On Fractional Analogs of Dirichlet and Neumann Problems for the Laplace Equation
- 31 Downloads
In this paper, we investigate solvability of fractional analogs of the Dirichlet and Neumann boundary-value problems for the Laplace equation. Operators of fractional differentiation in the Riemann–Liouville and Caputo sense are considered as boundary operators. The considered problems are solved by reducing them to Fredholm integral equations. Theorems on existence and uniqueness of solutions of the problems are proved.
KeywordsLaplace equation Dirichlet problem Neumann problem Fractional derivative Riemann–Liouville operator Caputo operator uniqueness existence
Mathematics Subject ClassificationPrimary 31A05 Secondary 35J05
The work was supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant no. AP05131268).
- 1.Akhmedov, T., Veliev, E., Ivakhnychenko, M.: Fractional operators approach in electromagnetic wave reflection problems. J. Electromagn. Waves Appl. 21, 1787–1802 (2007)Google Scholar
- 2.Akhmedov, T., Veliev, E., Ivakhnychenko, M.: Description of the boundaries in scattering problems using fractional operators. Radio Phys. Electron. 14, 133–141 (2009)Google Scholar