Inverse Problem for a Pseudoparabolic Equation with a Non-Self-Adjoint Involutive Second-Order Differential Operator
Beisebayeva A. Mussirepova E. Sarsenbi A.
February 2026Multidisciplinary Digital Publishing Institute (MDPI)
Mathematics
2026#14Issue 4
In this paper, we consider a partial differential equation with mixed derivatives of first order in time and second order in the spatial variable. Such equations are usually referred to as one-dimensional pseudoparabolic equations. We prove the existence and uniqueness of a classical solution to problems for a pseudoparabolic equation with a second-order differential operator involving pure involution, under certain requirements imposed on the initial data. The possibility of applying the Fourier method is based on the Riesz basis property of the eigenfunctions of the considered non-self-adjoint second-order differential operator with pure involution. Bessel-type inequalities are established for new systems of functions. The presence of a Bessel inequality for Fourier coefficients facilitates the proof of the uniform convergence of differentiated Fourier series. The solutions are obtained explicitly in the form of a Fourier series. Such representations can be used for performing numerical computations.
biorthogonal systems , boundary condition , differential equation with reflection of the arguments , differential operator with involution , operator eigenvalues , pseudoparabolic equation , uniqueness of solution
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Department of Mathematics, M. Auezov South Kazakhstan University, Shymkent, 160012, Kazakhstan
Department of Mathematics
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