On a pseudo-parabolic equations with a non-local term of the Kirchhoff type with random Gaussian white noise

This paper is concerned with a solution of backward pseudo-Parabolic equation$u_t-\alpha \Delta u_t+G\left(\parallel \nabla u\parallel {}_{L^2\left(\Omega \right)}\right)(-\Delta ){}^{\beta }u=K(t,x)$subject to the final data which is blurred by random Gaussian white noise. The primary aim of this work is to obtain the solution $u$. This problem is severely ill-posed (the solution's behavior does not change continuously with the final condition). By using a statistical method, we estimate the value data from observation data and the Fourier truncation method is used to propose a regularized solution. Under some priori assumptions, we derive an error estimate between a mild solution and its regularized solution. Finally, a numerical example is given to illuminate the effect of our method.