Identification of unknown diffusion coefficient in pure diffusive linear model of chronoamperometry. I. The theory

Coefficient identification problem for diffusion equation u (t) (x, t) = (D(x)u (x) (x, t)) (x) arising in chronoamperometry is studied. The adjoint problem approach is developed for the case when the output measured data is given in the form of left/right flux. Analytical formulas for determination of the values D(0), D(L) at the endpoints x = 0; L, of the unknown coefficient D(x), via the solution v(x, t) of the constant coefficient equation v (t) (x, t) = D v (xx) (x, t) is obtained. The integral identity relating solutions of the forward and corresponding adjoint problems is derived. This integral identity permits one to prove the monotonicity and invertibility of input-output map, as well as formulate the gradient of the cost functional via the solutions of the direct and adjoint problems.