Boundary-value problems for weakly singular integral equations

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March 2022, 27(3): 1379-1395. doi: 10.3934/dcdsb.2021094

Boundary-value problems for weakly singular integral equations

Oleksandr Boichuk and Victor Feruk

Institute of Mathematics of the National Academy of Sciences of Ukraine, 3, Tereshchenkivska Str., Kyiv, 01024, Ukraine

Received April 2020 Revised August 2020 Published March 2022 Early access March 2021

Fund Project: The present work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology)

We consider a perturbed linear boundary-value problem for a weakly singular integral equation. Assume that the generating boundary-value problem is unsolvable for arbitrary inhomogeneities. Efficient conditions for the coefficients guaranteeing the appearance of the family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter $ \varepsilon $ with singularity at the point $ \varepsilon = 0 $ are established.

Keywords: Boundary-value problem, weakly singular integral equation, pseudoinverse Moore-–Penrose matrix, bifurcation, Fredholm operator, matrix projector.

Mathematics Subject Classification: Primary: 45B05, 45E99; Secondary: 47G10.

Citation: Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094

References:

[1]	P. L. Auer and C. S. Gardner, Note on singular integral equations of the Kirkwood–Riseman type, J. Chem. Phys., 23 (1955), 1545-1546. doi: 10.1063/1.1742352.
[2]	A. Boichuk, J. Diblík, D. Khusainov and M. Růžičková, Boundary-value problems for weakly nonlinear delay differential systems, Abstr. Appl. Anal., 2011 (2011), Art. ID 631412, 19 pp. doi: 10.1155/2011/631412.
[3]	O. A. Boichuk and V. A. Feruk, Linear boundary-value problems for weakly singular integral equations, J. Math. Sci., 247 (2020), 248-257. doi: 10.1007/s10958-020-04800-6.
[4]	A. A. Boichuk, I. A. Korostil and M. Fečkan, Bifurcation conditions for a solution of an abstract wave equation, Differential Equations, 43 (2007), 495-502. doi: 10.1134/S0012266107040076.
[5]	O. A. Boichuk, N. O. Kozlova and V. A. Feruk, Weakly perturbed integral equations, J. Math. Sci., 223 (2017), 199-209. doi: 10.1007/s10958-017-3348-x.
[6]	A. A. Boichuk, N. A. Kozlova and V. A. Feruk, Weakly nonlinear integral equations of the Hammerstein type, Nonlin. Dynam. Syst. Theory, 19 (2019), 289-301.
[7]	A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, 2$^nd$ edition, Walter de Gruyter, Berlin, Boston, 2016. doi: 10.1515/9783110378443.
[8]	A. A. Boichuk and V. F. Zhuravlev, Solvability criterion for integro-differential equations with degenerate kernel in Banach spaces, Nonlin. Dynam. Syst. Theory, 18 (2018), 331-341.
[9]	C. Constanda and S. Potapenko (eds.), Integral Methods in Science and Engineering. Techniques and Applications, Birkhäuser, Boston, 2008. doi: 10.1007/978-0-8176-4671-4.
[10]	E. A. Galperin, E. J. Kansa, A. Makroglou and S. A. Nelson, Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations, J. Comput. Appl. Math., 115 (2000), 193-211. doi: 10.1016/S0377-0427(99)00297-6.
[11]	I. Golovatska, Weakly perturbed boundary-value problems for systems of integro-differential equations, Tatra Mt. Math. Publ., 54 (2013), 61-71. doi: 10.2478/tmmp-2013-0005.
[12]	O. Gonzalez and J. Li, A convergence theorem for a class of Nystrom methods for weakly singular integral equations on surfaces in $\mathbb{R}^{3}$, Math. Comp., 84 (2015), 675-714. doi: 10.1090/S0025-5718-2014-02869-X.
[13]	E. Goursat, A Course in Mathematical Analysis. Vol. III. Part 2, Dover Publications, Inc., New York, 1964.
[14]	I. G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comp., 39 (1982), 519-533. doi: 10.1090/S0025-5718-1982-0669644-3.
[15]	E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods in the Analysis of Nonlinear Systems, Nauka, Moscow, 1979.
[16]	D. Hilbert, Selected Works, Vol. 2, Faktorial, Moscow, 1998.
[17]	C. Huang, T. Tang and Z. Zhang, Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions, J. Comput. Math., 29 (2011), 698-719. doi: 10.4208/jcm.1110-m11si06.
[18]	S. G. Krein, Linear Equations in Banach Spaces, Nauka, Moscow, 1971.
[19]	I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, Gos. Izdt. Tekh.-Teor. Lit., Moscow, 1956.
[20]	S. G. Mikhlin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2$^{nd}$ edition, Pergamon, New York, 1964.
[21]	G. R. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl., 55 (1976), 32-42. doi: 10.1016/0022-247X(76)90275-4.
[22]	A. Samoilenko, A. Boichuk and S. Chuiko, Hybrid difference-differential boundary-value problem, Miskolc Mathematical Notes, 18 (2017), 1015-1031. doi: 10.18514/MMN.2017.2280.
[23]	C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory, 2 (1979), 62-68. doi: 10.1007/BF01729361.
[24]	J. Shen, C. Sheng and Z. Wang, Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels, J. Math. Study, 48 (2015), 315-329. doi: 10.4208/jms.v48n4.15.01.
[25]	V. I. Smirnov, A Course of Higher Mathematics. Vol. IV. Part 1, Nauka, Moscow, 1974.
[26]	E. Vainikko and G. Vainikko, A spline product quasi-interpolation method for weakly singular Fredholm integral equations, SIAM J. Numer. Anal., 46 (2008), 1799-1820. doi: 10.1137/070693308.
[27]	M. I. Vishik and L. A. Lyusternik, Solution of some perturbation problems in the case of matrices and self-adjoint or nonself-adjoint differential equations. I, Russ. Math. Surveys, 15 (1960), 1-73. doi: 10.1070/RM1960v015n03ABEH004092.
[28]	V. F. Zhuravlev and N. P. Fomin, Weakly perturbed boundary-value problems for the Fredholm integral equations with degenerate kernel in Banach spaces, J. Math. Sci., 238 (2019), 248-262. doi: 10.1007/s10958-019-04233-w.

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References:

[1]	P. L. Auer and C. S. Gardner, Note on singular integral equations of the Kirkwood–Riseman type, J. Chem. Phys., 23 (1955), 1545-1546. doi: 10.1063/1.1742352.
[2]	A. Boichuk, J. Diblík, D. Khusainov and M. Růžičková, Boundary-value problems for weakly nonlinear delay differential systems, Abstr. Appl. Anal., 2011 (2011), Art. ID 631412, 19 pp. doi: 10.1155/2011/631412.
[3]	O. A. Boichuk and V. A. Feruk, Linear boundary-value problems for weakly singular integral equations, J. Math. Sci., 247 (2020), 248-257. doi: 10.1007/s10958-020-04800-6.
[4]	A. A. Boichuk, I. A. Korostil and M. Fečkan, Bifurcation conditions for a solution of an abstract wave equation, Differential Equations, 43 (2007), 495-502. doi: 10.1134/S0012266107040076.
[5]	O. A. Boichuk, N. O. Kozlova and V. A. Feruk, Weakly perturbed integral equations, J. Math. Sci., 223 (2017), 199-209. doi: 10.1007/s10958-017-3348-x.
[6]	A. A. Boichuk, N. A. Kozlova and V. A. Feruk, Weakly nonlinear integral equations of the Hammerstein type, Nonlin. Dynam. Syst. Theory, 19 (2019), 289-301.
[7]	A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, 2$^nd$ edition, Walter de Gruyter, Berlin, Boston, 2016. doi: 10.1515/9783110378443.
[8]	A. A. Boichuk and V. F. Zhuravlev, Solvability criterion for integro-differential equations with degenerate kernel in Banach spaces, Nonlin. Dynam. Syst. Theory, 18 (2018), 331-341.
[9]	C. Constanda and S. Potapenko (eds.), Integral Methods in Science and Engineering. Techniques and Applications, Birkhäuser, Boston, 2008. doi: 10.1007/978-0-8176-4671-4.
[10]	E. A. Galperin, E. J. Kansa, A. Makroglou and S. A. Nelson, Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations, J. Comput. Appl. Math., 115 (2000), 193-211. doi: 10.1016/S0377-0427(99)00297-6.
[11]	I. Golovatska, Weakly perturbed boundary-value problems for systems of integro-differential equations, Tatra Mt. Math. Publ., 54 (2013), 61-71. doi: 10.2478/tmmp-2013-0005.
[12]	O. Gonzalez and J. Li, A convergence theorem for a class of Nystrom methods for weakly singular integral equations on surfaces in $\mathbb{R}^{3}$, Math. Comp., 84 (2015), 675-714. doi: 10.1090/S0025-5718-2014-02869-X.
[13]	E. Goursat, A Course in Mathematical Analysis. Vol. III. Part 2, Dover Publications, Inc., New York, 1964.
[14]	I. G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comp., 39 (1982), 519-533. doi: 10.1090/S0025-5718-1982-0669644-3.
[15]	E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods in the Analysis of Nonlinear Systems, Nauka, Moscow, 1979.
[16]	D. Hilbert, Selected Works, Vol. 2, Faktorial, Moscow, 1998.
[17]	C. Huang, T. Tang and Z. Zhang, Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions, J. Comput. Math., 29 (2011), 698-719. doi: 10.4208/jcm.1110-m11si06.
[18]	S. G. Krein, Linear Equations in Banach Spaces, Nauka, Moscow, 1971.
[19]	I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, Gos. Izdt. Tekh.-Teor. Lit., Moscow, 1956.
[20]	S. G. Mikhlin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2$^{nd}$ edition, Pergamon, New York, 1964.
[21]	G. R. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl., 55 (1976), 32-42. doi: 10.1016/0022-247X(76)90275-4.
[22]	A. Samoilenko, A. Boichuk and S. Chuiko, Hybrid difference-differential boundary-value problem, Miskolc Mathematical Notes, 18 (2017), 1015-1031. doi: 10.18514/MMN.2017.2280.
[23]	C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory, 2 (1979), 62-68. doi: 10.1007/BF01729361.
[24]	J. Shen, C. Sheng and Z. Wang, Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels, J. Math. Study, 48 (2015), 315-329. doi: 10.4208/jms.v48n4.15.01.
[25]	V. I. Smirnov, A Course of Higher Mathematics. Vol. IV. Part 1, Nauka, Moscow, 1974.
[26]	E. Vainikko and G. Vainikko, A spline product quasi-interpolation method for weakly singular Fredholm integral equations, SIAM J. Numer. Anal., 46 (2008), 1799-1820. doi: 10.1137/070693308.
[27]	M. I. Vishik and L. A. Lyusternik, Solution of some perturbation problems in the case of matrices and self-adjoint or nonself-adjoint differential equations. I, Russ. Math. Surveys, 15 (1960), 1-73. doi: 10.1070/RM1960v015n03ABEH004092.
[28]	V. F. Zhuravlev and N. P. Fomin, Weakly perturbed boundary-value problems for the Fredholm integral equations with degenerate kernel in Banach spaces, J. Math. Sci., 238 (2019), 248-262. doi: 10.1007/s10958-019-04233-w.

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