Abstract
The questions of solvability of a nonlocal inverse boundary value problem for a mixed pseudohyperbolic-pseudoelliptic integro-differential equation with spectral parameters are considered. Using the method of the Fourier series, a system of countable systems of ordinary integro-differential equations is obtained. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system regular and irregular values of the spectral parameters were calculated. The unique solvability of the inverse boundary value problem for regular values of spectral parameters is proved. For irregular values of spectral parameters is established a criterion of existence of an infinite set of solutions of the inverse boundary value problem. The results are formulated as a theorem.
Highlights
The questions of solvability of a nonlocal inverse boundary value problem for a mixed pseudohyperbolic-pseudoelliptic integro-differential equation with spectral parameters are considered
From the point of applications, partial differential and integro-differential equations are of great interest [1,2]
Important to study the spectral questions of solvability of the differential and integro-differential equations [5–10]
Summary
Taking conditions (17) and (18) into account from representations (15) and (16) we obtain. Taking formula (7) into account we will rewrite conditions (2) and (3) in the following forms u n 1 , . Relations (23) and (24) are considered as a system of algebraic equations (SAE) with respect to unknown coefficients B 1 n 1 , . Solving this system from (19) and (20) we arrive at the following representations u n 1 , . We substitute representations (37) and (38) into the Fourier series (6) and obtain the following formal solution of the direct problem (1)–(4). Substituting representations (44) and (45) into the Fourier series (8), we obtain g1 ( x, ω, ν)) =.
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