Abstract
We study the properties of fractional integro-differential operators. As an application, we analyze the solvability of some boundary value problems for the inhomogeneous polyharmonic equation in the unit ball. These problems generalize the Dirichlet and Neumann problems to the case of fractional boundary operators.
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Kal’menov, T.Sh. and Suragan, D., On a new method for constructing the Green function of the Dirichlet problem for the polyharmonic equation, Differ. Equations, 2012, vol. 48, no. 3, pp. 441–445.
Karachik, V.V., Solution of the Dirichlet problem with polynomial data for the polyharmonic equation in a ball, Differ. Equations, 2015, vol. 51, no. 8, pp. 1038–1047.
Karachik, V.V. and Antropova, N.A., Polynomial solutions of the Dirichlet problem for the biharmonic equation in the ball, Differ. Equations, 2013, vol. 49, no. 2, pp. 251–256.
Karachik, V.V., Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball, Comput. Math. Math. Phys., 2014, vol. 54, no. 7, pp. 1122–1143.
Bitsadze, A.V., Some properties of polyharmonic functions, Differ. Equations, 1988, vol. 24, no. 5, pp. 543–548.
Kanguzhin, B.E. and Koshanov, B.D., Necessary and sufficient conditions for the solvability of boundary value problems for the inhomogeneous polyharmonic equations in the ball, Ufimsk. Mat. Zh., 2010, vol. 2, no. 2, pp. 41–52.
Karachik, V.V., On solvability conditions for the Neumann problem for a polyharmonic equation in the unit ball, J. Appl. Ind. Math., 2014, vol. 8, no. 1, pp. 63–75.
Karachik, V.V., Solvability conditions for the Neumann problem for the homogeneous polyharmonic equation, Differ. Equations, 2014, vol. 50, no. 11, pp. 1455–1461.
Karachik, V.V., Turmetov, B.Kh., and Bekaeva, A.E., Solvability conditions of the Neumann boundary value problem for the biharmonic equation in the unit ball, Int. J. Pure Appl. Math., 2012, vol. 81, no. 3, pp. 487–495.
Turmetov, B.Kh. and Ashurov, R.R., On solvability of the Neumann boundary value problem for a nonhomogeneous polyharmonic equation in a ball, Boundary Value Problems, 2013, vol. 2013, no. 162. doi:10.1186/1687-2770-2013-162.
Berdyshev, A.S., Cabada, A., and Turmetov B.Kh., On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order, Acta Math. Sci., 2014, vol. 34B, no. 6, pp. 1695–1706.
Krasnoschok, M. and Vasylyeva, N., On a nonclassical fractional boundary-value problem for the Laplace operator, J. Differential Equations, 2014, vol. 257, no. 6, pp. 1814–1839.
Turmetov, B.Kh., Solvability of fractional analogues of the Neumann problem for a nonhomogeneous biharmonic equation, Electronic J. Differential Equations, 2015, vol. 2015, no. 82, pp. 1–21.
Bavrin, I.I., Operators for harmonic functions and their applications, Differ. Uravn., 1985, vol. 21, no. 1, pp. 9–15.
Turmetov, B.Kh. and Torebek, B.T., Modified Bavrin operators and their applications, Differ. Equations, 2015, vol. 51, no. 2, pp. 243–254.
Berdyshev, A.S, Cabada, A., and Turmetov, B.Kh., On solvability of some boundary value problem for polyharmonic equation with boundary operator of a fractional order, Appl. Math. Modeling, 2015, vol. 39, pp. 4548–4569.
Karachik, V.V., Turmetov, B.Kh., and Bekaeva, A., Solvability of some boundary-value problems for polyharmonic equation with Hadamard–Marchaud boundary operator, Russ. Math., 2014, vol. 58, no. 7, pp. 11–24.
Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
Karachik, V.V., Polynomial solutions of Dirichlet boundary value problem for the 3-harmonic equation in a ball, J. Sib. Fed. Univ. Math. Phys., 2012, vol. 5, no. 4, pp. 527–546.
Agmon, S., Duglas, A., and Nirenberg, L., Estimates near the boundary for elliptic partial. Differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 1959, vol. 12, pp. 623–727.
Gazzola, F., Grunau, H.-Ch., and Guido, S., Polyharmonic Boundary Value Problems, Berlin, 2010.
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Original Russian Text © B.Kh. Turmetov, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 3, pp. 343–354.
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Turmetov, B.K. On the solvability of some boundary value problems for the inhomogeneous polyharmonic equation with boundary operators of the Hadamard type. Diff Equat 53, 333–344 (2017). https://doi.org/10.1134/S0012266117030053
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DOI: https://doi.org/10.1134/S0012266117030053