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Boundary-value problems for weakly singular integral equations
Institute of Mathematics of the National Academy of Sciences of Ukraine, 3, Tereshchenkivska Str., Kyiv, 01024, Ukraine |
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.
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. |
show all references
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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