On a non-local Kirchhoff type equation with random terminal observation

In this work, we are concerned with the terminal value problem for time fractional equation (in sense of Conformable derivative) a nonlocal term Kirchhoff type$ \partial_t^\alpha u = K\Big(\|\nabla u\|_{L^2(\mathcal{D})}\Big)\Delta + f(x,t), \quad (x,t) \in (0,T)\times \mathcal{D} $subject to final data which is blurred by random Gaussian white noise. The principal goal article recover solution $ $. This severely ill-posed in Hadamard, because violation continuous dependence on (the solution's behavior does not change continuously condition). By applying non-parametric estimates from observation and truncation method Fourier series, obtain regularized solution. Under some priori assumptions, derive an error estimate between mild its