Inverse Problem for a Pseudoparabolic Equation with a Non-Self-Adjoint Involutive Second-Order Differential Operator

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.