Abstract
The Dirichlet boundary value problem (BVP) for the linear stationary diffusion partial differential equation with a variable coefficient is considered. The PDE right-hand side belongs to the Sobolev spaces H −1(Ω), when neither classical nor canonical co-normal derivatives are well defined. Using an appropriate parametrix (Levi function) the problem is reduced to a direct boundary-domain integro-differential equation (BDIDE) or to a domain integral equation supplemented by the original boundary condition thus constituting a boundary-domain integro-differential problem (BDIDP). Solvability, solution uniqueness, and equivalence of the BDIDE/BDIDP to the original BVP are analysed in Sobolev (Bessel potential) spaces.
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Acknowledgements
The first author acknowledges the support of the grant EP/M013545/1: ‘Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs’ from the EPSRC, UK. The second author’s work on this paper was supported by ISP, Sweden. He would like also to thank his PhD advisor Dr. Tsegaye Gedif Ayele for discussing the results.
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Mikhailov, S.E., Woldemicheal, Z.W. (2019). On United Boundary-Domain Integro-Differential Equations for Variable Coefficient Dirichlet Problem with General Right-Hand Side. In: Constanda, C., Harris, P. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-16077-7_18
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DOI: https://doi.org/10.1007/978-3-030-16077-7_18
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