Abstract
Mathematical model of the longitudinal vibration of bars includes higher-order derivatives in the equation of motion under considering the effect of the lateral motion of a relatively thick bar. This paper considers such an inverse coefficient problem of determining time-dependent potential of a linear source together with the unknown longitudinal displacement from a Rayleigh-Love equation (containing the fourth-order space derivative) by using an additional measurement. Existence and uniqueness theorem of the considered inverse coefficient problem is proved for small times by using contraction principle.
Highlights
Longitudinal vibrations of elastic bars are often regarded as the classical model in mathematical physics which is described by the second order wave equation under the consideration that the bar is thin and relatively long
If the order is higher than two, the equations of longitudinal vibrations can be obtained by taking into account the e¤ects of the lateral motion by which cross section of a long and relatively thick bar becomes variable
The inverse problems for pseudo-hyperbolic equations connected with recovery of the coe¢ cient are scarce
Summary
Longitudinal vibrations of elastic bars are often regarded as the classical model in mathematical physics which is described by the second order wave equation under the consideration that the bar is thin and relatively long. The inverse problems for the second order wave equation with di¤erent boundary conditions and space dependent coe¢ cients are studied in [11,15] and more recently in [5,6,12]. The inverse problem for the wave equation with time dependent coe¢ cient is investigated in [1] and the time-dependent source function of a time-fractional wave equation with integral condition in a bounded domain is determined in [16]. The solvability of the problem of determining an unknown coe¢ cient for the fourth-order pseudo-hyperbolic equation is theoretically studied in [18] and the sixth-order linear Boussinesq type equation is theoretically and numerically investigated in [21].
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