On a generalization of the robin problem for the Laplace equation in the circle
Anahtar Kelimeler:
Robin problem, fractional analog, Hadamard derivative, involutive transformation, uniqueness of solution, existence of solutionÖzet
In this paper, we study the solvability of the fractional analogue of the Robin
problem for the Laplace equation. A modified fractional differentiation operator in the sense of
Hadamard is considered as a boundary operator. Boundary conditions are given in the form of a
connection between different values of the unknown function in a circle. The problem is solved
using the Fourier expansion method. For various values of the parameters of the boundary operators
involved, theorems on the existence and uniqueness of a solution to the problem under study are
proved.
Referanslar
СПИСОК ИСПОЛЬЗОВАННОЙ ЛИТЕРАТУРЫ
Hadamard J. Essai sur létude des fonctions données par leur développement de Taylor //
Journal de Mathématiques Pures et Appliquées. – 1892. – Vol. 8. – P. 101 – 186.
Butzer P.L. , Kilbas A.A., Trujillo J.J. Compositions of Hadamard-type fractional
integration operators and the semigroup property // Journal of Mathematical Analysis and
Applications. – 2002. – Vol.269. – P.387-400.
Jarad F., Baleanu D., Abdeljawad A. Caputo-type modification of the Hadamard
fractional derivatives // Advances in Difference Equations. – 2012. – Vol.2012, No.142. – P.1 – 8.
Kilbas A.A. Hadamard-type fractional calculus // Journal of the Korean Mathematical
Society. – 2001. – Vol.38. – P.1191 – 1204 .
Kilbas A.A., Srivastava H.M, Trujillo J.J.Theory and applications of fractional differential
equations.Elsevier, Amsterdam, 2006.
Arioua Y., Benhamidouche N. Boundary value problem for Caputo-Hadamard fractional
differential equations //Surveys in Mathematics and its Applications. – 2017. – Vol.12. – P.103 –
Boutiara A., Benbachir M., Guerbati K. Boundary value problem for nonlinear Caputo-
Hadamard fractional differential equation with Hadamard fractional integral and anti-periodic
conditions // Facta Universitatis Series Mathematics and Informatics. – 2021. – Vol. 36, No. 4. –
P.735 – 748.
Turmetov B.Kh. On the solvability of some boundary value problems for the
inhomogeneous polyharmonic equation with boundary operators of the Hadamard type //
Differential Equations. – 2017. – V. 53, № 3. Р. 333–344.
Turmetov B.Kh., Koshanova M., Usmanov K. About solvability of some boundary value
problems for Poisson equation in the ball conditions // Filomat . – 2018. –V. 32, No.3. – P. 939-946.
Evans LC. Partial differential equations. Vol. 19, Graduate studies in mathematics.
Қожа Ахмет Ясауи атындағы Халықаралық қазақ-түрік университетінің хабарлары
(математика, физика, информатика сериясы), №3 (22), 2022
Providence(RI): American Mathematical Society; 1998. 668 p.
Бицадзе А.В., Самарский А.А. О некоторых простейших обобщениях линейных эллиптических краевых задач // Доклады АН СССР. – 1969. – Т.185, № 4. – С.739 – 740.
Przeworska-Rolewicz D. Some boundary value problems with transformed argument // Commentat. Math. – 1974. – V.17. – P.451 – 457.
Karachik V.V., Turmetov B.Kh. Solvability of one nonlocal Dirichlet problem for the Poisson equation // Novi Sad Journal of Mathematics. – 2020. – V. 50, No. 1. – P. 67 – 88.
Turmetov B. Kh., Karachik V.V. Solvability of nonlocal Dirichlet problem for generalized Helmholtz equation in a unit ball // Complex Variables and Elliptic Equations. – 2022. – P. 1 – 15.
Turmetov B.Kh., Nazarova K.Dz. On a generalization of the Neumann problem for the Laplace equation// Mathematische Nachrichten. – 2020. – V. 293,No. 1. – P.169 – 177.
Turmetov B.Kh Generalization of the Robin Problem for the Laplace Equation // Differential equations. – 2019.– V.55, № 9. – P. 1134 – 1142.
REFERENCES
Hadamard J. Essai sur létude des fonctions données par leur développement de Taylor // Journal de Mathématiques Pures et Appliquées. – 1892. – Vol. 8. – P. 101 – 186.
Butzer P.L. , Kilbas A.A., Trujillo J.J. Compositions of Hadamard-type fractional integration operators and the semigroup property // Journal of Mathematical Analysis and Applications. – 2002. – Vol.269. – P.387-400.
Jarad F., Baleanu D., Abdeljawad A. Caputo-type modification of the Hadamard fractional derivatives // Advances in Difference Equations. – 2012. – Vol.2012, No.142. – P.1 – 8.
Kilbas A.A. Hadamard-type fractional calculus // Journal of the Korean Mathematical Society. – 2001. – Vol.38. – P.1191 – 1204 .
Kilbas A.A., Srivastava H.M, Trujillo J.J.Theory and applications of fractional differential equations.Elsevier, Amsterdam, 2006.
Arioua Y., Benhamidouche N. Boundary value problem for Caputo-Hadamard fractional differential equations //Surveys in Mathematics and its Applications. – 2017. – Vol.12. – P.103 – 115.
Boutiara A., Benbachir M., Guerbati K. Boundary value problem for nonlinear Caputo-Hadamard fractional differential equation with Hadamard fractional integral and anti-periodic conditions // Facta Universitatis Series Mathematics and Informatics. – 2021. – Vol. 36, No. 4. – P.735 – 748.
Turmetov B.Kh. On the solvability of some boundary value problems for the inhomogeneous polyharmonic equation with boundary operators of the Hadamard type // Differential Equations. – 2017. – V. 53, № 3. Р. 333–344.
Turmetov B.Kh., Koshanova M., Usmanov K. About solvability of some boundary value problems for Poisson equation in the ball conditions // Filomat . – 2018. –V. 32, No.3. – P. 939-946.
Evans LC. Partial differential equations. Vol. 19, Graduate studies in mathematics. Providence(RI): American Mathematical Society; 1998. 668 p.
Bisadze A.V., Samarskii A.A. O nekotoryh prosteishih obobsheniah lineinyh ellipticheskih kraevyh zadach [On some simplest generalizations of linear elliptic boundary value problems] // Doklady AN SSSR. – 1969. – Т.185, № 4. – С.739 – 740.
Przeworska-Rolewicz D. Some boundary value problems with transformed argument // Commentat. Math. – 1974. – V.17. – P.451 – 457.
Karachik V.V., Turmetov B.Kh. Solvability of one nonlocal Dirichlet problem for the Poisson equation // Novi Sad Journal of Mathematics. – 2020. – V. 50, No. 1. – P. 67 – 88.
Turmetov B. Kh., Karachik V.V. Solvability of nonlocal Dirichlet problem for
