On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient
Комплекс мәндi коэффициентi бар төртiншi реттi параболалық теңдеу үшiн керi есептiң шешiмдiлiгi туралы
О разрешимости обратной задачи для параболического уравнения четвертого порядка с комплекснозначным коэффициентом
Imanbetova A.B. Sarsenbi A. Seilbekov B.
2024E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2024#113Issue 160 - 72 pp.
In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2 (−1, 1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2 (−1, 1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.
classical solution , Fourier method , inverse problem , parabolic equation , Riesz basis , strongly regular boundary conditions
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Department of Mathematics, South Kazakhstan University of the name of M. Auezov, 68 A. Baitursynov street, Shymkent, 160012, Kazakhstan
Scientific Institute “Theoretical and Applied Mathematics”, South Kazakhstan University of the name of M. Auezov, Tauke Khana street, 5, Kazakhstan
Department of Mathematics and Informatics, Tashenev University, 21 Kunaev street, Shymkent, 160000, Kazakhstan
Department of Mathematics, South Kazakhstan State Pedagogical University, 13 A. Baitursynov street, Shymkent, 160000, Kazakhstan
Department of Mathematics
Scientific Institute “Theoretical and Applied Mathematics”
Department of Mathematics and Informatics
Department of Mathematics
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