An inverse problem for investigating the time-dependent coefficient in a higher-order equation

In this paper, we considered an inverse problem of recovering the time-dependent potential coefficient, for the first time, in the sixth-order Boussinesq-type equation from additional data as an over-specification condition. The unique solvability theorem for this inverse problem is supplied. 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 term), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown term. The sixth-order Boussinesq-Type problem is discretized using the Septic B-spline (SB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov Regularization (TR) function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both perturbed data and analytical solution are inverted. Numerical outcomes are reported and discussed. In addition, the Von Neumann stability analysis for the proposed numerical approach has also been discussed.