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. A class of inverse problems for restoring the right-hand side of the pseudo-parabolic equation for 1D Laplace operator is considered. The inverse problem is to be well-posed in the sense of Hadamard whenever an overdetermination condition of the final temperature is given. Mathematical statements involve inverse problems for the pseudo-parabolic equation in which, solving the equation, we have to find the unknown right-hand side depending only on the space variable. We prove the existence and uniqueness of the classical solutions. The proof of the existence and uniqueness results of the solutions is carried out by using L-Fourier analysis. The mentioned results are presented as well as for the fractional time pseudo–parabolic equation. Inverse problems of identifying the coefficients of right hand side of the pseudo-parabolic equation from the local overdetermination condition have important applications in various areas of applied science and engineering, also such problems can be modeled using common homogeneous left-invariant hypoelliptic operators on common graded Lie groups.

An inverse problem for the pseudo-parabolic equation for Laplace operator