An inverse problem of reconstructing the unknown coefficient in a third order time fractional pseudoparabolic equation

Abstract

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the potential term. The proposed problem is discretized using the cubic B-spline (CB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine $lsqnonlin$ tool has been used to expedite the numerical computations. Both perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.