Goursat-type nonlocal problem for a fourth-order loaded equation. (English) Zbl 1514.35004
Summary: Initial and boundary value problems for partial differential equations have been sufficiently studied. However, it has been recently become apparent that various processes and phenomena of modern natural science lead to nonclassical problems for differential equations. The class of nonclassical problems involves nonlocal problems. We turn our attention on nonlocal problems, to be exact, on some problems with nonlocal integral conditions for a loaded equation. The majority of the works concerned with boundary value problems with integral conditions deals with second-order equations. In this article we consider a nonlocal problem with integral conditions for a loaded fourth-order equation. The factorization enables to transfer this problem to two problems for second order equations: the Goursat problem for an integrodifferential equation and an integral analogue of the Goursat problem for a simple hyperbolic equation. The unique solvability of the considered problem is proved.
MSC:
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35G15 | Boundary value problems for linear higher-order PDEs |
| 35L10 | Second-order hyperbolic equations |
| 35R09 | Integro-partial differential equations |
Keywords:
nonlocal problem; fourth-order equation; integral conditions; Goursat problem; loaded equationReferences:
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