic methods in Criffits problem. Appl. Math. Mech. 53, 665–671 (1989)
V.M. Alexandrov, S.M. Mkitaryan, Contact Problems for Bodies with Thin Coverings and Interlayers (Nauka, Moscow, 1983)
S.M. Amer, On solution of non-linear singular integral equations with shift in generalized Holder space. Chaos, Solutions Fractals 12, 1323–1334 (2001)
K.E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind (SIAM, Philadelphia, 1976)
A.A. Badr, Integro-differential equations with Cauchy kernel. J. Comput. Appl. Math. 134, 191199 (2001)
S.M. Belotserkhovskii, I. K. Lifanov, Numerical Solutions of Singular Integral Equations (Nauka, Moscow, 1985)
L. Bers, L. Nirenberg, On a representation for linear elliptic systems with discontinuous coefficients and its application. Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali (Trieste, 1954)
B.V. Bojarskii, Quasiconformal mappings and general structural properties of system of non-linear elliptic in the sense of Lavrentev. Sympos. Math. 18, 485–499 (1976)
Y.G. Borisovich, V.G. Zvyagin, Non-linear Fredholm maps and the Leray-Schauder theory. Russ. Math. Surv. 32(4), 1–54 (1977)
D.L. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York 1983)
I.K. Daugavet, Introduction to Approximation Theory of Functions (Leningrad University, Leningrad, 1977) (in Russian)
G. David, Courbes corde-are et espaces de Hardy generalizes. Ann. Inst. Fourier 32(3), 227–239 (1982)
R. Duduchava, An application of singular integral equations to some problems of elasticity. Integr. Equ. Oper. Theory 5(1), 475–489 (1982)
R. Duduchava, S. Prösdorf, On the approximation of singular integral equations by equations with smooth kernels. Integr. Equ. Oper. Theory 21(2), 224–237 (1995)
L.R. Duduchava, D. Mitrea, M. Mitrea, Differential operators and boundary value problems on hypersurfaces. Math. Nachr. 279(9–10), 996–1023 (2006)
J. Frankel, A. Galerkin, Solution to a regularized Cauchy singular integro-differential equation. Q. Appl. Math. L11(2), 245–258 (1995)
B.G. Gabdulkhaev, Finite approximations of singular integrals, direct solution methods of singular integral and integro-differential equations. Itogi Nauki i Tekniki, VINITI AN SSSR, Math. Anal. 18, 25–31 (1980)
B.G. Gabdulkhaev, Optimal Approximation to Linear Problem (Kazan University Publications, Kazan, 1980)
B.G. Gabdulkhaev, V.E. Gorlov, On the optimal algorithm of the approximate solutions of singular integral equations. Izv. Vuzov Math. 11, 13–31 (1976)
F.D. Gakhov, Boundary Value Problems, English Edition (Pergamon Press, Oxford, 1966)
C.D. Green, Integral Equation Methods (Thomas Nelson, New York, 1969)
A.I. Gusseinov, K.S. Mukhtarov, Introduction to the Theory of Nonlinear Singular Integral Equations (Nauka, Moscow, 1980) (in Russian)
H. Hochstadt, Integral Equations (Wiley Interscience, New York, 1973)
V.V. Ivanov, The Theory of Approximate Methods and its Application to the Numerical Solution of Singular Integral Equations (Naukova Dumka, Kiev, 1968)
D.S. Jones, Integral equations for the exterior acoustic problem. Q. J. Mech. Appl. Math. 27, 129–142 (1974)
P. Junghanns, K. Müller, A collocation method for non-linear Cauchy singular integral equations. J. Comput. Appl. Math. 115, 283–300 (2000)
A.I. Kalandia, Mathematical Methods of the Two-Dimensional Elastics (Nauka, Moscow, 1973)
A.C. Kaya, F. Erdogan, On the solution of integral equation with strongly singular kernels. Q. Appl. Math. 45, 105–122 (1987)
E.G. Khalilov, Approximate methods for the solution of surface integral equations, Ph.D. thesis, Baku, 1999 (in Russian)
E.H. Khalilov, On an approximate solution of a boundary integral equation of mixed problem for Helmholtz equation. Proc. Inst. Math. Mach. NASA 31, 105–110 (2009)
R.E. Kleinman, G.F. Roach, Boundary integral equations for the three-dimensional Helmholtz equation. SIAM Rev. 16, 214–236 (1974)
R.E. Kleinman, G.F. Roach, On modified Green functions in exterior problems for the Helmholtzs equation. Proc. R. Soc. Lond. A 383, 313–333 (1982)
A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1981) (in Russian)
M.A. Krasnosel’skii, P.P. Zabreyko, Geometric Methods of Nonlinear Analysis (Nauka, Moscow, 1975)
M.A. Krasnosel’skii, G.M. Vainikko, P.P. Zabreiko, Ya. B. Rutitskii, V. Ya. Stetsenko, Approximate Solution of Operator Equations (Wolters-Noordhoff Publishing, Groningen, 1972)
E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978)
I.I. Kudish, Numerical solution methods of a one class nonlinear integral and integro-differential equations. J. Numer. Math. Math. Phys. 26, 14931511 (1986)
Y.A. Kustov, B.I. Musaev, A Cubature Formula for Double Singular Integral and its Application. VINITI, No 4281 (1981) (in Russian)
E. Lackau, W. Tutschke, Complex Analysis, Methods, Trends and Applications (Pergamon Press, London, 1985)
E.G. Ladopoulos, On the numerical solution of the finite-part singular integral equations of the first and the second kind used in fracture mechanics. Comput. Methods Appl. Mech. Eng. 65, 253–266 (1987)
E.G. Ladopoulos, Singular integral representation of three-dimensional plasticity problem. Theor. Appl. Fract. Mech. 8, 205–211 (1987)
E.G. Ladopoulos, On a new integration rule with the Gegenbauer polynomials for singular integral equations used in the theory of elasticity. Ing. Arch. 58, 35–46 (1988)
E.G. Ladopoulos, On the numerical evaluation of the general type of finite-part singular integrals and integral equations used in fracture mechanics. J. Eng. Fract. Mech. 31, 315–337 (1988)
E.G. Ladoloulos, The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics. Acta Mech. 75, 275–285 (1988)
E.G. Ladopoulos, On the solution of the finite-part singular integro-differential equations used in two-dimensional aerodynamics. Arch. Mech. 41, 925–936 (1989)
E.G. Ladopoulos, Singular integral operators method for two-dimensional plasticity problems. Comput. Struct. 33, 859–865 (1989)
E.G. Ladopoulos, Non-linear integro-differential equations used in orthotropic shallow spherical shell analysis. Mech. Res. Commun. 18, 111–119 (1991)
E.G. Ladopoulos, Relativistic elastic stress analysis for moving frames. Rev. Roum. Sci. Tech. Mech. Appl. 36, 195–209 (1991)
E.G. Ladopoulos, Singular integral operators method for three-dimensional elastoplastic stress analysis. Comput. Struct. 38, 1–8 (1991)
E.G. Ladopoulos, Singular integral operators method for two-dimensional elastoplastic stress analysis. Forsch. Ingenieurwes. 57, 152–158 (1991)
E.G. Ladopoulos, New aspects for generalization of the Sokhotski-Plemelj formulae for the solution of finite-part singular integrals used in fracture mechanics. Int. J. Fract. 54, 317–328 (1992)
E.G. Ladopoulos, Non-linear singular integral equations elastodynamics by using Hilbert transformations. J. Nonlinear Anal. Real World Appl. 6, 531–536 (2005)
E.G. Ladopoulos, V.A. Zisis, Existence and uniqueness for non-linear singular integral equations used in fluid mechanics. Appl. Math. 42, 345–367 (1997)
E.G. Ladopoulos, V.A. Zisis, Non-linear finite-part singular integral equations arising in two-dimensional fluid mechanics. J. Nonlinear Anal. 42, 277–290 (2000)
P. Linz, Analytical and Numerical Methods for Volterra Equations (SIAM, Philadelphia, 1985)
J.K. Lu, Boundary Value Problems for Analytic Functions (World Scientific, Singapore, 1993)
K. Mamedov, N. Kosar, Continuity of the scattering function and the Levinson type formula of the boundary value problem. Int. J. Contemp. Math. Sci. 5(4), 159–170 (2010)
K. Mamedov, H. Menken, On the inverse problem of the scattering theory for a boundary problem. Geom. Integrability Quantization 7, 226–237 (2006)
V.N. Monahov, Boundary Value Problems with Free Boundaries for Elliptic Systems (Nauka, Novosibirsk, 1977)
A.S. Mshim Ba, W. Tutschke, Functional-Analytic Methods in Complex Analysis and Applications to Partial Differential Equations (World Scientific, Singapore, 1990)
B.I. Musaev, On approximate solution of the singular integral equations. AN Az. SSR, Institute of Physics Preprint No 17 (1986)
B.I. Musaev, On the approximate solution of the singular integral equations. Izv. AN Az. SSSR, Fizik-Teknik Science 5, 1521 (1986)
B.I. Musaev, On the approximate solution of singular integral equations with negative index by Bubnov-Galerkin and collocation methods. Sov. Math. Dokl. 35(2), 411–416 (1987)
N.I. Muskelishvili, Singular Integral Equations (Noordhoff, Groningen, 1953)
N.I. Muskhelishvili, Singular Integral Equations, English Edition (Noordhoff, Groningen, 1968)
N. Mustafa, On the approximate solution of non-linear operator equations. Far East J. Appl. Math. 27(1), 121–136 (2007)
N. Mustafa, Fixed point theory and approximate solutions of non-linear singular integral equations. Complex Variables Elliptic Equ. 53(11), 1047–1058 (2008)
N. Mustafa, Non-linear singular integro-differential equations. Complex Variables Elliptic Equ. 53(9), 879–886 (2008)
N. Mustafa, On the approximate solution of singular integral equations with negative index. Complex Variables 55(7), 621–631 (2010)
N. Mustafa, Newton–Kantorovich method for two-dimensional non-linear singular integral equations. Maejo Int. J. Sci. Technol. 10(1), 41 (2016)
N. Mustafa, Some integral operators and their properties. Kuwait J. Sci. 43(4), 45–55 (2016)
N. Mustafa, C. Ardil, On the approximate solution of a non-linear singular integral equation. Int. J. Comput. Math. Sci. 3(1), 1–7 (2009)
N. Mustafa, E.H. Khalilov, The collocation method for the solution of boundary integral equations. Appl. Anal. 88(12), 1665–1675 (2009)
N. Mustafa, M.I. Yazar, On the approximate solution of a non-linear singular integral equation with Cauchy kernel. Far East J. Appl. Math. 27(1), 101–119 (2007)
N.M. Mustafaev, On the Approximate Solution of the Singular Integral Equation that is Defined on Closed Smooth Curve. Singular Integral Operators, vol. 1 (AGU Publications, Baku, 1987), pp. 91–99
N. Mustafaev, Approximate solution of non-linear singular integral equations. VINITI (338-B88), 1–36 (1988)
N.M. Mustafaev, Error of the approximation to the singular integral equation that is defined on closed smooth curve. In Az. NIINTI (338-B88), 137 (1988)
N.M. Mustafaev, Approximate formulas for singular integrals and their application to the approximate solution of singular integral equations that are defined on closed smooth curve, Ph.D. thesis, AN Az. SSR, Institute of Math. and Mech., Baku., 1991
V.V. Panasyuk, M.P. Savruk, Z.T. Nazarchuk, Singular Integral Equations Methods in Two-Dimensional Diffraction Problems (Naukova Dumka, Kiev, 1984)
V.Z. Parton, P.I. Perlin, Integral Equations of Elasticity Theory (Nauka, Moscow, 1977)
G.Y. Popov, Contact Problems for a Linearly Deformable Base (Kiev, Odessa, 1982)
S. Prösdorf, B. Silberman, Projektionsverfahren und die naherungsweise Losung Singularer (Gleichungen, Leipziq, 1977)
S. Prösdorf, Some Class Singular Integral Equations (Mir, Moscow, 1979)
H. Reşidoğlu (Kh. amedov), On the planar problem of the theory elasticity. Works SSU 1, 27–32 (2001)
M.H. Saleh, Basis of quadrature method for non-linear singular integral equations with Hilbert kernel in the spaceH φ,k. In Az. NIINTI 279, 1–40 (1984)
V.N. Seychuk, Direct methods of the solutions of singular integral equations that are defined on Lyapunov curve, Ph.D. thesis, Kishinev University, Kishinev, 1987
F.G. Trikomi, Integral Equations (Dover, New York, 1985)
W. Tutshke, Lözung nichtlinearer partieller differential-gleichungssysteme erster Ordnung in der Ebene cluch verwendung einer komlexen Normalform. Math. Nachr. 75, 283–298 (1976)
W. Tzong-Mou, Solving the non-linear equations by the Mewton-homotopy continuation method with adjustable auxiliary homotopy functions. J. Appl. Math. Comput. 173, 383–388 (2006)
F. Ursell, On the exterior problems of acoustics: II. Proc. Camb. Philol. Soc. 84, 545–548 (1978)
G.M. Vaynicco, The regular convergence of operators and the approximate solution of equations, Moscow V sb. Math. Anal. (Itogi nauki i tekniki VINITI AN SSSR) 16, 553 (1979) (in Russian)
I.N. Vekua, Generalized Analytic Functions (Pergamon Press, London, 1962)
V.A. Zolotaryevskii, On the Approximate solution of singular integral equations. Math. Res. Kishinev Shtiintsa 9(3), 82–94 (1974)
V.A. Zolotaryevskii, V.N. Seychuk, The solution of the singular integral equation that is defined on Lyapunov curve by collocation method. Differ. Equ. 19(6), 1056–1064 (1983)
A. Zygmund, A.P. Calderon, On the existence of singular integrals. Acta Math. 88, 85–139 (1952)
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Mustafa, N., Nezir, V. (2019). On Approximate Solutions of Linear and Nonlinear Singular Integral Equations. In: Dutta, H., Kočinac, L.D.R., Srivastava, H.M. (eds) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-15242-0_19
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