Recovery of the time‐dependent zero‐order coefficient in a fourth‐order parabolic problem

Abstract

AbstractThe determination of an unknown time‐dependent zero‐order coefficient in a fourth‐order parabolic problem is investigated form the integral‐type observation in this paper. Under certain assumptions, the well‐posedness of the weak solution to the inverse problem is obtained. Furthermore, its global solvability can be proved by applying some transformations. For the numerical reconstruction of the unknown coefficient, three methods are established, including the transformations used above, the time‐discrete method with the cubic spline function method, and the optimization method. The convergence and error estimates for the time‐discrete method are derived rigorously. Furthermore, the corresponding predictor–corrector scheme is introduced to determine the unknown quantity numerically. For the optimization method, the minimizer of the objective functional is applied to approximate the unknown coefficient. The convergence rates of the optimization problem is proved under some suitable source condition. In addition, the Fréchet derivative of the objective functional is obtained and utilized to establish the conjugate gradient algorithm. The numerical example shows that all the three methods can be applied to numerically recover the unknown quantity efficiently.