Geodesic
Construction of polynomial solutions to some boundary value problems for Poisson’s equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 9, pp. 1674-1694

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A polynomial solution of the inhomogeneous Dirichlet problem for Poisson’s equation with a polynomial right-hand side is found. An explicit representation of the harmonic functions in the Almansi formula is used. The solvability of a generalized third boundary value problem for Poisson’s equation is studied in the case when the value of a polynomial in normal derivatives is given on the boundary. A polynomial solution of the third boundary value problem for Poisson’s equation with polynomial data is found. 
@article{ZVMMF_2011_51_9_a8,
     author = {V. V. Karachik},
     title = {Construction of polynomial solutions to some boundary value problems for {Poisson{\textquoteright}s} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1674--1694},
     publisher = {mathdoc},
     volume = {51},
     number = {9},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_9_a8/}
}
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V. V. Karachik. Construction of polynomial solutions to some boundary value problems for Poisson’s equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 9, pp. 1674-1694. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_9_a8/