Inverse problem of determining the coefficient in a sixth order Boussinesq equation with additional nonlocal integral condition

Abstract

The work is devoted to the study of the solvability of the inverse boundary value problem with an unknown time depended coefficient for a sixth-order Boussinesq equation with an additional integral condition of the first kind is investigated. First, the given problem is  reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the  system of integral equations. The existence and uniqueness of a solution to the system of integral equation is proved by the contraction  mapping principle. This solution is also the unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence  and uniqueness of a classical solution to the given problem is proved. The sixth order Boussinesq-Type problem is discretized using the  Quintic B-spline (QB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov regularization  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.