A Final Value Problem for the Nonlinear Modified Helmholtz Equation Associated with the Nonlinear Wave Velocity

This study focuses on the development and analysis of the backward problem for the nonlinear modified Helmholtz equation associated with nonlinear wave velocity. The governing equation is given as follows

$$\triangle u\left ( x,y\right ) -k\left ( l_{0}\left ( u\right ) \left ( y\right ) \right ) u\left ( x,y\right ) =S\left ( x,y,u\left ( x,y\right ) \right ) ,~x\in \Omega ,~0< y< L. $$

To overcome the ill-posseness of the above problem, we apply the variational quasi-reversibility method. It is imperative to investigate the convergence analysis of this regularization method when we do not determine a formula of the exact solution. In this regard, we construct the approximate problem by adding the so-called perturbing operator to the original problem and by exploiting the Fourier reconstructed the final data. Then, we obtain the Hölder convergence rate of the proposed scheme under some certain assumptions on the exact solution. Finally, a numerical example is provided to corroborate the theoretical results.