Geodesic
Convergence of an approximate solution of the Showalter--Sidorov--Dirichlet problem for the modified Boussinesq equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 217 (2022), pp. 11-19

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In this paper, we obtain necessary and sufficient conditions for the existence of a unique solution of the Showalter–Sidorov–Dirichlet problem for a second-order, semilinear Sobolev-type equation. For the initial-boundary-value problem considered, using the Galerkin method, we construct an approximate solution as an expansion in the system of eigenfunctions of the homogeneous Dirichlet problem for the Laplace operator. The proof of the $*$-weak convergence of the Galerkin approximations to the exact solution is based on a priori estimates, embedding theorems, and the Gronwall lemma.
Mots-clés : Sobolev-type equation
Keywords: Showalter–Sidorov problem, Galerkin method, $*$-weak convergence. 
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     author = {E. V. Bychkov},
     title = {Convergence of an approximate solution of the {Showalter--Sidorov--Dirichlet} problem for the modified {Boussinesq} equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {11--19},
     publisher = {mathdoc},
     volume = {217},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_217_a1/}
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E. V. Bychkov. Convergence of an approximate solution of the Showalter--Sidorov--Dirichlet problem for the modified Boussinesq equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 217 (2022), pp. 11-19. http://geodesic.mathdoc.fr/item/INTO_2022_217_a1/